Mathieu Le Breton

Laurent Baillet

Éric Larose

Etienne Rey

Philippe Benech

Denis Jongmans

Fabrice Guyoton

Michel Jaboyedoff

Eric Larose

Jongmans Denis

Passive radio-frequency identification ranging, a dense and weather-robust technique for landslide displacement monitoring

Keywords: Wireless sensor network, Slope stability, Real-time location tracking system, Early warning, Monitoring, Radio-Frequency Identification RFID: Radio-Frequency Identification TD-Phase: Time-Domain Phase Difference of Arrival FD-Phase: Frequency-Domain Phase Difference of Arrival

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Ground deformations are monitored at different scales. On a scale of a few hundred meters, a landslide creates surface displacements that vary in both time and space. Motion monitoring to understand the landslide mechanism requires the best possible spatial and temporal resolution. However, obtaining a high spatial resolution is expensive at the scale of landslides, owing to either the station or the multiple measurement points. Large-zone monitoring techniques, such as ground-based radar interferometers, laser scanners, robotic total stations and fiber optics [START_REF] Monserrat | A review of ground-based SAR interferometry for deformation measurement[END_REF][START_REF] Jaboyedoff | Use of LIDAR in landslide investigations: a review[END_REF][START_REF] Iten | Landslide monitoring using a road-embedded optical fiber sensor[END_REF] have a high station cost (30-100 k€), which restrict their use in standard applications. Similarly, more localized techniques such as GPS and extensometers have a cost per point (1-10 k€) that make them economically inefficient for a spatially-dense network of sensors [START_REF] Gili | Using Global Positioning System techniques in landslide monitoring[END_REF][START_REF] Angeli | A critical review of landslide monitoring experiences[END_REF]. The development of low-cost single-frequency GPS [START_REF] Benoit | Monitoring landslide displacements with the Geocube wireless network of low-cost GPS[END_REF] and radio-frequency transponders [START_REF] Kenney | Precise positioning with wireless sensor nodes: Monitoring natural hazards in all terrains[END_REF]Intrieri et al., 2018a) reduced the cost to 100-1000 € per point, but each instrument still requires its own power source, which considerably increases the cost of the material, its installation and maintenance. Satellite Remote sensing techniques are economically efficient but their best current time resolution is 4-6 days (Intrieri et al., 2018b;[START_REF] Milillo | COSMO-SkyMed Spotlight Interferometry Over Rural Areas: The Slumgullion Landslide in Colorado, USA[END_REF][START_REF] Lacroix | Use of Sentinel-2 images for the detection of precursory motions before landslide failures[END_REF]. Similarly, ground-based optical correlation barely exceeds a 1-day resolution [START_REF] Travelletti | Correlation of multi-temporal ground-based optical images for landslide monitoring: Application, potential and limitations[END_REF] because of low light at night and other luminance variations.

Furthermore, some techniques are completely unusable in specific conditions: GPS signal may be blocked by steep slopes and high trees, optical techniques may be hindered by fog, rain, snow and vegetation, and extensometers may be obstructed by snow, ice or animals. In comparison, Radio-Frequency techniques continue to work in the presence of rain, fog, snow and vegetation, although environmental conditions may affect accuracy. Given these facts, Radio-Frequency Identification (RFID) [START_REF] Heidrich | The Roots, Rules, and Rise of RFID[END_REF] appears to be a viable alternative in terms of cost, spatio-temporal resolution and weather robustness.

Billions of passive targets, or tags, are produced by the RFID industry every year [START_REF] Das | RFID Forecasts, Players and Opportunities[END_REF] to identify goods remotely, leading to numerous research and business applications [START_REF] Ngai | RFID research: An academic literature review (1995-2005) and future research directions[END_REF][START_REF] Tzeng | Evaluating the business value of RFID: Evidence from five case studies[END_REF]. Tags typically comprise a passive microcircuit connected to an antenna. They communicate with a station that consists of a reader connected to another antenna. This communication relies on a continuous radio-frequency wave emitted by the station, this wave powers the tag, which then backscatters a modulated wave that encodes its own identification number, and the station reads this number (EPC Gen2, 2015). Passive tags and stations currently cost 0.01-20 € and 2-4 k€, respectively, and the reader can identify 30 to 800 tags per second (EPC Gen2, 2015;[START_REF] Klair | A Survey and Tutorial of RFID Anti-Collision Protocols[END_REF]. These advantages have already been exploited in RFID applications in earth-science research, such as to sense soil moisture levels or vibrations [START_REF] Pichorim | Two Solutions of Soil Moisture Sensing with RFID for Landslide Monitoring[END_REF][START_REF] Jayawardana | Measurement System With Accelerometer Integrated RFID Tag for Infrastructure Health Monitoring[END_REF], measure crack openings [START_REF] Cazeca | Passive wireless displacement sensor based on RFID technology[END_REF][START_REF] Caizzone | Wireless Passive RFID Crack Width Sensor for Structural Health Monitoring[END_REF], or trace alluvial and coastal sediment [START_REF] Lamarre | Using Passive Integrated Transponder (PIT) Tags to Investigate Sediment Transport in Gravel-Bed Rivers[END_REF][START_REF] Nichols | A Radio Frequency Identification System for Monitoring Coarse Sediment Particle Displacement[END_REF].

However, continuous displacement monitoring with RFID remains untested outdoors.

Several methods have been developed to locate passive tags [START_REF] Miesen | Where is the Tag[END_REF]. These methods usually estimate the 1D range or displacement between a tag and a station antenna, and optionally locate the tag in 2D or 3D using multiple station antennas. These ranging techniques can use the Received Signal Strength (RSS) [START_REF] Griffin | Complete Link Budgets for Backscatter-Radio and RFID Systems[END_REF][START_REF] Ni | LANDMARC: indoor location sensing using active RFID[END_REF], the Phase Difference of Arrival [START_REF] Vossiek | The Switched Injection-Locked Oscillator: A Novel Versatile Concept for Wireless Transponder and Localization Systems[END_REF][START_REF] Nikitin | Phase based spatial identification of UHF RFID tags[END_REF] or more recently the Time of Flight (ToF) [START_REF] Arnitz | UWB ranging in passive UHF RFID: proof of concept[END_REF][START_REF] Arthaber | Spread-Spectrum Based Ranging of Passive UHF EPC RFID Tags[END_REF]. Currently, phase-based methods are compatible with commercial readers and offer the best accuracy. Several studies used the phase to localize tags with an accuracy of one centimeter or less. However, these experiments were of short duration (<1 h), over a short range (<10 m) and were performed indoors. When these techniques are used outdoors, environmental fluctuations could affect their accuracy. A previous study showed that moisture and temperature variations could create a phase drift of up to 20 cm over a year [START_REF] Breton | Outdoor UHF RFID: Phase Stabilization for Real-World Applications[END_REF]. Corrections were proposed, which reduced that drift to less than 2 cm per year, with a reversible error of 7 mm induced by rain. However, the short tag-station distance (6 m) limited the effect of multipath propagation in that study. The effect of multipath propagation may be increased at long range and outdoors and passive RFID has never yet been studied in these conditions.

Hectometer-range outdoor radio-frequency ranging has nevertheless been assessed on landslides using active transponders. The ranging techniques were based either on RSS, ToF or Phase, with displacement accuracies of 100, 5 and 1 cm, respectively [START_REF] Lucianaz | Real time outdoor localization of buried RFID tags through statistical methods[END_REF]Intrieri et al., 2018a;[START_REF] Kenney | Precise positioning with wireless sensor nodes: Monitoring natural hazards in all terrains[END_REF]. Since these techniques use the same physical principles as passive techniques, phase-based RFID ranging is also expected to have a displacement accuracy of 1 cm outdoors. As to the maximal range attainable, manufacturers claim ranges of up to 60 m for passive tags (Confidex, 2014); this range may be sufficient for the goals of this study.

The previous studies then suggest that a dense network of passive RFID tags could be deployed on hectometer-scale landslides at low cost, to monitor 1D radial displacements with an accuracy of 1-2 cm in a variety of meteorological conditions. To test the performance of RFID, for five months, we monitored the displacements of 19 tags placed 5 m apart on a medium-sized landslide (Pont-Bourquin, Switzerland) and compared the results to those obtained with total station and extensometer data. We provide an example application that benefit from the RFID technique, studying the relation between the rainfall and the velocity of this landslide, and finally discuss the advantages and limitations of RFID displacement monitoring.

MATERIAL AND METHODS

Experimental site: Pont-Bourquin landslide

The Pont-Bourquin landslide lies in the Swiss Pre-Alps, 50 km southeast of Lausanne (Fig. 1.a). The landslide crosses a complex geological zone composed of Triassic to Jurassic sediments, including gypsum, cargneule, shale siltstones and black shales, separated by tectonic thrusts [START_REF] Badoux | Les Diablerets[END_REF]. Quaternary moraines partially cover the slope. The landslide material thus mixes all these sediments, with a dominant proportion of clay produced by degradation of black shale.

The landslide is about 240 m long, 30 m large (Fig. 1.b) and 10 m deep [START_REF] Jaboyedoff | Earth flow in a complex geological environment: the example of Pont Bourquin, Les Diablerets (Western Switzerland)[END_REF]. Its average slope is around 25° facing south. The total landslide volume is approximately 40 000 m 3 , which includes an active part of around 11 000 m 3 . The landslide mostly moves translationally, with a slight rotation at its top. Orthophotos from 1995, 1997 and 2004 revealed continuous degradation due to slope movements and erosion in the landslide area. In 2006, a scarp of 0.8 m was observed at the top of the landslide. On 5 July 2007 a sudden earthflow of around 3 000 to 6 000 m 3 occurred after 95 mm of rain fell over a 3-day period [START_REF] Jaboyedoff | Earth flow in a complex geological environment: the example of Pont Bourquin, Les Diablerets (Western Switzerland)[END_REF]. In subsequent years, the landslide permanently moved by a few meters per year, before suddenly accelerating its descent in summer 2010 leading to a mudflow of a few thousand cubic meters that blocked the road downslope on 19 August 2010 (Mainsant et al., 2012).

