Ryeonshi Kang
email: ryeonshi.kang@inserm.fr

Charlotte Laborde
email: laborde.c@chu-toulouse.fr

Lesia Savchenko
email: lesia.savchenko@inserm.fr

Audrey Swiader
email: audrey.swiader@inserm.fr

Nathal Pizzinat
email: nathalie.pizzinat@inserm.fr

Dimitri Marsal
email: dimitri.marsal@inserm.fr

Yannis Sainte-Marie
email: yannis.sainte-marie@inserm.fr

Frederic Boal
email: frederic.boal@inserm.fr

Helene Tronchere
email: helene.tronchere@inserm.fr

Jerome Roncalli
email: roncalli.j@chu-toulouse.fr

Oksana Kunduzova
email: oxana.koundouzova@inserm.fr

Age-Related Shift in Cardiac and Metabolic Phenotyping Linked to Inflammatory Cytokines and Antioxidant Status in Mice

Keywords: aging, cardiac function, metabolic status, antioxidant, inflammation and resistin

Age-related alterations in cardiac function, metabolic, inflammatory and antioxidant profiles are associated with an increased risk of cardiovascular mortality and morbidity. Here, we examined cardiac and metabolic phenotypes in relation to inflammatory status and antioxidant capacity in young, middle-aged and old mice. Real-time reverse transcription-polymerase chain reactions were performed on myocardium and immunoassays on plasma. Left ventricular (LV) structure and function were assessed by echocardiography using high-frequency ultrasound. Middleaged mice exhibited an altered metabolic profile and antioxidant capacity compared to young mice, whereas myocardial expression of inflammatory factors (TNFα, IL1β, IL6 and IL10) remained unchanged. In contrast, old mice exhibited increased expression of inflammatory cytokines and plasma levels of resistin compared to young and middle-aged mice (p < 0.05). The pro-inflammatory signature of aged hearts was associated with alterations in glutathione redox homeostasis and elevated contents of 4-hydroxynonenal (4-HNE), a marker of lipid peroxidation and oxidative stress. Furthermore, echocardiographic parameters of LV systolic and diastolic functions were significantly altered in old mice compared to young mice. Taken together, these findings suggest age-related shifts in cardiac phenotype encompass the spectrum of metabo-inflammatory abnormalities and altered redox homeostasis.

Introduction

Aging is an inevitable event of the lifecycle associated with an elevated risk of heart failure [START_REF] Tsao | Heart Disease and Stroke Statistics-2023 Update: A Report From the American Heart Association[END_REF]. Age-related progressive decline in cardiac function differs across species, organs and tissues [START_REF] Riehle | Small animal models of heart failure[END_REF][START_REF] Dai | Cardiac aging in mice and humans: The role of mitochondrial oxidative stress[END_REF][START_REF] Federspiel | Conservation and divergence of protein pathways in the vertebrate heart[END_REF]. As the global population faces a progressive shift towards a higher median age, decoding the pathophysiological picture of an aged heart has become of paramount importance to the preservation of cardiac function. Among the various diseases encountered during aging, heart disease is prevalent among older populations, affecting their healthy lifespan and quality of life. Cardiac aging is driven by a finite number of inter-connected mechanisms that ultimately lead to the emergence of specific phenotypes, calibrating increased susceptibility to multiple chronic manifestations and death. Among the many critical aspects of age-related myocardial remodeling processes, oxidative stress inflammation and functional reprogramming are by far the most relevant [START_REF] Triposkiadis | Cardiovascular Aging and Heart Failure: JACC Review Topic of the Week[END_REF]. The prevalence of heart disease is common in both humans and animals, manifesting typically as progressive loss in cardiac function, cytokine/adipokine dysfunctions and antioxidant reserve [START_REF] Heidenreich | AHA/ACC/HFSA Guideline for the Management of Heart Failure: A Report of the American College of Cardiology/American Heart Association Joint Committee on Clinical Practice Guidelines[END_REF][START_REF] Yan | Cardiac Aging: From Basic Research to Therapeutics[END_REF][START_REF] Feijóo-Bandín | Adipokines and Inflammation: Focus on Cardiovascular Diseases[END_REF]. In mice, cardiac aging closely recapitulates human heart aging, which involves hypertrophy, diastolic dysfunction, reduced functional reserve and adaptive capacity to stress [START_REF] Riehle | Small animal models of heart failure[END_REF][START_REF] Yan | Cardiac Aging: From Basic Research to Therapeutics[END_REF]. Echocardiographic imaging performed on a mouse longevity cohort demonstrated distinct age-dependent linear trends in increased left ventricular mass index (LVMI) and left atrial dimension, reduction in fractional shortening (FS) and diastolic function (Ea/Aa), as well as worsening of myocardial performance index [START_REF] Dai | Cardiac aging in mice and humans: The role of mitochondrial oxidative stress[END_REF]. Similar cardiac phenotypes have also been observed in aging of C57BL/6 mice [START_REF] De Lucia | Echocardiographic Strain Analysis for the Early Detection of Left Ventricular Systolic/Diastolic Dysfunction and Dyssynchrony in a Mouse Model of Physiological Aging[END_REF], suggesting that age-mediated cardiac alterations observed in mouse and human are comparable. However, it is still unclear how these imaging modalities can be used to explore aging with mice models. Age-dominated dynamic changes in ventricular function may be monitored using non-invasive cardiac imaging. Echocardiographic measurements are important in the detection of cardiac abnormalities and in defining personalized aetiology of heart failure by providing well-defined parameters on ventricular structure and function [START_REF] Riehle | Small animal models of heart failure[END_REF][START_REF] Dai | Overexpression of catalase targeted to mitochondria attenuates murine cardiac aging[END_REF]. In preclinical research, echocardiographic imaging of ventricular function and structure is a powerful approach to cardiac phenotyping, beneficial in the validation and characterization of disease progression, drug screens as well as exploration of heart damage and recovery [START_REF] Riehle | Small animal models of heart failure[END_REF][START_REF] Yan | Cardiac Aging: From Basic Research to Therapeutics[END_REF][START_REF] Dai | Overexpression of catalase targeted to mitochondria attenuates murine cardiac aging[END_REF]. In contrast to humans, mice have a naturally short lifespan, which makes a mouse a particularly interesting model for preclinical exploration of cardiac aging. Recent advances in imaging technology, with improved resolution, have made echocardiographic imaging in mice a universal tool for dissecting the physiological and pathophysiological alterations linked to age-associated phenotypes [START_REF] Zacchigna | Towards standardization of echocardiography for the evaluation of left ventricular function in adult rodents: A position paper of the ESC Working Group on Myocardial Function[END_REF]. Yet, there is little comprehensive study related to age-associated cascade of cardiovascular events calibrating systolic and diastolic functions in relation to metabolic status and antioxidant capacity. Aging is characterized by excessive generation of high levels of reactive oxygen species (ROS) and defects in antioxidant capacity that can affect the heart's physiological function [START_REF] Shields | Beneficial and Detrimental Effects of Reactive Oxygen Species on Lifespan: A Comprehensive Review of Comparative and Experimental Studies[END_REF][START_REF] Pagan | The Role of Oxidative Stress in the Aging Heart[END_REF]. The imbalance between antioxidant defense system and ROS production is the principal cause of oxidative damage, culminating in progressive loss of structural and functional parameters of the myocardium [START_REF] Pagan | The Role of Oxidative Stress in the Aging Heart[END_REF][START_REF] Moris | The role of reactive oxygen species in the pathophysiology of cardiovascular diseases and the clinical significance of myocardial redox[END_REF][START_REF] Van Der Pol | Treating oxidative stress in heart failure: Past, present and future[END_REF]. At the cellular level, the enhanced production of ROS leads to peroxidation of lipids, oxidation of proteins, damage to nucleic acids and activation of programmed cell death [START_REF] Moris | The role of reactive oxygen species in the pathophysiology of cardiovascular diseases and the clinical significance of myocardial redox[END_REF]. The first line of defense against ROS are catalase, superoxide dismutase (SOD) and glutathione peroxidase which play the capital roles in protecting the heart from oxidative stress and endorsing delayed cardiac aging [START_REF] Dai | Overexpression of catalase targeted to mitochondria attenuates murine cardiac aging[END_REF][START_REF] Romuk | Superoxide dismutase activity as a predictor of adverse outcomes in patients with nonischemic dilated cardiomyopathy[END_REF]. Several studies suggest that catalase and SOD may alleviate cardiac disease and aging in a mouse model [START_REF] Pendergrass | Temporal effects of catalase overexpression on healing after myocardial infarction[END_REF][START_REF] Yoshida | Targeted disruption of the mouse Sod I gene makes the hearts vulnerable to ischemic reperfusion injury[END_REF][START_REF] Van Deel | Extracellular superoxide dismutase protects the heart against oxidative stress and hypertrophy after myocardial infarction[END_REF], suggesting a principal role of these enzymes in age-mediated myocardial remodeling. The purpose of the present study was to characterize age-associated progressive changes in cardiac phenotype in relation to metabolic, antioxidant and inflammatory status in mice.

Results

Age-Dependent Cardiac Phenotype in C57BL/6 Mice

To characterize age-dependent cardiac phenotypes in a cohort of male C57BL/6 mice, we compared physiological, metabolic and echocardiographic parameters in three age groups: young (3 months old), middle-aged (12 months old) and old (24 months old). Metabo-physiological status was the first characteristic assessed due to its high human relevance as the established parameters of clinical outcomes (Table 1).

Group differences in body weight (BW) and whole-body fat were noted, with a significant increase in middle-aged mice and a decline thereafter in aged mice (Table 1). Compared to the young group, whole-body lean tissue was largely preserved in aged mice but significantly elevated in middle-aged animals. In addition, in aged mice, free body fluid was significantly increased as compared to young and middle-aged mice. Furthermore, age-associated alterations in metabolic status in mice were accompanied by increased mRNA and protein expression levels of adipokine resistin in mice (Figure 1). Group differences in body weight (BW) and whole-body fat were noted, with a significant increase in middle-aged mice and a decline thereafter in aged mice (Table 1). Compared to the young group, whole-body lean tissue was largely preserved in aged mice but significantly elevated in middle-aged animals. In addition, in aged mice, free body fluid was significantly increased as compared to young and middle-aged mice. Furthermore, age-associated alterations in metabolic status in mice were accompanied by increased mRNA and protein expression levels of adipokine resistin in mice (Figure 1).

Left Ventricular Dimensions in Young Middle-Aged and Aged Mice

Comparing mice according to age, absolute measurements of the left ventricle in the parasternal short-axis view demonstrated that the middle-aged group had significantly larger left ventricular anterior wall thickness (LVAW) and left ventricular posterior wall thickness (LVPW) versus the young group (Table 2).

Left Ventricular Dimensions in Young Middle-Aged and Aged Mice

Comparing mice according to age, absolute measurements of the left ventricle in the parasternal short-axis view demonstrated that the middle-aged group had significantly larger left ventricular anterior wall thickness (LVAW) and left ventricular posterior wall thickness (LVPW) versus the young group (Table 2). As shown in Figure 2a-e, the left ventricular internal diameter (LVID) at diastole and systole measured from M-mode images revealed significant differences between young, middle-aged and old mice. The LV mass was markedly increased in middle-aged mice compared to young mice and was subsequently maintained in old mice (Figure 2f).

The Echocardiographic Systolic Function in Young, Middle-Aged and Aged Mice

In order to evaluate LV systolic function in young, middle-aged and old mice, we analyzed ejection fraction (EF) in the parasternal long axis view and FS in the parasternal short axis view of the heart. As shown in Figure 3, the assessment of LV systolic function in mice revealed age-dependent decline of myocardial function. The mean EF was 56.87% for young, 57.92% for middle-aged, and 40.93% for old mice (Figure 3). Both EF and FS were preserved in the middle-aged mice compared to the young mice, whereas declines in EF and FS were found in the old mice as compared to the young and middle-aged mice (Figure 3d,e). Measurements of stroke volume (SV) demonstrated a similar profile of changes with preservation of SV in middle-aged mice compared to young mice and then progressive decline in the old group. In contrast to EF, FS and SV, there was no significant difference in cardiac output (CO) between young and old mice (Table 2). There were no significant differences in heart rate between the three groups (Table 2).

The Echocardiographic Diastolic Function in Young, Middle-Aged and Aged Mice

To assess the LV diastolic function in young, middle-aged and old mice, we obtained the mitral valve (MV) inflow velocity and tissue Doppler imaging from the apical fourchamber views. The representative apical four-chamber view images are presented, and MV E from MV inflow imaging and E' from tissue Doppler imaging in young, middle-aged and old mice are shown in Figure 4.

The mean velocity of MV E was 690.0 mm/s for the young, 587.7 mm/s for middle, and 581.2 mm/s for old mice. The mean velocity of E' in young, middle-aged, and old mice was 34.09 mm/s, 19.65 mm/s, and 18.91 mm/s, respectively. The velocity of MV E was significantly reduced in middle-aged mice compared to young mice and thereafter maintained in old mice, but there was no significant difference between young and old mice (Figure 4d). The velocity of E' was significantly reduced in middle-aged mice compared to young mice and subsequently preserved in old mice (Figure 4e). In contrast to MV E, E' was significantly lower in old mice than in young mice. The other parameters of diastolic function, including MV A, E/A, deceleration time (DT), isovolumic relaxation time (IVRT), IVRT/R-R interval (RR) and E/E', are depicted in Table 2. In contrast to E/E', we found no significant differences between young, middle and old groups. Compared to young mice, the E/E' was significantly increased in middle-aged mice and then maintained in old mice. There was no significant difference in E/E' between young and old mice.

Antioxidant Status in Young, Middle-Aged and Aged Mice

To evaluate the antioxidative capacity, plasma levels of SOD and catalase activities were measured in young, middle-aged and old mice. The mean SOD activity in young, middle, and old mice was 83.9%, 86.8%, and 84.6%, respectively. The SOD activity was significantly higher in middle-aged mice compared to young mice, whereas there were no significant differences in the SOD activity of old mice compared to middle-aged and young mice (Figure 5a). A similar reduced activity of catalase was observed in middle-aged mice as compared to the young group (Figure 5b). As shown in Figure 5b, no significant differences were found between middle-aged and old mice. We also examined the myocardial expression profile of 4-HNE, a sensitive marker of oxidative damage and lipid peroxidation, in young, middle-aged and old mice. As shown in Figure 6a,b, old mice displayed an increased level of 4-HNE in the myocardium as compared to young mice. In addition, we examined reduced (GSH) and oxidized (GSSG) glutathione levels, representing a dynamic balance between oxidants and antioxidants in mice. Compared to the young group, the ratio of GSH/GSSG was markedly reduced in middle-aged and old mice (Figure 6c). We also examined the myocardial expression profile of 4-HNE, a sensitive marker of oxidative damage and lipid peroxidation, in young, middle-aged and old mice. As shown in Figure 6a,b, old mice displayed an increased level of 4-HNE in the myocardium as compared to young mice. In addition, we examined reduced (GSH) and oxidized (GSSG) glutathione levels, representing a dynamic balance between oxidants and antioxidants in mice. Compared to the young group, the ratio of GSH/GSSG was markedly reduced in middle-aged and old mice (Figure 6c). We also examined the myocardial expression profile of 4-HNE, a sensitive marker of oxidative damage and lipid peroxidation, in young, middle-aged and old mice. As shown in Figure 6a,b, old mice displayed an increased level of 4-HNE in the myocardium as compared to young mice. In addition, we examined reduced (GSH) and oxidized (GSSG) glutathione levels, representing a dynamic balance between oxidants and antioxidants in mice. Compared to the young group, the ratio of GSH/GSSG was markedly reduced in middle-aged and old mice (Figure 6c). Ratio of reduced glutathione to oxidized glutathione (GSH/GSSG) in plasma from young, middle-aged and old mice (c). * p < 0.05 between the indicated groups, ns, not significant, one-way ANOVA.

Profile of Inflammatory Factors in Young, Middle-Aged and Aged Mice

Chronic inflammation plays a fundamental role in disease by favorizing the initiation and progression of age-specific cardiac abnormalities [START_REF] Dick | Chronic Heart Failure and Inflammation: What Do We Really Know?[END_REF][START_REF] Hanna | Inflammatory Cytokines and Chemokines as Therapeutic Targets in Heart Failure[END_REF]. We next examined the myocardial expression of inflammatory cytokines in young, middle-aged and old mice. As shown in Figure 7, we found no significant differences in the myocardial expression of inflammatory factors including tumor necrosis factor-α (TNFα), interleukin-1β (IL1β), interleukin-6 (IL6) and interleukin-10 (IL10) between young and middle-aged mice. In contrast,