Due to its continuous activity, this landslide appears to be a good candidate to test the RFID technique over several months. 109

Monitoring instruments

On 3 July 2017, the Pont-Bourquin landslide was equipped with the RFID material proposed previously by Le [START_REF] Breton | Outdoor UHF RFID: Phase Stabilization for Real-World Applications[END_REF] that reduces the influence of water and temperature on the phase. The acquisition system was composed of a stable station, 19 moving tags, and a remote server. The station was installed on a tree 3 m above the landslide's surface and comprised a reader, an antenna and a computer (SR420 from Impinj and model 80010643 from Kathrein). It typically required 20 W power under continuous operations, which was provided through the Swiss power grid. The station collected about 20 Mo of data per day by measuring the phase difference of arrival and the temperature on 30 tags per second, using a randomized tag interrogation order (EPC Gen2, 2015). This corresponds to the slowest and most accurate mode of interrogation available on the reader (Dense M8). The reader was chosen because it provided the most precise phase measurements and the best sensitivity (and therefore reading range) and is well adapted for use with slow-moving objects. The phase-shift measured was limited within the range [0, π] (Miesen et al., 2013b). The tags were installed on 0.9-m-high fiber-glass stakes that were planted in the unstable slope (Fig.

1.d). The tags were elevated above the ground because preliminary tests indicated that placing tags near the ground considerably reduced the RSS, down to -30 dB at ground-level. These tags were placed within the zone covered by the RFID antenna, which measured approximately 40x10 m (Fig. 1.b-c), and remained within the line-of-sight of the station antenna during the experiment. The tags (Survivor B, from Confidex) comprised a solid plastic casing (155×26×14 mm) protecting a patch antenna (+0.6 dBi effective gain and ±65 °E / ±80 °H beam-width at 3 dB) and a battery-assisted microcircuit (EM4325 from EM Microelectronic). While assisted by a tiny button cell to boost their reading range, the tags remain passive: they backscatter the interrogator carrier wave the without amplification, and the exactly same technique work with batteryless tag. In term of lifetime, the tag's battery may last 5 to 20 years under continuous interrogation, given the cell of 290 mAh and the typical micro-circuit current of 1.7-6 μA (EM 4325 Spec., 2015). This lifetime may be shortened by battery aging and by the extra temperature measurements, but still last several years according to the tag manufacturer (Confidex, 2014) and to our own experience.

To validate the RFID measurements, we installed a wire extensometer (ASM WS17KT) between a metallic stake (0.5-m-high, 0.7-m-deep and 1 meter away from tag No. 10) and the bottom of the station antenna. We extended the original 2.5-meter Invar wire with a 20-meter Kevlar wire. An acquisition card (Campbell CR1000) measured the length of the extensometer wire every minute, representing a radial displacement along the line of sight of the antenna.

During processing, the displacement from both the extensometer and the RFID was averaged every hour.

We also measured the 3D coordinates of the tags and base station on 27 July and 6 October using two manual total station surveys (Leica TCR805), to compute the 1D tag-station radial distance and its difference between the two surveys. The positions of the extensometer and tags No. 10 and 11 were measured using a fixed reflector, with an accuracy of ± 5 mm. For the other tags, the manual positioning of the prism resulted in lower accuracy, estimated at ± 2 cm. Hence, the accuracy of the relative displacement between two total station surveys was ± 1 cm for tags No. 10 and 11, and ± 4 cm for the other tags.

Finally, we used data provided by MétéoSuisse related to fresh snow height and total snow height (measured daily at 6 am at Diablerets village, 1 km away), precipitations, humidity, air temperature (every 10 minutes at Col des Mosses, 6 km away) and pressure (every 10 minutes at Les Diablerets).

RFID ranging methods

This study first compares two absolute ranging techniques (FD-Phase and RSS) with a relative displacement technique (TD-Phase). Firstly, the relative radial displacement of a backscattering tag can be computed in free space using the Time-Domain Phase Difference of Arrival technique (TD-Phase) [START_REF] Nikitin | Phase based spatial identification of UHF RFID tags[END_REF], as presented in Fig. 2 f carrier frequency (= 865.7, 866.3, 866.9 or 867.5 MHz as defined by ETSI-EN 302-208 (2016)).

In this application, the reader measures the phase difference of arrival between [0, π] modulo π. Using equation ( 1), this phase interval is equivalent to a distance of 8.6 cm, or ± 4.3 cm. Larger displacements are ambiguous to measure directly and require phase unwrapping, by adding or subtracting 8.6 cm when the difference between two consecutive measurements exceeds ± 4.3 cm.

Secondly, the absolute distance r can be computed by measuring the phase at different frequencies, based on the Frequency-Domain Phase Difference of Arrival (FD-Phase) technique [START_REF] Nikitin | Phase based spatial identification of UHF RFID tags[END_REF] : The resulting distance r was computed for each of the three available frequency intervals and averaged. As long as tags are placed at less than 125 m (minus 0 r ) away from the station antenna, this technique provides an absolute distance without ambiguity. The unique offset 0 r used on all the tags was initially calibrated from tag No. 10 as 11 m, for which the range to the station (22 m) was measured using the total station on July 27.

0 4 air v r r f δϕ δ π = - (2)
Thirdly, the absolute distance r can be estimated from the attenuation of the received power (RSS) over a two-ways propagation [START_REF] Griffin | Complete Link Budgets for Backscatter-Radio and RFID Systems[END_REF]. As the RSS depends on the station and tag gain, their relative orientation, and the environmental conditions [START_REF] Dobkin | Environmental effects on RFID tag antennas[END_REF], it requires calibration based on reference tags placed at a known distance [START_REF] Ni | LANDMARC: indoor location sensing using active RFID[END_REF]. We computed one virtual reference tag, from the average RSS and tacheometric distances of all the tags over the first month. The absolute range i r of tag i can then be estimated by comparing its received power P to that of the reference tag, using:

Stable station

Moving tag RFID Reader

1 ϕ 2 ϕ 4 i ref ref i
r r P P ≈

(3)

These three absolute and relative ranging techniques are compared in section 3.1. The remainder of the article then focuses solely on the TD-Phase relative displacement measurements.

RESULTS

Performance of the different RFID ranging techniques

The absolute distance covered by each tag was computed from the Frequency-Domain Phase (using ( 2)) and the RSS (using (3)) and compared to the total station laser measurements (Fig. 3). The range values were computed every hour and averaged over a 5-month period. The vertical bars represent the drift over this whole period, based on the 90% confidence interval. The curve shows that the RSS-based ranging measurements follow an increasing trend with distance from the station, but that some strong outliers are found, as are large drifts of up to ±15 m. These datapoints were due to the strong impact of uncontrolled parameters: environmental variations, antenna orientation and ground reflection interference. Therefore, the method cannot be considered reliable. In contrast, the distances obtained with the FD-Phase method gave a good fit with the distances measured with the total station, with a maximum drift of ± 2 m.

This level of error is still too high for our monitoring purposes, but it can be reduced by using relative displacement techniques.

The relative displacements of tag No. 10 over time, computed from variations in FD-Phase, RSS, and TD-Phase are compared in Fig. 4, along with the total station measurements. The FD-Phase and RSS show significant variations over time, which can be up to 4 m. In contrast, the TD-Phase measurement appears much more stable and accurate than the RSS and FD-Phase ranging techniques when monitoring relative displacements. Indeed, the relative displacement is more sensitive to phase or frequency errors in equation ( 2) compared when applying (1). The TD-Phase results appear coherent with those provided by the total station, and are validated in more detail in the next section by comparison with standard ranging techniques.

Validation of the TD-Phase RFID technique

The TD-Phase displacements are compared with the automatic wire extensometer measurements over 4 months and with the two manual total station surveys (Fig. 5). The tag studied (No. 10) was installed next to the extensometer stake. The RFID curve appears to match the extensometer data although a slight discrepancy appeared in September, which progressively increased to around 5 cm difference at the end of the observation period. The total station data showed that this discrepancy was caused by a true differential displacement between the tag and the extensometer, which were installed on separate stakes. Another difference was that the wire extensometer curve was noisier and exhibited several strong spikes from mid-October, which appear to be linked to snowfall (Fig. 5b): snow accumulation bent the wire down, pulling the cable along by several decimeters, such as on October 20. This extra length remained until snowmelt or until the cable was manually dug out from under the snow. RFID tracking thus provided better continuity and accuracy over time than the wire extensometer. In particular, the error due to snowfall was smaller and restricted in time to the precipitation episode. Each velocity was derived from the displacement, after the application of a low-pass filter (1-day moving average) to reduce the cyclic daily drift. The RFID velocity curve appears more stable than the extensometer velocity curve.

Indeed, rain-and snow-fall (Fig. 6.c-d) created numerous positive and negative peaks on the extensometer velocity curve. That resulted from the additional water or snow weight that temporarily increasesd curvature in the wire. In comparison, the RFID velocity curve shows almost no perturbation due to rain and the artifact caused by snow was much smaller (± 1 cm/day). To illustrate its capacity to monitor multiple tags, the RFID cumulative displacement (Fig. 7) and the velocity for all tags (Fig. 8) were plotted over the five-month period. The cumulated displacement ranged between 0.69 and 0.94 m (extremes measured for tags No. 12 and No. 7, respectively). The shape of the displacement curves was similar on all tags, suggesting that the zone studied moved in the same way during the observation period. Locally, some displacement measurements may have been affected by a rotational motion of the stick, observed from pictures as ± 7°

during the 5-month period. Such motion of the stick would cause a ± 12 cm displacement of the tags, which is the same magnitude as the motion discrepancy between tags. However, the similarity between all the curves indicates that this effect was progressive and smaller than the ground displacement. The velocity for all tags is shown in Fig. 8. The curves were ordered according to a noise indicator, computed as the standard deviation of the detrended total displacement, with additional weight for the heavy rainfall recorded between September 30 and October 3. As a result of this ordering, the more accurate curves are placed at the bottom of the figure. Each tag measured the true displacement of the landslide surface, added to an instrumental artifact due to rain. The amplitude of this rain artifact was quite distinct from one tag to another (e.g. between September 30 and October 3) and results in heterogeneous accuracies.

Precision, trueness and robustness of the TD-Phase technique

To determine the validity limits of the RFID TD-Phase technique, we tried to quantify its precision, trueness and robustness from the five-month experiment. The accuracy indicators are computed from the experimental data and summarized in Table I for precision and trueness, and Table II for robustness. The accuracy is then given by the combination of the precision and the trueness.

Precision

The precision is given by the random measurement error estimated from the phase standard deviation over one minute, a timespan that is short enough to allow systematic error due to meteorological variations to be neglected. This random measurement error was then plotted against the RSS (Fig. 9). The random measurement error appears to obey the relation (4) with the received power P, using an interrogator-dependent coefficient α=9.5•10 -9 m•W 1/2 . The curve obtained is coherent to the phase signal-to-noise ratio observed by [START_REF] Vossiek | The Switched Injection-Locked Oscillator: A Novel Versatile Concept for Wireless Transponder and Localization Systems[END_REF] that depends on the distance. On a single measurement, our random measurement error 3σ represent 0.1 to 1.3 cm (Table I), for a received power of 10 -8.3 to 10 -6 mW. This random error can be statistically decreased to 0.01-0.17 cm by averaging 60 measurements over one minute, it thus becomes negligible compared to the systematic measurement error presented below.