Profile of Inflammatory Factors in Young, Middle-Aged and Aged Mice

Chronic inflammation plays a fundamental role in disease by favorizing the initiation and progression of age-specific cardiac abnormalities [START_REF] Dick | Chronic Heart Failure and Inflammation: What Do We Really Know?[END_REF][START_REF] Hanna | Inflammatory Cytokines and Chemokines as Therapeutic Targets in Heart Failure[END_REF]. We next examined the myocardial expression of inflammatory cytokines in young, middle-aged and old mice. As shown in Figure 7, we found no significant differences in the myocardial expression of inflammatory factors including tumor necrosis factor-α (TNFα), interleukin-1β (IL1β), interleukin-6 (IL6) and interleukin-10 (IL10) between young and middle-aged mice. In contrast, a significantly elevated expression of TNFα, IL1β and IL10 was detected in oldaged mice compared to those in young and middle-aged mice, whereas the IL6 level remained unchanged. The transcription factor NF-κB regulates the expression of multiple proinflammatory cytokines [START_REF] Lawrence | The nuclear factor NF-kappaB pathway in inflammation[END_REF][START_REF] Christian | The Regulation of NF-kappaB Subunits by Phosphorylation[END_REF][START_REF] Liu | NF-kappaB signaling in inflammation[END_REF]. We next analyzed levels of the phosphorylated form of NF-κB by immunoblot using whole tissue extract from young, middle-aged and old mice (Figure 8). Middle-aged mice showed increased NF-kB p65 (Ser 276) phosphorylation as compared to young mice, whereas there were no significant differences in NF-κB p65 subunit phos- The transcription factor NF-κB regulates the expression of multiple proinflammatory cytokines [START_REF] Lawrence | The nuclear factor NF-kappaB pathway in inflammation[END_REF][START_REF] Christian | The Regulation of NF-kappaB Subunits by Phosphorylation[END_REF][START_REF] Liu | NF-kappaB signaling in inflammation[END_REF]. We next analyzed levels of the phosphorylated form of NF-κB by immunoblot using whole tissue extract from young, middle-aged and old mice (Figure 8). Middle-aged mice showed increased NF-kB p65 (Ser 276) phosphorylation as compared to young mice, whereas there were no significant differences in NF-κB p65 subunit phosphorylation between young and old mice (Figure 8). The transcription factor NF-κB regulates the expression of multiple proinflammatory cytokines [START_REF] Lawrence | The nuclear factor NF-kappaB pathway in inflammation[END_REF][START_REF] Christian | The Regulation of NF-kappaB Subunits by Phosphorylation[END_REF][START_REF] Liu | NF-kappaB signaling in inflammation[END_REF]. We next analyzed levels of the phosphorylated form of NF-κB by immunoblot using whole tissue extract from young, middle-aged and old mice (Figure 8). Middle-aged mice showed increased NF-kB p65 (Ser 276) phosphorylation as compared to young mice, whereas there were no significant differences in NF-κB p65 subunit phosphorylation between young and old mice (Figure 8).

Discussion

Age-associated cardiac phenotype in the murine model closely reflects age-related functional and metabolic alterations in humans. Data from the current study demonstrate that aging progressively alters cardiac phenotype, metabolic, inflammatory and antioxidant status in mice. Elevated expression of several cytokines including IL1, IL10 and

Discussion

Age-associated cardiac phenotype in the murine model closely reflects age-related functional and metabolic alterations in humans. Data from the current study demonstrate that aging progressively alters cardiac phenotype, metabolic, inflammatory and antioxidant status in mice. Elevated expression of several cytokines including IL1, IL10 and TNFα was associated with age-related progressive decline in cardiac function. While systolic activity was preserved in the middle-aged group, both EF and FS were reduced in aged mice, suggesting age-dependent progression to systolic dysfunction. Diastolic function evaluated by echocardiography also declined in aged mice as compared with young or middle-aged groups. In mice without apparent cardiovascular complications, both systolic and diastolic functions are progressively compromised with aging, suggesting that a decline in cardiac performance can predispose to the development of heart failure in aged mice. Interestingly, age-dominated impairments in cardiac function and metabolic parameters were accompanied by increased oxidative stress. These findings highlight the role of several cytokines and antioxidants in age-related events of cardiac dysfunction and may have important implications for tailored treatment of heart disease in aging.

Alterations in the structure and function of the heart accompany aging and predispose to the development and progression of heart failure [START_REF] Li | Targeting Age-Related Pathways in Heart Failure[END_REF]. Using echocardiography, the present study provides an integrated, non-invasive approach to the evaluation of systolic and diastolic cardiac function in mice, contributing to the pathophysiology of aging in the heart. Importantly, both LVID and LV mass had a progressive increase in aged mice. Previous studies have suggested that the aged C57BL/6 male mouse captures major aspects of cardiac remodeling processes that have been implicated as core pathophysiologic mediators of heart failure [START_REF] De Lucia | Echocardiographic Strain Analysis for the Early Detection of Left Ventricular Systolic/Diastolic Dysfunction and Dyssynchrony in a Mouse Model of Physiological Aging[END_REF][START_REF] Schnelle | Echocardiographic evaluation of diastolic function in mouse models of heart disease[END_REF][START_REF] Kiper | Mouse strain determines cardiac growth potential[END_REF][START_REF] Zi | Cardiac hypertrophy or failure?-A systematic evaluation of the transverse aortic constriction model in C57BL/6NTac and C57BL/6J substrains[END_REF]. The mouse models phenocopy the human cardiovascular system. However, several distinctions between humans and mice include their different lifespans (70+ years versus 2+ years) [START_REF] Dutta | Men and mice: Relating their ages[END_REF]30] and body size [START_REF] Dutta | Men and mice: Relating their ages[END_REF]. Ventricular contraction includes cardiac output (5 L/min for humans [START_REF] Brandfonbrener | Changes in cardiac output with age[END_REF], 15 mL/min for mice [START_REF] Doevendans | Cardiovascular phenotyping in mice[END_REF]), and heart rate (60 bpm for humans, 600 bpm for mice [START_REF] Papadimitriou | The use of mice and rats as animal models for cardiopulmonary resuscitation research[END_REF]). Nevertheless, despite the higher heart rate in mice, the number of cardiac cycles over a lifetime is still greater in humans (2.2 × 10 9 ) than in mice (6.3 × 10 8 ), which can impact age-associated cardiac remodeling processes [START_REF] Hopper | Comparative Study of Human and Murine Aortic Biomechanics and Hemodynamics in Vascular Aging[END_REF]. Indeed, cardiac tissue aging in mice can be attributed to the accumulation of remodeled collagen and a progressive fall in structural integrity [START_REF] Horn | Aging and the cardiac collagen matrix: Novel mediators of fibrotic remodelling[END_REF][START_REF] Tao | Collagen XIV is important for growth and structural integrity of the myocardium[END_REF]. This reason, which distinguishes mouse models of compromised collagen fiber integrity, can be useful in cardiac aging research [START_REF] Horn | Aging and the cardiac collagen matrix: Novel mediators of fibrotic remodelling[END_REF]. Further diversity between human and murine cardiac aging remains to be established. In relation to both the acquisition of aortic stiffness and ventricular remodeling, heart failure with preserved ejection fraction (HFpEF) plagues elderly patients [START_REF] Redfield | Heart Failure with Preserved Ejection Fraction[END_REF]. In clinical practice, the accentuation of HFpEF in elderly patients is linked to adverse outcomes in the aging patient population, with regard to morbidity and reduced life quality [START_REF] Redfield | Heart Failure with Preserved Ejection Fraction[END_REF]. To date, we have limited evidence-based interventions to stem the development of HFpEF in patients. Myocardial ischemia, due to either ischemic cardiomyopathy or subendocardial ischemia in the hypertrophied LV, can predispose to heart failure with reduced EF [START_REF] Redfield | Heart Failure with Preserved Ejection Fraction[END_REF]. Thus, aging can promote the development of both major forms of heart failure, which represent a crucial challenge to the quality of life in an aging population.

Oxidative damage in the cardiovascular system may occur when antioxidant defense is perturbed to limit ROS production. Oxidant stress has been linked to the pathogenesis of age-associated cardiovascular complications and heart disease [START_REF] Pagan | The Role of Oxidative Stress in the Aging Heart[END_REF][START_REF] Voigt | Cytokine-induced oxidative stress in cardiac inflammation and heart failurehow the ubiquitin proteasome system targets this vicious cycle[END_REF][START_REF] Saad | The Pathogenic Role of Oxidative Stress, Cytokine Expression, and Impaired Hematological Indices in Diabetic Cardiovascular Diseases[END_REF][START_REF] Li | Reduction of inflammatory cytokine expression and oxidative damage by erythropoietin in chronic heart failure[END_REF]. In this context, early observational studies centered on dietary antioxidants (β-carotene, ascorbic acid, α-tocopherol) and showed an inverse relationship between intake of these antioxidants and adverse cardiovascular events [START_REF] Van Der Pol | Treating oxidative stress in heart failure: Past, present and future[END_REF][START_REF] Stephens | Randomised controlled trial of vitamin E in patients with coronary disease: Cambridge Heart Antioxidant Study (CHAOS)[END_REF]. Importantly, these data supported a number of randomized trials of selected antioxidants as primary and secondary prevention strategies to limit cardiovascular risk; however, the majority of these studies reported disappointing results with little or no observed risk reduction in antioxidant-treated subjects [START_REF] Myung | Korean Meta-Analysis Study, G. Efficacy of vitamin and antioxidant supplements in prevention of cardiovascular disease: Systematic review and meta-analysis of randomised controlled trials[END_REF]. Multiple explanations for these findings have been explored, including maladaptive antioxidant dose or choice, dietary versus synthetic antioxidant as the intervention, and patient selection, all of which will be important to consider for designing future clinical trials.

A shift of the prooxidant/antioxidant balance towards excessive formation of ROS leads to a wide spectrum of cardiovascular manifestations [START_REF] Pagan | The Role of Oxidative Stress in the Aging Heart[END_REF][START_REF] Moris | The role of reactive oxygen species in the pathophysiology of cardiovascular diseases and the clinical significance of myocardial redox[END_REF][START_REF] Van Der Pol | Treating oxidative stress in heart failure: Past, present and future[END_REF]. To investigate the significance of antioxidant systems in cardiac aging we measured the activities of critical antioxidant enzymes, catalase and SOD in young, middle-aged and old mice. A rise of total SOD activity which plays a role to dismutate superoxide into H 2 O 2 , was detected in middle-aged mice, while old group exhibited SOD values similar to levels found in young controls. At the same time, no differences in the activity of another antioxidant enzyme, catalase, were observed between the young, middle-aged and old groups of mice. These results attest to the predominant role of SOD in active periods of life, which can serve as the source for the maintenance of dynamic metabolism at a high level in the course of natural aging. Among the three distinct isoforms of SODs, SOD2 specifically localizes in the mitochondrial matrix [START_REF] Palma | Mitochondrial Superoxide Dismutase: What the Established, the Intriguing, and the Novel Reveal About a Key Cellular Redox Switch[END_REF]. Several studies have demonstrated the role of SOD2 in cardiac aging. Deletion of the SOD2 gene results in early postnatal lethality in mice [START_REF] Li | Dilated cardiomyopathy and neonatal lethality in mutant mice lacking manganese superoxide dismutase[END_REF]. Deficiency in SOD2 in mice is viable but demonstrates increased susceptibility to oxidative stress, defective mitochondrial function and elevated sensitivity to cell death [START_REF] Kokoszka | Increased mitochondrial oxidative stress in the Sod2 (+/-) mouse results in the age-related decline of mitochondrial function culminating in increased apoptosis[END_REF][START_REF] Van Remmen | Knockout mice heterozygous for Sod2 show alterations in cardiac mitochondrial function and apoptosis[END_REF]. A recent study has reported that SOD2 deficiency over a lifetime is enough to induce cardiac dysfunction [START_REF] Sharma | SOD2 deficiency in cardiomyocytes defines defective mitochondrial bioenergetics as a cause of lethal dilated cardiomyopathy[END_REF]. In addition, SOD2 deficiency also increases collagen I in aged smooth muscle cells [START_REF] Zhou | Mitochondrial oxidative stress in aortic stiffening with age: The role of smooth muscle cell function[END_REF]. Further studies are required to explore the precise roles of three distinct isoforms of SOD in the course of human-animal aging.

Numerous experimental and clinical studies report that inflammatory cytokines strongly predict heart failure and multiple adverse myocardial events [START_REF] Dick | Chronic Heart Failure and Inflammation: What Do We Really Know?[END_REF][START_REF] Hanna | Inflammatory Cytokines and Chemokines as Therapeutic Targets in Heart Failure[END_REF]. Cytokine dysregulation is believed to play a dominant role in the remodeling of the cardiac function at older age, with evidence pointing to an inadequancy to fine-control systemic inflammation reflecting unhealthy aging [START_REF] Barcena | Cardiovascular Inflammaging: Mechanisms and Translational Aspects[END_REF][START_REF] Smykiewicz | Proinflammatory cytokines and ageing of the cardiovascular-renal system[END_REF]. This reshaping of cytokine expression pattern, with a progressive decline in antioxidant capacity can calibrate the functional performance of the heart [START_REF] Voigt | Cytokine-induced oxidative stress in cardiac inflammation and heart failurehow the ubiquitin proteasome system targets this vicious cycle[END_REF][START_REF] Saad | The Pathogenic Role of Oxidative Stress, Cytokine Expression, and Impaired Hematological Indices in Diabetic Cardiovascular Diseases[END_REF][START_REF] Li | Reduction of inflammatory cytokine expression and oxidative damage by erythropoietin in chronic heart failure[END_REF]. Chronic low-grade systemic inflammation is a well-established hallmark of aging [START_REF] Barcena | Cardiovascular Inflammaging: Mechanisms and Translational Aspects[END_REF][START_REF] Lazzeroni | The Aging Heart: A Molecular and Clinical Challenge[END_REF] and is characterized by high levels of circulating cytokines in the aged population in the absence of general pathophysiological stress [START_REF] Barcena | Cardiovascular Inflammaging: Mechanisms and Translational Aspects[END_REF][START_REF] Smykiewicz | Proinflammatory cytokines and ageing of the cardiovascular-renal system[END_REF]. Chronic inflammation is involved in accelerated biological aging and age-related heart disease [START_REF] Barcena | Cardiovascular Inflammaging: Mechanisms and Translational Aspects[END_REF][START_REF] Howcroft | The role of inflammation in age-related disease[END_REF][START_REF] Ferrucci | Inflammageing: Chronic inflammation in ageing, cardiovascular disease, and frailty[END_REF]. Accumulating evidence suggest an age-related increase in blood inflammatory markers including TNFα, IL1β, and IL10 [START_REF] Hanna | Inflammatory Cytokines and Chemokines as Therapeutic Targets in Heart Failure[END_REF][START_REF] Barcena | Cardiovascular Inflammaging: Mechanisms and Translational Aspects[END_REF][START_REF] Smykiewicz | Proinflammatory cytokines and ageing of the cardiovascular-renal system[END_REF]. This condition is known as inflammaging and a cause of accelerated aging and a comprehensive marker of multimorbidity, disability, frailty, and premature death in old adults [START_REF] Ferrucci | Inflammageing: Chronic inflammation in ageing, cardiovascular disease, and frailty[END_REF][START_REF] Franceschi | Inflammaging and 'Garb-aging[END_REF]. In our study, we have demonstrated an age-associated increase in plasma levels of cytokines, including IL1, IL10 and TNFα, suggesting inflammaging-related cardiac remodeling. Further studies are needed to clarify the molecular age-related mechanisms leading to cellular senescence, including impaired mitochondrial function. In cardiac tissue, inflammatory cytokines are not stored intracellularly, and their secretion depends on new protein synthesis. As a consequence, the elaboration of cytokines in response to an inflammatory stimulus is significantly or predominantly regulated by the transcription rates of cytokine genes. Since transcriptional regulation is critical for the production of many cytokines, transcription factors, including NF-kB, may play key roles in regulating cytokine-mediated inflammation. In an aged heart, inflammatory and oxidative stresses stimulate the NF-κB family of transcription factors [START_REF] Moris | The role of reactive oxygen species in the pathophysiology of cardiovascular diseases and the clinical significance of myocardial redox[END_REF][START_REF] Howcroft | The role of inflammation in age-related disease[END_REF]. Therefore, as a common responder to varied stress stimuli, NF-κB is well positioned to play a capital role in driving aging. Indeed, NF-κB has been directly implicated in the aging process. For example, using motif mapping, NF-κB was determined to be the transcription factor most associated with aging [START_REF] Adler | Motif module map reveals enforcement of aging by continual NF-kappaB activity[END_REF]. Furthermore, biologic pathways implicated in aging, including immune responses, cell senescence, apoptosis and metabolism, are regulated at least in part by NF-κB. Additionally, other cellular processes implicated in regulating lifespan, including insulin/IGF-1 and growth hormone pathways, SIRT, FoxO and mTOR, are all interconnected with NF-κB signaling. Finally, the role of aberrant NF-κB signaling is well documented in numerous age-associated diseases, including neurodegeneration, osteoporosis, diabetes, sarcopenia and atherosclerosis [START_REF] Adler | Motif module map reveals enforcement of aging by continual NF-kappaB activity[END_REF].

Furthermore, the results of our study highlight the possible role of adipokine resistin, in age-related cardiac dysfunction. Interestingly, we found that age-mediated changes in metabolic parameters associated with cardiovascular risk in mice were accompanied by elevated levels of resistin. Alterations in resistin levels in aged mice depend on the metabolic parameters of the animals [START_REF] Jamaluddin | Resistin: Functional roles and therapeutic considerations for cardiovascular disease[END_REF][START_REF] Acquarone | Resistin: A reappraisal[END_REF]. In humans, resistin levels are linked to the metabolic and cardiovascular parameters in cases of metabolic disorders [START_REF] Franceschi | Inflammaging and 'Garb-aging[END_REF][START_REF] Jamaluddin | Resistin: Functional roles and therapeutic considerations for cardiovascular disease[END_REF]. Resistin demonstrates its inflammatory role by intracellular activation of NF-kB, which calibrates the trajectories of inflammatory cytokine secretion [START_REF] Acquarone | Resistin: A reappraisal[END_REF][START_REF] Filkova | The role of resistin as a regulator of inflammation: Implications for various human pathologies[END_REF]. An extensive study is needed to establish correlations between resistin and antioxidant capacity in aged hearts.