Trueness

The trueness is given by the systematic measurement error, estimated as the bias between the true value and the mean of the measured values. It includes the drift over time (days to months) and the inaccuracy over spatial displacements. Table I shows the displacement accuracy computed from the difference between the RFID technique and the total station surveys used as reference measurements. The average displacement measured for all tags by both methods differed by only 6 mm. Hence, the displacement of all the RFID tags appears coherent and, on average, unbiased compared to the total station data. However, the results are spread statistically over ± 53 mm within the 99.7% confidence interval. This spread mostly corresponds to the error when placing the optical reflector on the tags for the manual total station surveys: we have estimated this survey error to be ± 40 mm for most of the tags. The tags No. 10 and 11 were installed on a single stick, which improved the estimated survey error to ± 10 mm. For these two tags, the difference compared to the RFID measurements remained below 0.5 mm, which, given the small number of samples, corresponds to a potential error of ± 10 mm within the 99.7% confidence interval. In fact, the reference technique -manual total station -appears as or less accurate than the RFID method that is tested, and thus another method is required to assess the trueness.

The systematic measurement error with RFID is mostly due to environmental variations such as temperature, humidity and moisture, which vary over the course of a day (Le [START_REF] Breton | Outdoor UHF RFID: Phase Stabilization for Real-World Applications[END_REF]. To estimate this systematic measurement error, we computed the RFID residual displacement amplitude after removing the 1.0 (centered on -0.05) Versus 24h-smoothed data 60 000 1.1 During the first snowfall (c) 19 8.0 Radial Precision of a single reading (d) 7x10 7 0.1-1.3 over one minute 4x10 6 0.01-0.17 (a) The 99.7% confidence interval was computed from the Student law when there were less than 2000 samples and directly from the equivalent quantile when there were more than 60 000 samples. The distributions were verified to ensure they were close to a normal law. (b) Tacheometric surveys were more accurate for tags No. 10 & 11, and their fixation on a single sealed stick was more stable.

(c) The first snow fell between October 21, 18:00 and October 24, midnight, when tags showed a strong measurement bias. This was the worst bias accuracy observed.

(d) The precision of a single reading was linear with the signal amplitude, and followed the law

/ P ϕ σ α = , with α=1•10 -8 m•W 1/2
1-day-moving-averaged trend. These calculations showed a daily drift of less than 11 mm 99.7% of the time. Finally, the worst errors were observed during the first snowfall (see photo in Fig. 11), which triggered reversible displacement peaks of up to 79 mm, either positive or negative.

Hence, we estimate the trueness of this RFID method to be approximately 1 cm on a real landslide application under mixed meteorological conditions (dry, rain, snow), except for the first snowfall which appeared to limit the trueness to about 8 cm. Because the systematic error is much larger than the random one after averaging, we consider that this trueness also represent the total accuracy.

The measurement error could be partly related to the tag-station distance. To verify this relation, the standard deviation was plotted against the distance (Fig. 10). This analysis shows that the systematic measurement error tends to increase the measured distance by about 50 parts per millions (ppm), or 150 ppm for 3σ. This trend could be related to the following distance-dependent effects: Firstly, the atmospheric conditions influence on wave velocity [START_REF] Gage | On the scattering and reflection mechanisms contributing to clear air radar echoes from the troposphere, stratosphere, and mesophere[END_REF] could induce a +/-3.4 ppm standard deviation (after detrending), computed from the meteorological data. Secondly, the interrogator frequency instability could result in variations of +/-10 ppm, in line with the regulations (EPC Gen2, 2015). Thirdly, the amount of vegetation crossed by ground-reflected waves will increase with the distance, and its properties (such as height or water content) may vary over time and affect the overall signal (Kim et al., 2012). Fourthly, the greater the distance between the tag and the station, the more opportunities for multipathing due to the topography, and therefore potentially the greater the systematic error due to multipathing.

Robustness

The robustness can be defined as the ratio between the total monitoring time and the time when the RFID is operational. This is an important indicator for operational monitoring. Failure to identify the tags disqualifies the monitoring. Table II shows the availability ratio for each tag over the observation period; 14 out of 19 tags were available 97.5% of the time. The 2.5% unavailability was the result of software and network errors on the station. The availability of the other tags ranged from 96 to 55%. Snowfall clearly affected communication with tags No. 16 and 20.

For other tags, the RSS was close to the limit of interrogator sensitivity (-83 dBm) due to either their distance (No. 12 and No. 14) or their angle relative to the station's antenna (No. 17). Signal strength may have been reduced by destructive multipath interference at a specific tag position, and increased again once the tag moved forward.

SAMPLE APPLICATIONS: COUPLING BETWEEN DISPLACEMENT AND PRECIPITATIONS.

This example demonstrates how all-weather, dense displacement measurements can help to better characterize landslide behavior, by analyzing the relation between rainfall and velocity data.

Fig. 11. Photo illustrating the two main sources of inacurracies in this experiment the snow (particularly the first snowfall) and the angle of the tag supports, which created a reversible error of 4.9 cm (3σ) at the end of October and a slow drift roughly estimated at up to 5 cm per month, respectively. Apart from those two specific problems, the accuracy remained below 1.1 cm (3σ).

Computation of the rainfall-related impulse response

Precipitations often control the displacements of reactivated landslides by the intermediate of groundwater infiltration [START_REF] Corominas | Prediction of ground displacements and velocities from groundwater level changes at the Vallcebre landslide (Eastern Pyrenees, Spain)[END_REF] and subsequent changes of rheology, we will examine this effect in this section.

The velocity/precipitation curves shown in Fig. 6 suggest that Pont-Bourquin landslide reacts directly to precipitation.

The Impulse Response (IR) of velocity to rainfall on a reactivated landslide [START_REF] Belle | The application of an innovative inverse model for understanding and predicting landslide movements (Salazie cirque landslides, Reunion Island)[END_REF][START_REF] Bernardie | Prediction of changes in landslide rates induced by rainfall[END_REF] can provide empirical insights into the behavior of this type of landslide, as long as no dramatic rupture occurs

The shape of the IR was first estimated by cross-correlating the precipitations with the velocity of the tag No. 10 between 3 July and 1 October. The cross-correlation on Fig. 12 shows a peak velocity approximately 1.2 days after the rainfall followed by a decrease in velocity until day 5-6, as already observed on this site [START_REF] Bièvre | Influence of environmental parameters on the seismic velocity changes in a clayey mudflow (Pont-Bourquin Landslide[END_REF]. However, the correlation technique results in a poorly defined IR, producing an unrealistic reaction of the landslide's velocity before the start of the precipitations.

To improve the resolution, the IR was deconvoluted using the linear Lasso regression [START_REF] Tibshirani | Regression Shrinkage and Selection via the Lasso[END_REF] implemented in the scikit-learn library [START_REF] Pedregosa | Scikit-learn: Machine Learning in Python[END_REF], to minimize the following cost function:

2 2 1 0.5 J Y X β λ β = - + (5)
where β IR vector; each corresponds to the response after 0,1 … hours of lag, N=120 here;

Y velocity vector; each corresponds to the velocity during the 0,1 … hours of the observation period, 3 months here;

X precipitation matrix, where each , corresponds to the precipitations measured hours after the start of the observation period;

λ regularization parameter, to avoid over-fitting and causing the β parameters to tend toward zero. A value of 0.04 was chosen based on preliminary cross-validation tests.

This process provided a slight under-estimation of the IR amplitude because the regularization factor causes the values of β to tend toward zero. For this reason, the IR was later corrected with a factor (1.7 in this study) to equalize the measured and modeled velocities (the modeled velocity is presented in the next section).

Impulse response and velocity model

The IR (Fig. 12) was estimated using data collected from tag No. 10 between 3 July and 1 October (before snow).

This response suggests that the landslide remained stable for five hours after rainfall, then accelerated rapidly until it reached a peak velocity after 18-24 h, and finally returned to a null velocity after 4.5 days. This pattern might correspond to an initial infiltration in the superficial layers while the landslide remains stable, followed by a pore pressure increase and then decrease in the aquifer as the landslide accelerates then decelerates. The lag before the velocity peak was coherent with the 20-hour delay prior to elevation of the water table, measured previously in this zone [START_REF] Brönnimann | Effect of Groundwater on Landslide Triggering[END_REF]. In addition, the shape of the IR curve is similar to a typical water infiltration process [START_REF] Iverson | Landslide triggering by rain infiltration[END_REF].

As the landslide may behave differently depending on the point measured or instrument used, the IR for all tags and the extensometer are shown in Fig. 13. For the RFID, the darkest curves correspond to less noisy data, based on a noise indicator, which represents both the instrumental sensitivity to rain and the random noise (see section 3.2). All the tags showed a similar IR, and the discrepancy observed mostly depended on the instrumental bias due to rain, which is added to the landslide IR. In contrast, the IR computed from the extensometer data was very different for the first two days, showing a maximum peak almost during the rain event, before decreasing a null value and finally showing similar results as from RFID after the second day. This pattern of landslide behavior is unrealistic, and can be explained by a domination of the instrumental response, affected by the rain. This domination consequently prevents the extraction of synthetic information from the landslide, such as the IR peak velocity, the lag time before it is reached and the deceleration rate.

20

Convolution of the IR with the precipitations can be used to compute a rain-based velocity model [START_REF] Belle | The application of an innovative inverse model for understanding and predicting landslide movements (Salazie cirque landslides, Reunion Island)[END_REF][START_REF] Bernardie | Prediction of changes in landslide rates induced by rainfall[END_REF]. Fig. 14 shows this velocity model superimposed on the velocity measured. Visually, the model appears coherent with the measurements, except on October 23, when the first snowfall was recorded. This coherence is confirmed by the cross-correlation of both curves that reach their maximum coefficient of 0.76 after a zero-hour lag.