Limitations of the Study

Our study has several limitations. We did not execute a longitudinal study design to explore trajectories of cardiac function decline with age in relation to the inflammatory and oxidant/antioxidant status. Rather, we performed a cross-sectional study design that compared different ages of mice to examine age-specific cardiac and metabolic phenotypes. While longitudinal analyses have their strengths (i.e., repeated measurements on the same animal), they are certainly not without obstacles and have their own sets of limitations including in vivo complications of performing assays and potential effects on the outcome of subsequent repeated tests. Another critical issue linked to longitudinal designs is that measurements cannot be taken at the same time but instead require a comparison of data collected at different time points, corresponding to 2 years apart in our study. Therefore, longitudinal studies may not naturally generate more robust values of agerelated trajectories in mouse populations than population estimates derived from crosssectional data collected at the same time and under the same well-controlled conditions. According to these considerations, it is common practice to explore aging-associated cardiac phenotypes in studies using cross-sectional designs. An additional critical point is that we did not standardize LV mass to BW in mice since there is controversy regarding the optimal method for indexing LV mass to body BW in the clinical setting. Indeed, the prediction of values of LV by BW is more accurate at birth and progressively less precise with increasing age. Moreover, in this work, we did not take into consideration the sexdependent differences in cardiac phenotype in mice; in fact, there is growing experimental and clinical evidence supporting the existence of sex-specific patterns of cardiac aging [START_REF] Ji | Sex Differences in Myocardial and Vascular Aging[END_REF]. Thus, these aspects need to be further investigated.

Materials and Methods

Animals

All animal work described in this investigation was performed in accordance with institutional guidelines for animal research and were approved by the French Ministry of Research in agreement with European Union guidelines according to the Directive 2010/063 EU [60] and French Decree 2013-118 [61]. C57BL/6 male mice aged 3 months (young group, n = 11), 12 months (middle-aged group, n = 15) and 24 months (old group, n = 4) (Envigo RMS, Gannat, France) were used for the echocardiographic measurements of cardiac structure and function. Mice were housed in groups of five in polycarbonate cages, enriched with paper and were allowed free access to standard food and water.

Echocardiography

Mice were anesthetized with isoflurane and underwent echocardiography in the supine position using a Vevo 2100 high-frequency high-resolution ultrasound System (VisualSonics, Toronto, ON, Canada). Electrocardiogram (ECG) and heart rate (HR) were monitored using limb electrodes. Body temperature was measured using a rectal probe and maintained at 37 • C. Chest hair was removed with a depilatory cream or a shaver. The parasternal long-axis imaging, the parasternal short-axis B-mode and M-mode imaging, and the apical four-chamber imaging were recorded. Left ventricular end-diastolic (LVEDV) and end-systolic volume (LVESV), SV, CO, and EF were obtained from the parasternal longaxis image by tracing the left ventricle in diastole and systole. FS, LVID in diastole (LVIDd) and in systole (LVIDs), LVAW in diastole (LVAWd) and in systole (LVAWs), LVPW in diastole (LVPWd) and in systole (LVPWs), and LV mass were obtained from the parasternal short-axis M-mode image. The MV inflow velocity was acquired using the pulsed-wave Doppler and placing the sample volume at the tip of the mitral valve leaflet in the apical four-chamber view. Mitral inflow E wave velocity (E), mitral inflow A wave velocity (A), E velocity DT, IVRT, and isovolumic contraction time (IVCT) were measured from the mitral valve inflow imaging. The tissue Doppler imaging was acquired with the sample volume placed on the septal mitral valve annulus in the apical four-chamber view. The tissue Doppler-derived septal mitral annular E' (E') and A' wave velocity (A') were obtained from the tissue Doppler imaging.

Metabolic Parameters

To determine the amount of body fat mass, lean mass, and free body fluid weight we used a nuclear magnetic resonance (NMR) based instrument (Minispec, Bruker, MA, USA). This approach provides the measurements non-invasively and without the need for anesthesia.

Tail-Tip Whole Blood Sampling

Whole blood was collected from the tail vein of conscious mice using EDTA-coated microvettes up to a maximum volume of 100-150 µL. The samples were then centrifuged at 3800 rpm for phase separation, and the plasma was frozen and stored at -80 • C for future assays. After collection, the bleeding was stopped by applying slight pressure on the incision using a cotton pad. Catalase (707002, Cayman Chemical Company, Ann Arbor, MI, USA), SOD (19160, SIGMA Life Science, Buchs, Switzerland) and resistin (A05178, SPIbio, Montigny le Bretonneux, France) levels were measured using a commercially available ELISA kits according to the manufacturer's protocol.

Immunoblotting Analysis

For immunoblot analyses, heart tissues were lysed in RIPA buffer (89900, Pierce, Appleton, WI, USA) supplemented with protease inhibitor cocktail (P8340, Sigma-Aldrich, St. Louis, MO, USA) and Halt Phosphatase inhibitor cocktail (78426, Thermo Scientific, Waltham, MA, USA). Protein concentrations were determined by BCA protein assay reagent (23225, Thermo Scientific). Protein extracts were separated on 12% SDS-PAGE gels and transferred to PVDF membranes (IPVH00010, Millipore, Burlington, MA, USA). Membranes were incubated with the primary antibodies anti-4-Hydroxynonenal, (MA5-27570, Invitrogen, Carlsbad, CA, USA), anti-phospho-NF-kB, (ab106129, Abcam, Cambridge, UK) or α-Tubulin (T6074, Sigma-Aldrich) overnight at 4 • C, then for 2 h with HRP-conjugated secondary antibodies (Cell Signaling Technologies, Danvers, MA, USA). Membranes were revealed by chemiluminescence using Clarity Western ECL substrate (170-5060, BioRad, Hercules, CA, USA) on a Chemidoc Touch (BioRad).

GSH/GSSG Ratio Assay

The ratio of reduced (GSH) to oxidized glutathione (GSSG) was determined in young, middle-aged and old mice plasma using the GSH/GSSG Ratio Detection Assay Kit (ab138881, Abcam) according to manufacturer.

Quantitative RT-PCR Analysis

The expression of genes was assessed using quantitative polymerase chain reaction (qRT-PCR). Total RNAs were isolated from cardiac tissue using the RNeasy mini kit (Qiagen, Hilden, Germany). The sequences of the primers used are as follow and were given in the 5 -3 orientation: TNFa ATCTCATACCAGGAGAAAGTCAACCT (forward) and ATGGGCT-CATACCAGGGTTTG (reverse); IL 1β CTGCACTACAGGCTCCGAGAT (forward) and TGTTGGTTGATATTCTGTCCATTGA (reverse); IL 6 AAGTGCATCGTTGTTCATACA (forward) and GAGGATACCACTCCCAACAGACC (reverse); IL-10 TGAGGCGCTGTCATC-GATTT (forward) and TGGCCTTGTAGACACCTTGG (reverse); resistin TCATTTCCC-CTCCTTTTCCTTT(forward) and TGGGACACAGTGGCATGCT (reverse); 36b4 GCT-TCA-TTG-TGG-GAG-CAG-AC (forward) and ATG-GTG-TTC-TTG-CCC-ATC-AG (reverse). The expression of target gene was normalized to 36b4 housekeeping gene expression.

Statistical Analysis

Continuous variables were expressed as the mean ± SEM. A one-way analysis of variance (ANOVA) followed by Tukey's multiple comparisons test for multiple pairwise comparisons was performed using GraphPad Prism software (version 9.5.0, GraphPad Software, Inc., La Jolla, CA, USA). A p value of less than 0.05 was considered to be statistically significant.

Conclusions

In conclusion, this study demonstrated that age-dependent trajectories of cardiac function decline are associated with dynamic changes in oxidative stress and antioxidant status markers, as well as metabolic and cytokine dysregulation. We show that in mice without apparent cardiovascular manifestations, both systolic and diastolic functions are progressively compromised with age, suggesting that a decline in cardiac activity can predispose to the development of heart failure in aged mice. Age-specific differences in cardiac phenotype linked to the inflammatory and antioxidant status in middle-and aged animals may have important implications for tailored treatment of cardiovascular defects in aging.

Figure 1 .

 1 Figure 1. Age-dependent profile of resistin expression in mice. Plasma level (a) and quantitative RT-PCR analysis of mRNA expression (b) of resistin in young, middle-aged and old mice. * p < 0.05 and *** p < 0.001 between the indicated groups, ns, not significant, one-way ANOVA.

Figure 1 .

 1 Figure 1. Age-dependent profile of resistin expression in mice. Plasma level (a) and quantitative RT-PCR analysis of mRNA expression (b) of resistin in young, middle-aged and old mice. * p < 0.05 and *** p < 0.001 between the indicated groups, ns, not significant, one-way ANOVA.

Figure 2 .