Advantages of RFID data for the IR approach

This empiric IR method produced a relatively good fit between the measured and modeled velocity, for which the advantages and limitations are discussed in [START_REF] Belle | The application of an innovative inverse model for understanding and predicting landslide movements (Salazie cirque landslides, Reunion Island)[END_REF][START_REF] Capparelli | FLaIR and SUSHI: two mathematical models for early warning of landslides induced by rainfall[END_REF]. This example emphasizes the potential of dense and continuous in situ displacement data, as provided by the RFID technique. As the IR is likely to be heterogeneous across a landslide area, placing multiple sensors enables the delimitation of these zones and provides a redundancy benefit. In this example, continuous data unaffected by rain and fog, with an hourly sampling time and sub-centimetric accuracy was also required to compute the IR. Observation of how this response changes over time could provide an additional risk indicator, and would require continuous time series covering several months or years. RFID is once again compatible with these measurements because of the low investment necessary and the reduced maintenance cost. This application therefore highlights several advantages of the RFID technique, which will be more fully discussed in the next section

ADVANTAGES AND LIMITATIONS

The assessment of RFID relative ranging on a landslide has shed light on its advantages and limitations, which will help guide future monitoring applications.

The major advantage identified in this study is that RFID works at night and under all meteorological conditions. Indeed, radio-frequency waves are known to propagate in rain, fog, snow [START_REF] Monserrat | A review of ground-based SAR interferometry for deformation measurement[END_REF] and vegetation (Intrieri et al., 2018a). Passive RFID in particular has been tested without line-of-sight (Wang andKatabi, 2013), outdoors (Le Breton et al., 2017) and with snow on the tags [START_REF] Nummela | Passive UHF RFID tags in arctic environment[END_REF]. This real-world test confirms that RFID adapts very well to natural outdoor areas despite periods of fog and precipitation, although minor accuracy and readability degradations were noted in the presence of snow. This degradation may be due to detuning effects in the vicinity of the tag and station antenna [START_REF] Dobkin | Environmental effects on RFID tag antennas[END_REF] and to changes in the propagation paths (Kim et al., 2012) due to snow [START_REF] Larson | Can we measure snow depth with GPS receivers?[END_REF] or vegetation (Kim et al., 2012). Displacement accuracy outdoors was found to be ±1 cm with vegetation, precipitations, fog or frost, and in rare cases ±8 cm after snowfall. The technique is thus well adapted for outdoor applications to measure displacements ranging from 1 cm to several meters.

The second advantage of RFID is its low cost, allowing displacements to be measured with a high spatio-temporal resolution. Current prices are about 20 € per tag and 4 k per station. Moreover, the instruments were found to be robust and reliable, requiring little maintenance during the experiment (zero maintenance on the tags, three reboot of the station operated remotely, and one replacement of network cables degraded by animals) and providing a data completeness rate of 97% over five months for most tags. Because of their low cost, a large number of tags can be deployed on landslides, providing a high spatial resolution to delineate and characterize moving zones. From a technical point of view, unique tag identifiers and anti-collision algorithms make simultaneous reading of hundreds of densely stacked tags possible [START_REF] Caizzone | RFID Grids: Part II -Experimentations[END_REF] at a rate of up to 800 tags per second (EPC Gen2, 2015; [START_REF] Klair | A Survey and Tutorial of RFID Anti-Collision Protocols[END_REF]. This capacity results in a high spatio-temporal resolution with lightweight time series that are easy to transmit and process almost in real-time. Because of these characteristics, the method could easily be incorporated in an early-warning-system [START_REF] Intrieri | Design and implementation of a landslide early warning system[END_REF] and appears to be well suited for low-cost, dense monitoring applications in high-risk environments such as landslides where devices may be destroyed.

The final advantage is the continuous development of tags to become multi-parameter sensors [START_REF] Zhang | A Review of Passive RFID Tag Antenna-Based Sensors and Systems for Structural Health Monitoring Applications[END_REF].

In this study, the tags were equipped with a temperature sensor that we used for phase calibration. Recently, other sensors have been incorporated into tags, to gather data related to tilt [START_REF] Lai | A Novel Displacement and Tilt Detection Method Using Passive UHF RFID Technology[END_REF], soil moisture content [START_REF] Aroca | Calibration of Passive UHF RFID Tags Using Neural Networks to Measure Soil Moisture[END_REF][START_REF] Pichorim | Two Solutions of Soil Moisture Sensing with RFID for Landslide Monitoring[END_REF] and vibrations [START_REF] Jayawardana | Measurement System With Accelerometer Integrated RFID Tag for Infrastructure Health Monitoring[END_REF]. All these parameters could be of interest when monitoring landsides, opening new perspectives in dense sensing.

As for the limitations, the first one on landslides is currently the tag's reading range, which was limited to 60 m in this application. However, this limit [START_REF] Nikitin | Antennas and propagation in UHF RFID systems[END_REF] should rapidly evolve with the development of ultra-sensitive tags [START_REF] Amato | RFID Backscattering in Long-Range Scenarios[END_REF][START_REF] Durgin | RF thermoelectric generation for passive RFID[END_REF] and directive tag antennas [START_REF] Kim | Dual-Band Long-Range Passive RFID Tag Antenna Using an AMC Ground Plane[END_REF]. Therefore, a range of a few hundred meters seems to be an achievable goal in the near future.

Secondly, the measurement is limited to 1D radial displacement. Measuring a 2D or 3D translational displacement field is technically possible, but would require additional antennas and may alter measurement accuracy (Miesen et al., 2013a;[START_REF] Wang | Dude, Where's My Card?: RFID Positioning That Works with Multipath and Non-line of Sight[END_REF]. Furthermore, it is not enough to measure a purely translational displacement, as a progressive tilt of the tag's supporting stick can create a discrepancy between the ground and tag displacements that must be accounted for. This tilt could be measured in real-time by joining multiple tags [START_REF] Lai | A Novel Displacement and Tilt Detection Method Using Passive UHF RFID Technology[END_REF] and corrected in the next future.

Finally, the displacements measured are affected by an ambiguity of 8.6 cm that originates from phase difference measurements. To address this problem will require first a high time resolution without missing data to avoid differential displacements exceeding this ambiguity, and second an unwrapping of the phase [START_REF] Zuo | Temporal phase unwrapping algorithms for fringe projection profilometry: A comparative review[END_REF].

Unwrapping noisy or incomplete data produced by a tag could be guided by nearby high-quality data if a large number of tags are deployed, or by a previously defined rain-based model such as that presented in the application section.

CONCLUSIONS

RFID displacement tracking appears effective for monitoring surface deformations on a landslide, with unpreceded spatio-temporal sampling, at a significantly lower cost than GPS or ground-based radar and with an overall 1-cm accuracy within a range of 60 m including a distance-dependent error of around 150 ppm.

In the future, the technique could be enhanced by monitoring displacements in 3D, increasing the reading range, correcting for tilt of the tag support, and assessing performance in the presence of obstacles (e.g. snow, vegetation, rocks or concrete). New environmental applications could be developed by exploiting sensor tags (e.g. temperature, tilt, soil moisture or vibrations), developing a real-time surveillance system that exploits the rapid time sampling

(3-30 ms), measuring very fast displacements, and testing new outdoor applications (e.g. volcanoes, civil infrastructure, rocks, snow or vegetation). Finally, the fast pace of technological developments in RFID should lead to considerable improvements to the technique in the near future.

Fig. 1 .

 1 Fig. 1. (a) Pont-Bourquin landslide location in Switzerland (46.3518N, 7.1780E), (b) General aerial view of the Pont-Bourquin landslide, (c) zoom on the zone where instruments were placed (black rectangle in b), showing the tags and the RFID station, and (d) photograph showing tags No. 10 and No.°2 installed on the landslide, and the station which is placed on a stable zone.

  resulting from propagation through air; r distance between the station and the tag; v RF wave velocity in the medium (≈ 2.998•10 8 m/s);

  wave propagation in cables and instruments (11 m).

Fig. 2 .

 2 Fig. 2. Schematic representation of how tag displacement is tracked by a static station. The phase variation measures the radial displacement dr between two acquisitions.

Fig. 3 .

 3 Fig. 3. Comparison of the radial distances between the tags and the station antenna estimated by RFID ranging techniques (RSS and FD-Phase) with this same distance measured by a manual total station, for the 19 tags deployed. The vertical bars represent the amplitude of the drift over the 5-month period, and the width of the 1:1 ratio line represents the true displacement over 5 months. The centimeter-scale total station error is negligible, and was therefore not represented.

Fig. 4 .

 4 Fig. 4. Relative radial displacement between RFID tag No. 10 and the station antenna over a 5-month period, using the Received Signal Strength (RSS), the Frequency-Domain Phase Difference of Arrival (FD-Phase) and the Time-Domain Phase Difference of Arrival (TD-Phase) techniques, compared to the radial displacement computed from total station measurements. The darker lines for RSS and FD-Phase represent a 24-h moving averaged.

Fig. 5 .

 5 Fig. 5. (a) Cumulative radial displacement measured over 4 months on the Pont-Bourquin landslide, using a wire extensometer, the phase of RFID tag No. 10, and a manual laser total station. (b) Total snow height. The snow considerably perturb the wire extensometer measurements.

Fig. 6 .

 6 Fig. 6. Comparison of radial velocities between (a) RFID tag n°10 and (b) extensometer after filterAing for daily variations. (c) Rainfall and (d) fresh snowfall are also shown.

Fig. 7 .

 7 Fig. 7. Radial displacement curves for all tags over 5 months. The curves are ordered based on their cumulative displacement using a vertical offset to facilitate vizualization. The inset indicates the snow height.

Fig. 8 .

 8 Fig. 8. Radial velocity curves for all tags, ranked by decreasing noise levels from top to bottom.

Fig. 9 .

 9 Fig. 9. Random measurement error represented as the standard deviation σr computed every minute for each tag, plotted against the received power P, on a logarithmic scale. The curve shows a clear linear relation between √ and σr.

  2

Fig. 10 .

 10 Fig. 10. Standard deviation of the 1-hour resampled RFID displacement after trend removal, plotted against the tag radial distance, for each tag. The deviation tends to increase with distance, at a rate of about 50 parts per million.

Fig. 12 .

 12 Fig. 12. Impulse response of landslide velocity (cm/day) following precipitation (cm/day) estimated by cross-correlation and Lasso regression deconvolution on rainfall and displacement data from tag No.10 over the 5-month period.

Fig. 13 .

 13 Fig. 13. Impulse response of the landslide velocity (cm/day) to precipitation (cm/day) deconvoluted using a Lasso regression on the velocity measured by all the RFID tags and the extensometer. For the RFID, darker lines represent more accurate tags, presenting lower noise and a lower sensitivity to rain.

ACKNOWLEDGMENTS

This work was supported by Géolithe, the ANRT, the Labex OSUG@2020, the VOR program from Univ. Grenoble Alpes, and Tagsys. The authors thank Benjamin Vial for instrumentation, Antoine Guillemot for field surveys, Pascal Lacroix for insights into deformation monitoring techniques, and Hannah Taylor and Maighread Gallagher for proofreading.