 2 Figure 2. Parasternal short-axis M-mode images and age-associated comparisons of the left ventricular dimensions. Representative M-mode images of the parasternal short-axis view with measurements of the left ventricular dimensions in young (a), middle-aged (b) and old mice (c). Measurements of the left ventricular internal diameter in diastole (LVIDd) (d), left ventricular internal diameter in systole (LVIDs) (e), and left ventricular mass (f) in young, middle-aged and old mice. The echocardiographic examinations were performed and analyzed using the Vevo 2100 Imaging System (VisualSonics, Toronto, ON, Canada). * p < 0.05, ** p < 0.01, *** p < 0.001, and **** p < 0.0001 between the indicated groups, ns, not significant, one-way ANOVA.

Figure 3 .

 3 Figure 3. Parasternal long-axis images and LV systolic function in young, middle-aged and old mice. Representative images from parasternal long-axis view with the left ventricular tracing in systole (green) in young (a), middle-aged (b), and old mice (c). Measurements of the left ventricular ejection fraction (EF) (d) and left ventricular fractional shortening (FS) (e) in young, middle-aged and old mice. The echocardiographic examinations were performed and analyzed using the Vevo 2100 Imaging System (VisualSonics, Toronto, ON, Canada). **** p < 0.0001 between the indicated groups, ns, not significant, one-way ANOVA.

Figure 4 .

 4 Figure 4. Pulse Doppler waveform of mitral inflow velocity and LV diastolic function in young, middle-aged and old mice. Representative images of pulse Doppler waveforms of mitral inflow velocity (white wave) and tissue Doppler imaging (yellow wave) in young (a), middle-aged (b), and old mice (c). Measurements of mitral valve inflow E wave (MV E) (d) and tissue Doppler-derived septal mitral annular E' wave (E') (e) in young, middle-aged and old mice. The echocardiographic examinations were performed and analyzed using the Vevo 2100 Imaging System (VisualSonics, Toronto, ON, Canada). * p < 0.05, *** p < 0.001, and **** p < 0.0001 between the indicated groups, ns, not significant, one-way ANOVA.

Figure 5 .

 5 Figure 5. Age-dependent antioxidant capacity in mice. Total SOD (a) and catalase (b) activities in young, middle-aged and old mice. Plasma levels of SOD and catalase were measured using ELISA kits. * p < 0.05 and ** p < 0.01 between the indicated groups, ns, not significant, one-way ANOVA.

Figure 5 .

 5 Figure 5. Age-dependent antioxidant capacity in mice. Total SOD (a) and catalase (b) activities in young, middle-aged and old mice. Plasma levels of SOD and catalase were measured using ELISA kits. * p < 0.05 and ** p < 0.01 between the indicated groups, ns, not significant, one-way ANOVA.

Figure 5 .

 5 Figure 5. Age-dependent antioxidant capacity in mice. Total SOD (a) and catalase (b) activities in young, middle-aged and old mice. Plasma levels of SOD and catalase were measured using ELISA kits. * p < 0.05 and ** p < 0.01 between the indicated groups, ns, not significant, one-way ANOVA.

Figure 6 .

 6 Figure 6. Profile of oxidative stress markers in young, middle-aged and old mice. Quantification analysis (a) and Western blot image (b) of 4-HNE in cardiac tissue from young, middle-aged and old mice. Ratio of reduced glutathione to oxidized glutathione (GSH/GSSG) in plasma from young, middle-aged and old mice (c). * p < 0.05 between the indicated groups, ns, not significant, one-way ANOVA.

Figure 6 .

 6 Figure 6. Profile of oxidative stress markers in young, middle-aged and old mice. Quantification analysis (a) and Western blot image (b) of 4-HNE in cardiac tissue from young, middle-aged and old mice. Ratio of reduced glutathione to oxidized glutathione (GSH/GSSG) in plasma from young, middle-aged and old mice (c). * p < 0.05 between the indicated groups, ns, not significant, one-way ANOVA.

Figure 7 .

 7 Figure 7. Age-dependent mRNA expression of inflammatory factors in mice. Quantitative RT-PCR analysis of mRNA expression levels of TNFα (a), IL1β (b), IL6 (c), and IL10 (d). * p < 0.05, ** p < 0.01, and *** p < 0.001 between the indicated groups, ns, not significant, one-way ANOVA.

Figure 7 .

 7 Figure 7. Age-dependent mRNA expression of inflammatory factors in mice. Quantitative RT-PCR analysis of mRNA expression levels of TNFα (a), IL1β (b), IL6 (c), and IL10 (d). * p < 0.05, ** p < 0.01, and *** p < 0.001 between the indicated groups, ns, not significant, one-way ANOVA.

Figure 7 .

 7 Figure 7. Age-dependent mRNA expression of inflammatory factors in mice. Quantitative RT-PCR analysis of mRNA expression levels of TNFα (a), IL1β (b), IL6 (c), and IL10 (d). * p < 0.05, ** p < 0.01, and *** p < 0.001 between the indicated groups, ns, not significant, one-way ANOVA.

Figure 8 .

 8 Figure 8. NF-κB-p65 Ser276 phosphorylation in young, middle-aged and old mice. Heart total protein extracts were resolved by SDS-PAGE and probed with antibody against p-NF-κB-p65-Ser276. Western blot image (a) and quantification analysis from a (b). * p < 0.05 between the indicated groups, ns, not significant, one-way ANOVA.

Figure 8 .

 8 Figure 8. NF-κB-p65 Ser276 phosphorylation in young, middle-aged and old mice. Heart total protein extracts were resolved by SDS-PAGE and probed with antibody against p-NF-κB-p65-Ser276. Western blot image (a) and quantification analysis from a (b). * p < 0.05 between the indicated groups, ns, not significant, one-way ANOVA.

Table 1 .

 1 The metabolic characteristics and their comparisons by age in C57BL/6 mice.All values are represented as mean ± SEM. BW, body weight; HR. * p < 0.05, ** p < 0.01, and **** p < 0.0001 vs. young mice. § p < 0.05, § § p < 0.01, and § § § p < 0.001 vs. middle-aged mice.

	Parameters	Young	Middle	Old
	BW (g)	33.18 ± 1.00	46.20 ± 1.51 ****	34.33 ± 3.84 § §
	Fat (g)	4.18 ± 0.80	12.86 ± 0.82 ****	4.00 ± 1.98 § § §
	Lean tissue (g)	23.65 ± 0.45	25.73 ± 0.56 *	23.91 ± 1.94
	Free body fluid (g) 0.92 ± 0.05	1.36 ± 0.06	2.91 ± 1.56 ** §
	Parameters	Young	Middle	Old
	BW (g)	33.18 ± 1.00	46.20 ± 1.51 ****	34.33 ± 3.84 § §
	Fat (g)	4.18 ± 0.80	12.86 ± 0.82 ****	4.00 ± 1.98 § § §
	Lean tissue (g)	23.65 ± 0.45	25.73 ± 0.56 *	23.91 ± 1.94
	Free body fluid (g)	0.92 ± 0.05	1.36 ± 0.06	2.91 ± 1.56 ** §

All values are represented as mean ± SEM. BW, body weight; HR. * p < 0.05, ** p < 0.01, and **** p < 0.0001 vs. young mice. § p < 0.05, § § p < 0.01, and § § § p < 0.001 vs. middle-aged mice.

Table 2 . Echocardiographic parameters in young, middle-aged and aged mice.

 2

	Parameters	Young	Middle	Old
	HR (bpm)	472.7 ± 20.56	456.0 ± 11.52	455.5 ± 30.73
	SV (μL)	36.63 ± 0.91	38.49 ± 1.14	30.45 ± 3.21 * § §
	CO (mL/min)	17.17 ± 0.50	17.45 ± 0.65	13.93 ± 1.31 §
	LVEDV (μL)	64.41 ± 0.95	66.63 ± 2.00	73.78 ± 3.57 *
	LVESV (μL)	27.78 ± 0.79	28.14 ± 1.27	43.32 ± 0.96 **** § § § §
	LVAWd (mm)	0.67 ± 0.01	0.75 ± 0.02 **	0.65 ± 0.04 § §
	LVAWs (mm)	1.03 ± 0.03	1.09 ± 0.02	0.88 ± 0.03 * § § §
	LVPWd (mm)	0.71 ± 0.02	0.80 ± 0.02 ***	0.72 ± 0.01 §
	LVPWs (mm)	1.00 ± 0.02	1.15 ± 0.02 ***	1.01 ± 0.02 §

Table 2 . Echocardiographic parameters in young, middle-aged and aged mice. Parameters Young Middle Old

 2

	HR (bpm)	472.7 ± 20.56	456.0 ± 11.52	455.5 ± 30.73
	SV (µL)	36.63 ± 0.91	38.49 ± 1.14	30.45 ± 3.21 * § §
	CO (mL/min)	17.17 ± 0.50	17.45 ± 0.65	13.93 ± 1.31 §
	LVEDV (µL)	64.41 ± 0.95	66.63 ± 2.00	73.78 ± 3.57 *
	LVESV (µL)	27.78 ± 0.79	28.14 ± 1.27	43.32 ± 0.96 **** § § § §
	LVAWd (mm)	0.67 ± 0.01	0.75 ± 0.02 **	0.65 ± 0.04 § §
	LVAWs (mm)	1.03 ± 0.03	1.09 ± 0.02	0.88 ± 0.03 * § § §
	LVPWd (mm)	0.71 ± 0.02	0.80 ± 0.02 ***	0.72 ± 0.01 §
	LVPWs (mm)	1.00 ± 0.02	1.15 ± 0.02 ***	1.01 ± 0.02 §
	A (mm/s)	469.8 ± 17.7	430.4 ± 31.3	426.4 ± 60.4
	E/A	1.49 ± 0.06	1.42 ± 0.07	1.39 ± 0.11
	DT (ms)	19.27 ± 1.31	19.87 ± 1.67	20.74 ± 2.09
	IVRT (ms)	17.35 ± 0.62	15.88 ± 0.96	20.00 ± 1.53

Table 2 .

 2 Cont.

	Parameters	Young	Middle	Old
	IVRT/RR	0.13 ± 0.01	0.12 ± 0.01	0.14 ± 0.004
	E/E'	21.71 ± 1.49	31.71 ± 2.56 *	31.79 ± 4.65
	All values are represented as mean ± SEM. A, mitral inflow A wave velocity; CO, cardiac output; DT, E velocity
	deceleration time; E, mitral inflow E wave velocity; E', tissue Doppler-derived septal mitral annular E' wave
	velocity; HR, heart rate; IVRT, isovolumic relaxation time; LVAWd, left ventricular anterior wall thickness in
	diastole; LVAWs, left ventricular anterior wall thickness in systole; LVEDV, left ventricular end-diastolic volume;
	LVESV, left ventricular end-systolic volume; LVPWd, left ventricular posterior wall thickness in diastole; LVPWs,
	left ventricular posterior wall thickness in systole; RR, R-R interval; SV, stroke volume. * p < 0.05, ** p < 0.01,
	*** p < 0.001, and **** p < 0.0001 vs. young mice. § p < 0.05, § § p < 0.01, § § § p < 0.001, and § § § § p < 0.0001 vs.
	middle-aged mice.

Funding: This research was funded by INSERM 1297.

Acknowledgments:

We are grateful to US006 Anexplo team for the technical support.

Conflicts of Interest:

The authors declare no conflict of interest.


Karim Adiprasito

AND Joel Hakavuori

Minas Margaritis

SUBADDITIVITY OF SHIFTS, EILENBERG-ZILBER SHUFFLE PRODUCTS AND COHOMOLOGY OF LATTICES

We show that the maximal shifts in the minimal resolution of a monomial ideal form a subadditive function, answering a conjecture of Avramov, Conca and Iyengar. To do so, we study dierent models for the cohomology rings of posets and lattices, introduce the Eilenberg-Zilber shue product for lattices and use it to establish vanishing theorems and models for homology for lattices.

Preliminaries

For a homogeneous ideal I in the polynomial ring S := K[x 1 , . . . , x n ] over some eld K we study β ij (S/I) := dim K Tor S i (S/I, K) j , the degree j part of the ith Betti number of the minimal free resolution of S/I over S. The ith maximal shift t i (S/I) is the maximal j such that β ij (S/I) ̸ = 0. If β ij (S/I) = 0 for all j we set t i (S/I) = 0. The following question has been asked (see [START_REF] Avramov | Subadditivity of Syzygies of Koszul algebras[END_REF][START_REF] Eisenbud | The regularity of Tor and graded Betti numbers[END_REF]) for general graded ideals I and their graded Betti numbers.

Question 1.1. In which generality is it true that

t i 1 +i 2 ≤ t i 1 + t i 2 for all 0 ≤ i 1 , i 2 .
Avramov, Conca and Iyengar in particular conjectured this for monomial ideals [START_REF] Avramov | Subadditivity of Syzygies of Koszul algebras[END_REF], and the case has been investigated extensively [START_REF] Abdelfatah | Some results on the subadditivity condition of syzygies[END_REF][START_REF] Herzog | A note on the subadditivity problem for maximal shifts in free resolutions[END_REF][START_REF] Eisenbud | The regularity of Tor and graded Betti numbers[END_REF][START_REF] Dao | Bounds on the regularity and projective dimension of ideals associated to graphs[END_REF].

Since t 0 = 0 the only interesting case is i 1 , i 2 ≥ 1. The question has a negative answer for general ideals, indeed it fails already for binomial ideals, see for example the results on ideals of Veronese embeddings in [START_REF] Ein | Asymptotic syzygies of algebraic varieties[END_REF]. But for the case of monomial ideals there is evidence that the question has a positive answer; for example the case i 1 = 1 is known by [START_REF] Herzog | A note on the subadditivity problem for maximal shifts in free resolutions[END_REF] and the papers [START_REF] Eisenbud | The regularity of Tor and graded Betti numbers[END_REF][START_REF] Dao | Bounds on the regularity and projective dimension of ideals associated to graphs[END_REF] provides further supportive evidence.

For a monomial ideal I with minimal monomial generating set G the LCMlattice L(I) is the set of all least common multiples lcm m∈B m for B ⊆ G(I) where lcm m∈∅ m = 1. Ordered by divisibility the set L(I) becomes an atomic lattice. We denote the proper part of any lattice, that is, the lattice without its maximum (henceforth denoted by 1) and minimum ( 0), by Ľ.

Date : August 27, 2023.

To answer Question 1.1 for monomial ideals and our combinatorial approach via the LCM-lattice, and their topology. For the latter, when speaking of the topology of a poset, we are meaning the topology of its order complex, the simplicial complex of (order) chains, that is, the simplicial complex of totally ordered subsets 1 .

For p ≤ q in P we denote by [p, q] the interval {x ∈ P | p ≤ x ≤ q} in P . Open and halfopen intervals in P are dened analogously. We shall also be interested in the halfopen intervals P ≤x and analogous subposets. An ideal is a downclosed set in a poset, that is, if y ∈ a and x ≤ y then x ∈ a. An ideal generated by a single element is a principal ideal.

Our approach to the algebraic question is based on the known relation (see [START_REF] Gasharov | The lcm-lattice in monomial resolutions[END_REF]) between Betti numbers β ij (S/I) and homologies of the lattice of least common multiples L = L(I) of order complexes Ľ<m given by

β ij (S/I) = m∈L(I) deg(m)=j dim H i-2 ( Ľ<m ),
where homology is homology with K-coecients. Moreover, every nonempty simplicial complex contains naturally an empty set, and therefore homology is naturally reduced homology. Thus if t i 1 +i 2 > 0 then H i 1 +i 2 -2 ( Ľ<m ) ̸ = 0 for some monomial m of degree ℓ := t i 1 +i 2 . If Question 1.1 has a positive answer then there are monomials n 1 and n 2 in our lattice such that H i 1 -2 ( Ľ<n 1 ) ̸ = 0 and H i 2 -2 ( Ľ<n 1 ) ̸ = 0 and ℓ is bounded from above by the degree of the lcm of n 1 and n 2 . Hence, the positive answer to the question for monomial ideals is implied by the following theorem.

Theorem 1.2. Let L be a lattice 2 with an element m such that

H i 1 +i 2 -2 ( Ľ<m ) ̸ = 0.
Then there are elements n 1 , n 2 ∈ L such that

H i 1 -2 ( Ľ<n 1 ) ̸ = 0 and H i 2 -2 ( Ľ<n 2 ) ̸ = 0 and n 1 ∨ n 2 ≥ m.
Here ∨ denotes the join of two elements. Of course, we may assume that m = 1 for this result. Let us note that this statement is natural, and is immediate for several classes of lattices, such as face lattices of strongly regular CW complexes.

To prove this theorem, we shall introduce a new way to generate the cohomology of a poset, and the cohomology ring of a lattice. We close this paper by concluding some further algebraic consequences of this vanishing theorem.

1

We speak of order chains to distinguish them from chains in homology.

2

All our lattices shall be nite. their interest. This paper was written while the rst author was a member at the Institute of Advanced Study in Princeton. An earlier, far more complicated argument for the main theorem was found jointly by the rst author and Volkmar Welker, and we thank Volkmar for his input and work on the predecessor on this note. Funding for his membership has been provided by The Ambrose Monell Foundation and Horizon Europe ERC Grant number: 101045750 / Project acronym: HodgeGeoComb.

Posets, homology and the shuffle product in lattices

For a poset P we call a order multichain c = (c 0 ≤ c 1 ≤ • • • ≤ c k ) of elements of P a order multichain of length k or a order k-multichain. The empty multichain c = () is the unique multichain of length -1.