Andrei Pajitnov
email: andrei.pajitnov@univ-nantes.fr

TWISTED MONODROMY HOMOMORPHISMS AND MASSEY PRODUCTS

Keywords: 2010 Mathematics Subject Classification. 55N25, 55T99, 32Q15 mapping torus, Massey products, twisted cohomology, Kähler manifolds

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

The relation between non-vanishing Massey products of length 2 in the cohomology of mapping tori and the Jordan blocks of size greater than 1 of the monodromy homomorphism was discovered in the work of M. Fern ández, A. Gray, J.

Morgan [START_REF] Fern Ández | Compact symplectic manifolds with free circle actions and Massey products[END_REF]. This relation was used by these authors to prove that certain mapping tori do not admit a structure of a Kähler manifold. In the work of G. Bazzoni, M. Fern ández, V. Mu ñoz [START_REF] Bazzoni | Non-formal co-symplectic manifolds[END_REF] it was proved that the existence of Jordan blocks of size 2 implies the existence of a non-zero triple Massey product of the form xξ, ξ, ay.

In the paper [START_REF] Pajitnov | Massey products in mapping tori[END_REF] we began a systematic treatment of this phenomena, relating the maximal length of non-zero Massey products to the maximal size of Jordan blocks of the monodromy homomorphism. The both numbers turn out to be less by a unit than the number of the sheet where the formal deformation spectral sequence degenerates. In that paper we dealt with the case of Massey products of the form xξ, . . . , ξ, xy where ξ is the 1-dimensional cohomology class determined by the fibration of the mapping torus X over the circle, and x is an element in the cohomology of X with coefficients in a 1-dimensional local system.

In the present paper we continue the study of this phenomena, and prove a theorem relating the Jordan blocks of the monodromy homomorphisms to the Massey products of the form xξ, . . . , ξ, xy where x is an element in the twisted cohomology of X corresponding to an arbitrary representation π 1 pXq Ñ GLpn, Cq. One technical problem here is that there is no immediate definition of the homomorphism induced in the twisted homology of M by the diffeomorphism φ : M Ñ M . We construct such homomorphism in the present paper (Section 4). The construction is based on the techniques developed by P. Kirk and C. Livingston [START_REF] Kirk | Twisted Alexander invariants, Reidemeister torsion, and Casson-Gordon invariants[END_REF] in the context of twisted Alexander polynomials.

One corollary is that if X is a compact Kähler manifold, then all the Jordan blocks of this twisted monodromy homomorphism are of size 1, that is, the twisted monodromy homomorphism is diagonalizable. This result imposes new constraints on the homology of Kähler manifolds.

OVERVIEW OF THE AR TICLE

Let φ : M Ñ M be a diffeomorphism of a C 8 compact connected manifold, and X its mapping torus. Choose a point x 0 P M , and put H " π 1 pM, x 0 q, G " π 1 pX, x 0 q. We have an exact sequence (1)

1 / / H i / / G p / / Z / / 1
Let V be a finite dimensional vector space over C and ρ : G Ñ GLpV q be a representation; denote by ρ 0 its restriction to H. Put V ˚" HompV, Cq and let ρ ˚: G Ñ GLpV ˚q be the antirepresentation of G conjugate to ρ. In Section 4 we construct from these data an automorphism φ ˚of the group H ˚pM, ρ 0 q; we call it the twisted monodromy homomorphism. This homomorphism can be considered as a generalization of the map induced by φ in the ordinary homology. Observe however, that the homomorphism φ ˚is not entirely determined by φ and ρ 0 , but depends also on the values of the representation ρ on the elements of GzH.

In the particular case when ρ 0 is the trivial 1-dimensional representation, and the representation ρ sends the positive generator u of G{H « Z to λ P C ˚, the map λφ ˚equals the homomorphism induced by φ in the ordinary homology (see the details in Subsection 4.2).

The main result of the paper is the theorem A below. To state it we need some terminology. Denote by H ˚pX, ρq the twisted cohomology of X with respect to the representation ρ. We have a natural pairing H ˚pX, Cq b H ˚pX, ρq Ñ H ˚pX, ρq; one can define Massey products of the form xξ, . . . , ξ, xy, where x P H ˚pX, ρq.

The Massey product containing r terms ξ will be denoted by xξ, xy r ; we say that the length of this product is equal to r. Denote by M k pρq the maximal length r of a non-zero Massey product xξ, xy r for x P H k pX, ρq. For a number λ P C define a representation ρ λ : π 1 pX, x 0 q Ñ GLpV q as follows: ρ λ pgq " λ ξpgq ¨ρpgq. Denote by J k pφ ˚, λq the maximal size of a Jordan block of eigenvalue λ of the homomorphism φ ˚of H k pM, ρ 0 q.

Theorem A. We have J k pφ ˚, λq " M k pρ λ q for every k and λ.

This theorem implies that the monodromy homomorphism φ ˚has only Jordan blocks of size 1 (that is, φ ˚is diagonalizable) provided that the space X satisfies a suitable formality condition. Such formality conditions are discussed in details in Subsection 3.3. The main application of these ideas is the following theorem.

Theorem B. Assume that the mapping torus X of a diffeomorphism φ : M Ñ M is a compact Kähler manifold. Let ρ : π 1 pX, x 0 q Ñ GLpn, Cq be a semisimple representation. Then the twisted monodromy homomorphism φ ˚of H ˚pM, ρ 0 q is a diagonalizable linear map.

The proofs of these theorems are given in Section 6. They are based on the construction of the twisted monodromy homomorphism φ ˚(Section 4), and the computation of the Massey spectral sequences in terms of φ ˚(Section 5). In Subsection 4.2 we discuss a particular case of special interest. Let us say that φ is π 1 -split if the exact sequence (1) splits. For this case we give a natural geometric construction of an automorphism φ ˝of H ˚pM, ρ 0 q which is entirely determined by φ and ρ 0 (see the formula [START_REF] Pajitnov | Massey products in mapping tori[END_REF] and Remark 4.8 of the Subsection 4.2). Theorem 6.2 states the version of Theorem A for this particular case.

Let us mention a generalization of Theorem A to the case of an arbitrary manifold Y endowed with a non-zero cohomology class α P H 1 pY, Zq and a representation θ : π 1 pY, y 0 q Ñ GLpV q (the manifold Y is not assumed to be a mapping torus any more). Assume that α is indivisible and consider the corresponding infinite cyclic covering Y . Let L " Crt, t ´1s. Similarly to the above, denote by θ 0 the restriction of θ to π 1 pY q, and by θ 0 its conjugate antirepresentation. The homology H k pY , θ 0 q is a finitely generated L-module; denote by T k its torsion part. This is a finite dimensional vector space over C endowed with an action of L. In particular the element t P L represents an automorphism of T k .

Theorem C. Let Y be a connected compact manifold, For every k and λ P C the maximal length r of a Massey product of the form xα, xy r (where x P H k pY, θ λ q) equals the maximal size of a Jordan block of eigenvalue λ of the automorphism t :

T k Ñ T k .
Theorem D. Let Y be a connected compact Kähler manifold, θ : π 1 pY, y 0 q Ñ GLpV q a semisimple representation. Then the homomorphism t : T k Ñ T k is diagonalizable for every k.

2.1. About the terminology. We will keep the notations from Section 2 throughout the paper. Namely, X will always denote the mapping torus of a diffeomorphism φ of a compact connected manifold M ; the corresponding cohomology class in H 1 pX, Zq will be always denoted by ξ, the ring Crt, t ´1s is denoted by L, and Crrzss by Λ. There are two exceptions: in Subsection 3.3 G will denote any group, and in Subsection 5.1 G will denote the fundamental group of a topological space Y .

2.2.

Relations with other works. The case of the trivial representation ρ 0 was settled in the author's paper [START_REF] Pajitnov | Massey products in mapping tori[END_REF]. The diagonalizability of the monodromy homomorphism in the ordinary homology for Kähler manifolds was also proved by N. Budur, Y. Liu, B. Wang [START_REF] Budur | The monodromy theorem for compact Khler manifolds and smooth quasi-projective varieties[END_REF].

Another approach to the relation between the size of Jordan blocks and formality properties was developed by S. Papadima and A. Suciu [START_REF] Papadima | Algebraic monodromy and obstructions to formality[END_REF], [START_REF] Papadima | Geometric and algebraic aspects of 1-formality[END_REF]. They prove in particular that if the monodromy homomorphism has Jordan blocks of size greater than 1, then the fundamental group of the mapping torus is not a formal group.

FORMAL DEFORMATIONS AND MASSEY SPECTRAL SEQUENCES

The main aim of this section is to recall necessary definitions and results concerning the Massey spectral sequences. There are different versions of these spectral sequences in literature, see [START_REF] Farber | Exactness of Novikov inequalities[END_REF], [START_REF] Novikov | Bloch homology, critical points of functions and closed 1-forms Soviet Math[END_REF], [START_REF] Pajitnov | Proof of a conjecture of Novikov on homology with local coefficients over a field of finite characteristic[END_REF], [START_REF] Farber | Topology of closed 1-forms and their critical points[END_REF], [START_REF] Kohno | Novikov homology, jump loci and Massey products[END_REF]. We will recall here the versions described in [START_REF] Kohno | Novikov homology, jump loci and Massey products[END_REF], referring to this article for details and proofs.

The only new material in the present section is the definition of F -formal manifold, introduced in Subsection 3.3. This notion generalizes the classical notion of formality (D. Sullivan [START_REF] Sullivan | Infinitesimal computations in topology[END_REF]) incorporating to it differential forms with coefficients in flat bundles.

Formal deformations of differential graded algebras. Let

A ˚" tA k u kPN " tA 0 d / / A 1 d / / . . .u be a graded-commutative differential algebra (DGA) over C. Let N ˚be a graded differential module (DGM) over A ˚. We denote by A ˚rrzss the algebra of formal power series over A ˚endowed with the differential extended from the differential of A ˚. Let θ P A 1 be a cocycle. Consider the A ˚rrzss-module N ˚rrzss and endow it with the differential

D t x " dx `zθx.
Then N ˚rrzss is a DGM over A ˚rrzss, and we have an exact sequence of DGMs:

(2) 0 / / N ˚rrzss z / / N ˚rrzss π / / N ˚/ / 0
where π is the natural projection z / / 0. The induced long exact sequence in cohomology can be considered as an exact couple Thus E 1 " H ˚pN ˚q, and it is easy to see that E 2 « Ker L α {Im L α , where L α is the homomorphism of multiplication by α. The higher differentials in this spectral sequence can be computed in terms of special Massey products. Let a P H ˚pN ˚q. An r-chain starting from a is a sequence of elements ω 1 , . . . , ω r P N ˚such that dω 1 " 0, rω 1 s " a, dω 2 " θω 1 , . . . , dω r " θω r´1 .

Denote by M Z m

prq the subspace of all a P H m pN ˚q such that there exists an rchain starting from a. Denote by M B m prq the subspace of all β P H m pN ˚q such that there exists an pr ´1q-chain pω 1 , . . . , ω r´1 q with θω r´1 belonging to

β. It is clear that M B m piq Ă M Z m pjq for every i, j. Put M H m prq " M Z m prq M M B m prq .
In the next definition we omit the upper indices and write M H prq , M Z prq etc. in order to simplify the notation. Definition 3.2. Let a P H ˚pN ˚q, and r 1. We say that the r-tuple Massey product xθ, . . . , θ, ay is defined, if a P M Z prq . In this case choose any r-chain pω 1 , . . . , ω r q starting from a. The cohomology class of θω r is in M Z prq (actually it is in M Z pN q for every N ) and it is not difficult to show that it is well defined modulo M B prq . The image of θω r in M Z prq {M B prq is called the r-tuple Massey product of θ and a:

xθ, ay r " A θ, . . . , θ looomooon r , a E P M Z prq M M B prq .
The correspondence a / / xθ, ay r gives rise to a well-defined homomorphism

of degree 1 ∆ r : M H prq / / M H prq .
The following result is proved in [START_REF] Kohno | Novikov homology, jump loci and Massey products[END_REF], Theorem 2.5.

Theorem 3.3. 1)

For any r we have ∆ 2 r " 0, and the cohomology group

H ˚pM H prq , ∆ r q is isomorphic to M H pr`1q .
2) For any r there is an isomorphism

φ : M H prq « / / E r commuting with differentials.
Therefore the differentials in the spectral sequence E r are equal to the higher Massey products with the cohomology class of θ. Observe that these Massey products, defined above, have smaller indeterminacy than the usual Massey products. Now let us consider some cases when the spectral sequences constructed above, degenerate in their second term. Recall that a differential graded algebra A ˚is called formal if it has the same minimal model as its cohomology algebra. Theorem 3.4. [ [START_REF] Kohno | Novikov homology, jump loci and Massey products[END_REF], Th. 3.14 ] Let A ˚be a formal differential algebra, α P H 1 pA ˚q. Then the spectral sequence E r pA ˚, αq degenerates at its second term. Definition 3.5. A differential graded module N ˚over a differential graded algebra A ˚will be called formal if it is a direct summand of a formal differential graded algebra

B ˚over A ˚, that is, (4) B ˚" N ˚' K ˚,
where both N ˚and K ˚are differential graded A ˚-submodules of B ˚.

Proposition 3.6. [ [START_REF] Kohno | Novikov homology, jump loci and Massey products[END_REF], Th. 3.16 ] Let N ˚be a formal DG-module over A ˚, and α P H 1 pA ˚q. Then the spectral sequence E r pN ˚, αq degenerates at its second term.

In our applications N ˚will be a DGM of differential forms on a manifold. Let Y be a connected C 8 manifold, and ρ be a representation of the fundamental group of Y . Put A ˚" Ω ˚pY, Cq and let N ˚" Ω ˚pY, ρq be the DGM of differential forms with coefficients in the flat bundle E ρ , induced by ρ. Then N ˚is a DGM over A ˚, so for any cohomology class α P H 1 pY, Cq we obtain a spectral sequence starting with the twisted cohomology H ˚pY, ρq (see the next subsection for recollections about the twisted cohomology). The differentials in this spectral sequence are the Massey products with the class α. We denote this spectral sequence by E r pY, ρ, αq. Corollary 3.7. In the above notations assume that N ˚" Ω ˚pY, ρq is a formal differential graded module over A ˚" Ω ˚pY, Cq. Then the spectral sequence E r pY, ρ, αq degenerates at its second term.

Homology with local coefficients and spectral sequences.

Let us first recall the definition of homology and cohomology with local coefficients. Let R be a commutative ring, and K a free module over R; denote by GLpKq the group of Rautomorphisms of K. Let Y be a connected topological space, and ρ : π 1 pY, y 0 q Ñ GLpKq a representation. Define the R-module of ρ-twisted cochains of Y with coefficients in ρ as follows:

(5)

C ˚pY, ρq " Hom ρ `C˚p r Y q, K ˘.

Here r Y denotes the universal covering of Y ; we endow it with the canonical free left action of π 1 pY, y 0 q. We denote by C ˚pY q the group of singular chains of Y ; if Y is a CW-complex we can replace it by the group of cellular chains. The group C ˚p r Y q of r Y has a natural structure of a free left module over Zπ 1 pY, y 0 q. The cohomology H ˚pC ˚pY, ρqq is called twisted cohomology of Y with coefficients in ρ, or cohomology of Y with local coefficients in ρ.

Let β : π 1 pY, y 0 q Ñ GLpKq be an antirepresentation (that is, βpg 1 g 2 q " βpg 2 qβpg 1 q for every g 1 , g 2 P π 1 pY, y 0 q); it determines a right action of π 1 pY, y 0 q on K. Define the R-module of ρ-twisted chains of Y with coefficients in K as follows:

(6) C ˚pY, βq " K b β C ˚p r Y q.
The homology H ˚pC ˚pY, βqq is called twisted homology of Y with coefficients in β, or homology of Y with local coefficients in β.

In this paper we will be dealing with the cases when R is one of the following rings: C, L " Crt, t ´1s, Λ " Crrzss. Let V be a finite-dimensional vector space over C, put

V rt ˘s " V b C Crt, t ´1s, V rrzss " V b C Crrzss.
If α P H 1 pY, Zq is a non-zero cohomology class, we can define two representations:

rαs : π 1 pY, y 0 q Ñ L ' ; rαspgq " t xα,gy , xαy : π 1 pY, y 0 q Ñ Λ ' ; xαypgq " e zxα,gy .

For a representation ρ : π 1 pY, y 0 q Ñ GLpV q put (7)

ρ : π 1 pY, y 0 q Ñ GL `V rt ˘s˘, ρ " ρ b rαs, (8)
p ρ : π 1 pY, y 0 q Ñ GL `V rrzss ˘, p ρ " ρ b xαy.
The representation ρ is a basic tool in the theory of twisted Alexander polynomials (see [START_REF] Kirk | Twisted Alexander invariants, Reidemeister torsion, and Casson-Gordon invariants[END_REF], Section 2.1). Observe that p ρ " Exp ˝ρ, where Exp : L Ñ Λ is the homomorphism sending t to e z .

Applying the functor Hom to the exact sequence 0 Ñ Λ t / / Λ Ñ C Ñ 0 we obtain an exact sequence of groups of twisted cochains:

0 / / C ˚pY, p ρq z / / C ˚pY, p ρq / / C ˚pY, ρq Ñ 0.
The corresponding long exact sequence of twisted homomorphism modules can be considered as an exact couple In particular the differentials in the spectral sequence D r pY, ρ, αq are equal to the Massey products xα, ¨yr .

Formality with respect to a family of representations of the fundamental group.

Let us start with a definition. Definition 3.9. Let Y be a manifold, y 0 P Y , denote π 1 pY, y 0 q by G. Let F be a family of complex representations of G, such that F is closed under tensor products, that is, if

ρ 1 , ρ 2 P F then ρ 1 b ρ 2 P F . Put Ω ˚pY, F q " ' ρPF Ω ˚pY, ρq.
Then Ω ˚pY, F q has a natural structure of a DGA. We say that Y is F -formal, if this DGA is formal.

Examples.

1) If F is the trivial 1-dimensional representation, then the notion of F -formality is the same as the classical notion of formality as introduced by D. Sullivan [START_REF] Sullivan | Infinitesimal computations in topology[END_REF], [START_REF] Deligne | Real homotopy theory of Kahler manifolds[END_REF]. 2) Let F be the family of all 1-dimensional representations of G. Then the notion of F -formality is the same as the notion of strong formality introduced in [START_REF] Kohno | Novikov homology, jump loci and Massey products[END_REF], see also [START_REF] Kasuya | Minimal models, formality and hard Lefschetz properties of solvmanifolds with local systems[END_REF]. All compact connected Kähler manifolds are strongly formal, as it follows from C. Simpson's theorem [START_REF] Simpson | Higgs bundles and local systems[END_REF].

3) Let G be a fundamental group of a compact connected Kähler manifold, let ρ be a semisimple representation of G. Consider the family F consisting of all tensor powers of ρ (including the trivial representation of the same dimension as ρ). It follows from theorem of K. Corlette [START_REF] Corlette | Flat G -bundles with canonical metrics[END_REF] that F is closed under tensor products, see also [START_REF] Simpson | Higgs bundles and local systems[END_REF]. C. Simpson's theorem [START_REF] Simpson | Higgs bundles and local systems[END_REF] implies that Y is F -formal.

Theorem 3.10. Assume that a manifold Y is F -formal. Let ρ P F and α P H 1 pY, Cq. Then the formal deformation spectral sequence D r pY, ρ, αq degenerates at its second term. All Massey products xα, xy r vanish for every x P H ˚pY, ρq and r 2.

Proof. It suffices to apply Corollary 3.7 to the module Ω ˚pY, E ρ q.

The formality property of Example 3) above yields the following corollary.

Corollary 3.11. Assume that Y is a connected compact Kähler manifold, and α P H 1 pY, Cq a non-zero cohomology class. Let ρ : π 1 pY, y 0 q Ñ GLpn, Cq be a semisimple representation. Then the spectral sequence D r pY, ρ, αq degenerates at its second term.

TWISTED MONODROMY HOMOMORPHISMS

Let φ : M Ñ M be a diffeomorphism of a C 8 compact connected manifold, and X its mapping torus. Choose a point x 0 P M , and put H " π 1 pM, x 0 q, G " π 1 pX, x 0 q. Recall the exact sequence

1 / / H i / / G p / / Z / / 1.
Let W be a vector space of dimension n over C endowed with a right action of G. Such action can be described as a map β : G Ñ GLpW q « GLpn, Cq satisfying βpg 1 g 2 q " βpg 2 qβpg 1 q, that is, an antirepresentation of G. Set β 0 " β | H : H Ñ GLpW q. In this section we associate to these data an isomorphism φ ˚: H ˚pM, β 0 q Ñ H ˚pM, β 0 q of vector spaces that we call twisted monodromy homomorphism induced by φ. This homomorphism can be considered as a generalization of the map induced by φ in the ordinary homology. Observe however, that the homomorphism φ ˚is not entirely determined by φ and ρ 0 , but depends also on the values of ρ on the elements of GzH (see the details in Subsection 4.1). The constructions of this section will be applied in Section 6 to the map β : G Ñ GLpW q which is conjugate to the given representation ρ : G Ñ GLpW q.

4.1. Definition of the twisted monodromy homomorphism. Choose any path θ in M from x 0 to φpx 0 q. This choice determines three more geometric objects:

A) An element u P G such that ppuq " 1. Namely let u be a composition of the path θ with the image of the path φpx 0 q ˆr0, 1s in the mapping torus

X " M ˆr0, 1s M px, 0q " pφpxq, 1q.
Observe that any element u with ppuq " 1 can be obtained this way with a suitable choice of θ.

B) A lift of the map φ to a map r φ : Ă M Ñ Ă M . Namely, represent a point x P Ă M by a path γ in M starting at x 0 . The path φpγq " φ ˝γ joins the points φpx 0 q and φpxq. The composition of paths θ ¨φpγq joins x 0 and φpxq. Now put r φpγq " θ ¨φpγq.

C) A homomorphism K θ : H Ñ H defined by K θ pγq " θφpγqθ ´1 where γ is a loop starting at x 0 .

These objects satisfy the following easily checked properties:

(10) r φphxq " K θ phq r φpxq for every h P H and x P Ă M ;

(11) uhu ´1 " K θ phq for every h P H.

Now we can define the homomorphism φ

˚.

Definition 4.1. To simplify the notation, we shall abbreviate W b β0

C ˚p Ă M q to W b C ˚p Ă M q up to
the end of the present subsection.

Define a map

(12) α : W b C C ˚p Ă M q Ñ W b C C ˚p Ă M q; αpv b σq " vu ´1 b r φpσq,
where v P W , and σ is a simplex in C ˚p Ă M q (here and elsewhere we denote by vg the result of the action of g P G on the vector v P W ).

Lemma 4.2. 1) The map α factors to an endomorphism

A of W b C ˚p Ă M q.
2) The resulting map A is a chain map, and it does not depend on the choice of the path θ. Apply the formula [START_REF] Kohno | Novikov homology, jump loci and Massey products[END_REF] and the proof of the first part of Lemma is over.

2) Let θ 1 be another path joining x 0 and φpx 0 q, so that θ 1 " γθ where γ is a loop in M starting at x 0 . The corresponding element u 1 P G satisfies u 1 " γu, and r φ 1 " γφ, so that vu 1´1 b r φ 1 pσq " vuγ ´1 b γφpσq and the property 2) follows.

Definition 4.3. The map induced by A in the homology groups H ˚pM, βq will be denoted by φ ˚rβs and called the twisted monodromy homomorphism associated to φ and β (when the value of β is clear from the context we omit it in the notation).

Definition 4.4. For any antihomomorphism β : G Ñ GLpW q and λ P C ˚define an antihomomorphism β λ : G Ñ GLpW q as follows:

β λ pgq " λ xξ,gy ¨βpgq.
The proof of the following proposition follows immediately from the definition of φ ˚(see the formula ( 12)). Proposition 4.5. We have [START_REF] Pajitnov | Novikov homology, twisted Alexander polynomials, and Thurston cones[END_REF] φ ˚rβ λ s " 1 λ ¨φ˚r βs.

4.2.

The case G " H ˆZ. The algebralically simplest case occurs when the exact sequence (1) splits. This case can be characterized by the following simple lemma (the proof will be omitted).

Lemma 4.6. The three following properties are equivalent: 1) For some path θ from x 0 to φpx 0 q the homomorphism K θ : H Ñ H is an inner automorphism. 2) For every path θ from x 0 to φpx 0 q the homomorphism K θ : H Ñ H is an inner automorphism.

3) The extension (1) splits.

One can prove also that if the properties listed in the lemma hold for some choice of a base point x 0 , then they hold for any other choice of the base point. Assume that φ is π 1 -split. Choose an element u P G commuting with H, and such that ppuq " 1. Let β : H Ñ GLpW q be any antirepresentation, and let β 0 be its restriction to H. Put B " βpuq, then B P GLpW q. (Observe that in the split case any antirepresentation of H can be extended to an antirepresentation of G, sending u to a scalar matrix.)

1. Consider first the case when B is the identity map of W . The homomorphism φ ˚in this case has an especially simple definition. Namely, choose a path θ from x 0 to φpx 0 q in such a way that for any γ P π 1 pM, x 0 q we have θφpγqθ ´1 " γ. Then the corresponding lift r φ : Ă M Ñ Ă M has the property r φphxq " h r φpxq for every h P π 1 pM, x 0 q Denote such a lift by r φ ˝. The automorphism of H ˚pM, β 0 q corresponding to this choice will be denoted by φ ˝; it is defined by the following formula: [START_REF] Pajitnov | Massey products in mapping tori[END_REF] φ ˝pv b σq " v b r φ ˝σ (where σ is a singular simplex of Ă M ). This homomorphism φ ˝is entirely determined by φ and β 0 , and does not depend on the values of β on GzH. In the case when β 0 is the trivial representation the map φ ˝is just the induced map in the ordinary homology.

2. Now let B " λ ¨Id where λ P C ˚. Choosing for φ the same lift as in the previous case, we obtain the following formula for φ ˚:

φ ˚pv b σq " 1 λ v b r φ ˝pσq " 1 λ φ ˝pv b σq.
3. Now let B be an arbitrary element of GLpW q. Since B commutes with H, it induces a well-defined linear maps C ˚pM, β 0 q Ñ C ˚pM, β 0 q and H ˚pM, β 0 q Ñ H ˚pM, β 0 q, that will be denoted by the same letter B. We have then αpv b σq " vu ´1 b r φpσq " B ´1pv b r φpσqq and finally [START_REF] Papadima | Algebraic monodromy and obstructions to formality[END_REF] φ ˚" B ´1 ˝φ˝.

Remark 4.8. In the case λ " 1 it is possible to reformulate our definition of φ in terms of induced representations of fundamental groups. To explain this, let us proceed to a slightly more general framework. Let φ : X Ñ Y be a map of connected topological spaces, x 0 P X, y 0 P Y . Let ρ : π 1 pY, y 0 q Ñ GLpW q be a representation. Choose a path µ from y 0 to φpx 0 q. Define a representation ρ 1 : π 1 pX, x 0 q Ñ GLpW q as follows ρ 1 pgq " ρpµφpgqµ ´1q. It is easy to check that φ induces a homomorphism

φ ˚: H ˚pX, ρ 1 q Ñ H ˚pY, ρq, defined on the chain level as v b σ Þ Ñ v b r
φ ˝σ. This homomorphism depends obviously on µ. In the case when X " Y and φ is π 1 -split choose a path µ in such a way that µφphqµ ´1 " h for every h P π 1 pM, x 0 q. Then ρ 1 " ρ and we return to the homomorphism φ ˝of the above definition. 4.3. Relation wih Kirk-Livingston's setup. Let X be the infinite cyclic covering of X corresponding to ξ. Observe that π 1 pXq « H, so that the twisted homology H ˚pX , β 0 q of X is defined. We have the following simple lemma; the proof follows from the observation that X « M ˆR. Lemma 4.9. The inclusion i : M / / X induces an isomorphism

I : H ˚pM, β 0 q « / / H ˚pX , β 0 q.
In the work It is shown in [START_REF] Kirk | Twisted Alexander invariants, Reidemeister torsion, and Casson-Gordon invariants[END_REF] that that this action does not depend on the particular choice of u. The next proposition follows readily from the definition of the monodromy homomorphism φ ˚.

Proposition 4.10. The following diagram is commutative

H ˚pX , β 0 q t / / H ˚pX , β 0 q H ˚pM, β 0 q I O O φ˚/ / H ˚pM, β 0 q I O O Remark 4.11.
We worked in this section in the assumption that M is a compact C 8 manifold; the homology groups were in coefficients in C, in view of the applications to Kähler manifolds. However all the constructions and results of the section generalize without any changes to the case when M is any CW-complex; the coefficient field C can be replaced by an arbitrary field K.

TWISTED MONODROMY MAPS AND FORMAL DEFORMATION SPECTRAL SEQUENCES

We begin with a discussion of a universal coefficient theorem for twisted cohomology (Subsection 5.1). The next subsection contains the computation of the formal deformation spectral sequences in terms of the monodromy maps. The proof of the main theorem in Section 6 is based on these computations. 5.1. Universal coefficient theorem for twisted cohomology. Let Y be a connected topological space endowed with a non-zero cohomology class η P H 1 pY, Zq. Denote by G the fundamental group π 1 pY, y 0 q. Let V be a finite-dimensional vector space over C and ρ : G Ñ GLpV q a representation. Let L " Zrt, t ´1s, denote by V rt ˘s the free L-module V b (where w P W and σ P C ˚p r Y q). Applying this to the right action ρ ˚on W " V ˚we obtain the following isomorphism (see [START_REF] Kohno | Novikov homology, jump loci and Massey products[END_REF], Lemma 4.3) [START_REF] Papadima | Geometric and algebraic aspects of 1-formality[END_REF] H ˚pY, ρq « H ˚´Hom L pC ˚pY, V ˚q, Lq ¯.

The cohomology module in the right-hand side of ( 16) has the advantage that we can apply to it the universal coefficient theorem:

Proposition 5.1. For every k we have an exact sequence

(17) 0 Ñ Ext 1 L ´Hk´1 pY, ρ ˚q, L ¯Ñ H k pY, ρq Ñ Hom L ´Hk pY, ρ ˚q, L ¯Ñ 0.
We will now apply these results to mapping tori. In the rest of this subsection X is the mapping torus of a homeomorphism φ : M Ñ M (see Subsection 2.1 for the notations). Endow the vector space H k´1 pM, ρ 0 q with the action of L as follows: ta " φ ˚paq, where φ ˚is the twisted monodromy map from Subsection 4.1.

Proposition 5.2. We have an isomorphism of L-modules H k pX, ρq « H k´1 pM, ρ 0 q. Proof. By Theorem 2.1 of [START_REF] Kirk | Twisted Alexander invariants, Reidemeister torsion, and Casson-Gordon invariants[END_REF] we have H ˚pX, ρ ˚q « H ˚pX , ρ 0 q. Lemma 4.9 says that H ˚pX , ρ 0 q « H ˚pM, ρ 0 q, therefore this L-module is a finite dimensional vector space over C, and hence a finitely generated torsion L-module. Thus we have Hom L ´Hk pX, ρ ˚q, L ¯" 0 and Ext 1 L ´Hk´1 pX, ρ ˚q, L ¯« H k´1 pX, ρ 0 q « H k´1 pM, ρ 0 q.

The proposition follows.

Computation of deformation spectral sequences in terms of monodromy maps.

Our main aim here is to prove that the exact couples [START_REF] Sullivan | Infinitesimal computations in topology[END_REF] x x H ˚pX, ρq f f maps to pD 1 q via the homomorphism H ˚pX, ρq Ñ H ˚pX, p ρq induced by Exp, therefore the spectral sequences induced by pD 2 q and pD 0 q are isomorphic. Applying Proposition 5.2 we deduce that pD 2 q is isomorphic to the exact couple pD 3 q of the form (21) H ˚pM, ρ 0 q φ˚´1 / / H ˚pM, ρ 0 q j w w H ˚pX, ρq k g g where the maps j and k have the degrees respectively 1 and 0. We obtain finally a homomorphism pD 3 q Ñ pD 0 q of exact couples, which equals the identity map on the term H ˚pX, ρq. Therefore the exact sequences derived from these exact couples are isomorphic. Remark 5.3. Similary to the Section 4, all the constructions and results of the section generalize without any changes to the case when M is any CW-complex.

PROOFS OF THE MAIN RESULTS

Now we can complete the proofs of the main results. 6.1. Theorem A. Let us first prove Theorem A for the case λ " 1, that is, ρ λ " ρ. According to the previous section the exact couples pD 0 q and pD 3 q induce isomorphic spectral sequences. The differentials in the spectral sequence induced by pD 0 q are equal to Massey products: d r pxq " xξ, xy r , therefore the spectral sequence degenerates in degree k at the term number M k pρq `1. It suffices to prove that the spectral sequence induced by pD 3 q degenerates at the term J k pφ ˚q `1 in degree k. Denote by φ

δ ' τ : A ' B Ñ A ' B
where δ is nilpotent of degree m and τ is injective. 1) Let deg i " deg l " 0, deg j " 1. Then the spectral sequence induced by E degenerates at the step m `1 in degree k.

2) Let deg i " 0, deg l " 1, deg j " 0. Then the spectral sequence induced by E degenerates at the step m `1 in degree k ´1. Now let us consider the case when λ is an arbitrary non-zero complex number. According to the Proposition 4.5 the monodromy homomorphism φ ˚rρ λ s constructed from the representation ρ λ equals 1 λ φ ˚rρs. Therefore the exact couple pD 3 q for the case of the representation ρ λ has the following form

(23) H ˚pM, ρ 0 q 1 λ φ˚´1
/ / H ˚pM, ρ 0 q j 1 w w H ˚pX, ρ λ q k 1 g g (where φ ˚denotes the monodromy homomorphism corresponding to ρ). It remains to observe that J k p 1 λ φ ˚, 1q " J k pφ ˚, λq.

Theorem B.

Let X be a connected compact Kähler manifold, and ρ a semisimple representation. In view of Theorem A it suffices to prove that M k pρ λ q " 1 for every λ P C ˚, or, equivalently, that the spectral sequence associated to the exact couple (24) H ˚pX, p ρ λ q z / / H ˚pX, p ρ λ q w w H ˚pX, ρ λ q g g degenerates at its second term. Observe that the representation ρ λ is also semisimple; apply to it Corollary (3.11) and the proof of Theorem B is over. 6.3. The π 1 -split case. Recall from Subsection 4.2 that in this case we have an automorphism φ ˝determined by φ and by the representation of π 1 pM q. Choose an element u P G commuting with H and such that ppuq " 1. Let χ : π 1 pM, x 0 q Ñ GLpV q be any representation of the fundamental group of M . Let λ P C ˚. Extend χ to a representation χ λ : π 1 pX, x 0 q Ñ GLpV q by χpuq " λ (this is possible since u commutes with H). Theorem 6.2. We have 1) J k pφ ˝, λq " M k pχ λ q.

2) If moreover X is a compact Kähler manifold, and the representation χ is semisimple, then φ ˝is diagonalizable.

Proof. Part 1) follows immediately from Theorem A. As for the part 2), observe that the representation χ λ is also semisimple, so we can apply to it the Theorem B, and the proof is over. Remark 6.3. The particular case of the trivial representation χ corresponds to Theorems 3.1 and 5.1 of the paper [START_REF] Pajitnov | Massey products in mapping tori[END_REF]. 6.4. Theorem C. We need some more terminology. Definition 6.4. Let R be a finitely generated L-module, and a P L a polynomial of degree 1. Denote by R a the a-primary part of R, that is, the submodule of all x P R, such that a N x " 0 for some N . Denote by N ilpR, aq the minimal number N , such that a N R a " 0. The module R a is a finite-dimensional vector space, and a determines a linear map of this space. The number N ilpR, aq equals the maximal size of Jordan blocks of eigenvalue 0 of a.

Denote by M k pα, θ λ q the maximal length of a non-zero Massey product of the form xα, xy r where x P H k pY, θ λ q. Consider the spectral sequence associated to the exact couple (25) H ˚pY, θ λ q t´1 / / H ˚pY, θ λ q w w H ˚pY, θ λ q g g Applying the same argument as in the beginning of Subsection 6.1 we deduce that M k pα, θ λ q `1 equals the the number r of the sheet where this spectral sequence degenerates. By Lemma 6.1 this number r equals N il `Hk`1 pY, θ λ q, t´1 ˘. Observe that we have θ λ " g λ ˝θ, where g λ : L Ñ L is the isomorphism given by the formula g λ ptq " λ ¨t. Therefore the L-homomorphism t ´1 : H ˚pY, θ λ q / / H ˚pY, θ λ q is isomorphic to the L-homomorphism λ ´1t ´1 : H ˚pY, θq / / H ˚pY, θq, and we have N il `Hk`1 pY, θ λ q, t ´1˘" N il `Hk`1 pY, θq, t ´λ˘. The torsion submodule of H k`1 pY, θq is isomorphic to Ext `Hk pY, θ ˚q, L ˘which is in turn isomorphic to the torsion submodule T 1 k of H k pY, θ ˚q. A theorem of P. Kirk and C. Livingston ([9], Th. 2.1) says that we have an isomorphism H ˚pY, θ ˚q « H ˚pY , θ 0 q.

Therefore the module T 1 k is isomorphic to the torsion submodule T k of H ˚pY , θ 0 q, so, finally, M k pα, θ λ q " N ilpT k , t ´λq and the proof of Theorem C is complete. 6.5. Theorem D. It follows readily from Theorem C; the proof is similar to the argument of Subsection 6.2. 6.6. Acknowledgments. I am grateful to Professor F. Bogomolov for valuable discussions and support.

Definition 3 . 1 .

 31 One can prove that the spectral sequence induced by the exact couple (3) depends only on the cohomology class of θ ([10], Prop. 2.1). Put α " rθs. The spectral sequence associated to the exact couple (3) is called formal deformation spectral sequence and denoted by E r pN ˚, αq. If the couple pN ˚, αq is clear from the context, we suppress it in the notation and write just E r .

( 9 )

 9 H ˚pY, p ρq z / / H ˚pY, p ρq x x H ˚pY, ρq f f This exact couple induces a spectral sequence D r pY, ρ, αq starting from the module H ˚pY, ρq. We have the following theorem Theorem 3.8. [[10], Th. 5.4] The spectral sequences E r pY, ρ, αq and D r pY, ρ, αq are isomorphic.

Proof. 1 )

 1 We have to check that αpvh b σq and αpv b hσq give the same element in W b C ˚p Ă M q for every h P H. Observe that αpvh b σq " pvhqu ´1 b r φpσq " vu ´1 ¨uhu ´1 b r φpσq " vu ´1 ¨Kθ phq b r φpσq and this equals vu ´1 b K θ phq r φpσq in W b C ˚p Ă M q.

Definition 4 . 7 .

 47 If the map φ satisfies the three equivalent properties of Lemma (4.6) we say that φ is π 1 -split.

  [START_REF] Kirk | Twisted Alexander invariants, Reidemeister torsion, and Casson-Gordon invariants[END_REF] P.Kirk and C. Livingston constructed an action of the group Z on the space W b β0 C ˚pX q. Namely, choose any element u P G such that ppuq " 1, and let the generator t of Z act on W b β0 C ˚pX q by the following formula tpv b σq " vu ´1 b uσ.

CL.

  Recall from Subsection 3.2 the representation ρ : G Ñ GLpV rt ˘sq; ρpgq " ρpgqt ξpgq . The representation ρ determines an action from the left of G on V ; put V ˚" HompV, Cq, and consider the corresponding right action ρ ˚of G on V ˚. Similarly we obtain a right action ρ ˚of G on V rt ˘s. If we choose a basis in V , then ρ is identified with an antihomomorphism G Ñ GLpn, Cq obtained from ρ by transposition (similarly for ρ ˚). Associated to ρ ˚there is the L-module of ρ ˚-twisted chains V rt ˘s b ρ˚C ˚p r Y q, and its homology H ˚pX, ρ ˚q. Observe that for any right action χ of G on a free L-module W there is a natural isomorphism Hom L ˆW b χ C ˚p r Y q, L ˙Φ / / Hom ρ `C˚p r Y q, W ˚˘; the value of Φ on an L-homomorphism α : W b χ C ˚p r Y q Ñ L is defined by the following formula: ´Φpαqpσq ¯pwq " αpw b σq

Lemma 6 . 1 .

 61 pkq ˚the twisted monodromy homomorphism in degree k. Let A k be the invariant linear subspace of eigenvalue 1 of φ pkq ˚. Let B k be the sum of all invariant linear subspaces of φ pkq ˚corresponding to the eigenvalues different from 1. The restriction pφ pkq ˚´1q | A k is nilpotent of degree equal to J k pφ ˚, 1q, and the restriction pφ pkq ˚´1q | B k is an isomorphism of B k onto itself. The assertion of the theorem follows now from the following lemma ([14], Lemma 3.3). Let E be a graded exact couple: i : D k Ñ D k decomposes as follows:

  and (19) induce isomorphic spectral sequences. We have p ρ " Exp ˝ρ, where Exp : L Ñ Λ is the ring homomorphism sending t to e z . The exact couple pD 0 q

	(18)	H ˚pX, p ρq f f	z	/ / H ˚pX, p ρq
			x x
			H ˚pX, ρq
	is obviously isomorphic to the following exact couple pD 1 q
	(19)	H ˚pX, p ρq f f	e z ´1	/ / H ˚pX, p ρq
			x x
			H ˚pX, ρq
	The exact couple pD 2 q below
	(20)	H ˚pX, ρq

t´1

/ / H ˚pX, ρq