By M k (P) we denote kth (reduced) chain group of P with coecients in K; i.e. the free abelian group generated by the order k-multichains for k ∈ Z with the usual boundary operator. Of course, its homology is nothing but the simplicial homology of the underlying poset. Now, the next step is to understand how the join leads to a cup-product in the case of lattices; the Eilenberg-Zilber shue product [START_REF] Eilenberg | On the groups H(Π, n)[END_REF], which for lattices takes the following form: Let L be a partially ordered set which is a lattice. We dene τ : L k+1 → M k (L) as the operator which takes the (k + 1)-tuple (a 0 , . . . , a k ) ∈ L k+1 into the k-multichain (a 0 , a 0 ∨ a 1 , . . . , a 0 ∨ • • • ∨ a k ). For a permutation σ of {0, . . . , k} and a (k + 1)-tuple a = (a 0 , . . . , a k ) ∈ L k+1 we write a σ for (a σ(0) , . . . , a σ(n) ). We will consider an i-multichain c in L as an element of L i+1 . For an i-multichain c and a j-multichain c ′ we write c • c ′ for (i + j + 2) tuple in L i+j+1 that is the concatenation of c and c ′ . We say a permutation σ of {0, 1, . . . , i + j + 1} is an (i, j)-shue if σ(a) < σ(b) for all a and b such that either 0 ≤ a < b ≤ i or i + 1 ≤ a < b ≤ i + j + 1. We write S i,j for the set of all (i, j)-shues inside the set of permutations of {0, 1, . . . , i + j + 1}.

The shue operator ¡ : M i (L) × M j (L) → M i+j+1 (L) is then dened as the linear extension of the map sending an i-multichain c and a j-multichain

c ′ to c ¡ c ′ := σ∈S i,j sgn(σ)τ (c • c ′ ) σ .

Note that if

i = -1 then c ¡ c ′ = c ′ . Consider the second tensor power (D * (L), δ * ) = ( M * , ∂ * ) ⊗ ( M * , ∂ * ) of the chain complex ( M * , ∂ * ), with dierential δ n = i+j+1=n ((∂ i , id) + (-1) i+1 (id, ∂ j )). Since M i (L)⊗ M j (L)
has a basis consisting of the elementary tensors c ⊗ c ′ for poset i-multichains c and poset j-multichains c ′ , it follows that c⊗c ′ → c¡c ′ induces a map of free abelian groups ¡ : D n (L) → M n (L).

Proposition 2.1. For n ≥ 0 and α ∈ D n (L) we have

∂ n (¡(α)) = ¡(δ n (α)).
Proof. It suces to check the identity for α = c ⊗ c ′ where c is a poset i-multichain, c ′ a poset j-multichain and i

+ j = n + 1. Let c = (c 0 ≤ • • • ≤ c i ) and c ′ = (c i+1 ≤ • • • ≤ c i+j+1 ).
For a σ ∈ S i,j ) we write σ 1 (ℓ) for max{{0, . . . , i} ∩ {σ(0), . . . , σ(ℓ)}} and σ 2 (ℓ) for max{{i + 1, . . . , i + j + 1} ∩ {σ(0), . . . , σ(ℓ)}} where we consider the maximum over the empty set as -∞.

With this and setting

c -∞ = c ′ -∞ = 0 we get for σ ∈ S i,j that that τ ((c • c ′ ) σ ) = (c σ 1 (0) ∨ c ′ σ 2 (0) , . . . , c σ 1 (i+j+1) ∨ c ′ σ 2 (i+j+1) ).
Then

c ¡ c ′ = σ∈S i,j sgn(σ)τ ((cc ′ ) σ ) = σ∈S i,j sgn(σ)(c σ 1 (0) ∨ c ′ σ 2 (0) ≤ • • • ≤ c σ 1 (i+j+1) ∨ c ′ σ 2 (i+j+1) ).

Hence

(1)

∂ i+j+1 (c ¡ c ′ ) = i+j+1 ℓ=0 (-1) ℓ σ∈S i,j sgn(σ)g(c, c ′ , σ)
where

g(c, c ′ , σ) = (c σ 1 (0) ∨c ′ σ 2 (0) ≤ • • • ≤ c σ 1 (ℓ) ∨ c ′ σ 2 (ℓ) ≤ • • • ≤ c σ 1 (i+j+1) ∨c ′ σ 2 (i+j+1) ).
Fix some σ ∈ S i,j and 0 ≤ ℓ ≤ i + j.

Case: σ(ℓ) ∈ {0, . . . , i} and σ(ℓ

+ 1) ∈ {i + 1, • • • , i + j + 1}.
In this situation the permutation σ ′ dened by σ ′ (k) = σ(k) for k ̸ = ℓ, ℓ+1 and σ ′ (ℓ) = σ(ℓ+1), σ ′ (ℓ+1) = σ(ℓ) lies again in S i,j and sgn(σ) = -sgn(σ ′ ). Hence on the right hand side of (1) the terms:

(c σ 1 (0) ∨ c ′ σ 2 (0) ≤ • • • ≤ c σ 1 (ℓ) ∨ c ′ σ 2 (ℓ) ≤ • • • ≤ c σ 1 (i+j+1) ∨ c ′ σ 2 (i+j+1) )
and

(c σ ′ 1 (0) ∨ c ′ σ ′ 2 (0) ≤ • • • ≤ c σ ′ 1 (ℓ) ∨ c ′ σ ′ 2 (ℓ) ≤ • • • ≤ c σ ′ 1 (i+j+1) ∨ c ′ σ ′ 2 (i+j+1) )
cancel. Case: σ(ℓ + 1) ∈ {0, . . . , i} and σ(ℓ

) ∈ {i + 1, • • • , i + j + 1}.
With a similar argumentation as in the previous case one can show cancellation. Thus one is left with the terms

(c σ 1 (0) ∨ c ′ σ 2 (0) ≤ • • • ≤ c σ 1 (ℓ) ∨ c ′ σ 2 (ℓ) ≤ • • • ≤ c σ 1 (i+j+1) ∨ c ′ σ 2 (i+j+1) ) (2)
where σ(ℓ), σ(ℓ + 1) lie both in {0, . . . , i} or both in {i + 1, . . . , i + j + 1}.

We will show that these terms are exactly the terms appearing

¡(δ i+j+1 (c ⊗ c ′ )) = ¡(∂ i (c) ⊗ c ′ + (-1) i+1 c ⊗ ∂ j (c ′ )
and they appear with the same sign.

Consider the case where σ(ℓ), σ(ℓ + 1) both lie in {0, . . . , i}. We dene ρ : {0, . . . , i + j} → {0, . . . , i + j} by setting

ρ(k) = σ(k) if σ(k) < σ(ℓ) and ρ(k) = σ(k) -1 if σ(k) > σ(ℓ). It is easily checked that indeed ρ ∈ S i-1,j .
The inversions of ρ are in bijection with the inversions of σ not involving ℓ. Since σ is an (i, j)-shue this is #{σ(0), . . . , σ(ℓ -1)} ∩ {i + 1, . . . , i + j + 1}. Since {0, 1, . . . , σ(ℓ -1)} ⊆ {σ(0), σ(1), . . . , σ(ℓ -1)}, it follows that

#{σ(0), . . . , σ(ℓ -1)} ∩ {i + 1, . . . , i + j + 1} = ℓ -σ(ℓ)
and

sgn(ρ) = sgn(σ) • (-1) ℓ-σ(ℓ) .
(3) Set

c ′′ = (c ′′ 0 ≤ • • • ≤ c ′′ i-1 ) = (c 0 ≤ • • • ≤ c σ(ℓ) ≤ • • • c i ).
The term (2) appears as

(c ′′ ρ 1 (0) ∨ c ′ ρ 2 (0) ≤ • • • ≤ c σ 1 (ℓ) ∨ c ′ σ 2 (ℓ) ≤ • • • ≤ c ′′ ρ 1 (i+j) ∨ c ′ ρ 2 (i+j) ) in ∂ i (c) ¡ c ′ = ¡(∂ i (c) ⊗ c ′ ).
The chain c ′′ appears in ∂ i (c) with sign (-1) σ(ℓ) . Hence (2) appears in

∂ i (c) ¡ c ′ with sign sgn(ρ) • (-1) σ(ℓ) (3) = sgn(σ) • (-1) ℓ-σ(ℓ) • (-1) σ(ℓ) = (-1) ℓ • sgn(σ)
and by denition with the same sign in ∂ i+j+1 (c ¡ c ′ ).

A similar argumentation holds for the case σ(ℓ), σ(ℓ + 1)

∈ {i + 1, . . . , i + j + 1}.
Since all terms in ¡(δ i+j+1 (c ⊗ c ′ )) are of the form (2), we conclude. □ Corollary 2.2. For α ∈ M i (L) and β ∈ M j (L) we have

(∂ i α) ¡ β + (-1) i+1 α ¡ (∂ j β) = ∂ i+j+1 (α ¡ β).
Proof. By Proposition 2.1 we have

∂ i+j+1 (α ¡ β) = ∂ i+j+1 ¡ (α ⊗ β) = ¡ δ i+j+1 (α ⊗ β) = ¡ ∂ i (α) ⊗ β) + (-1) i+1 α ⊗ ∂ j (β) = ∂ i (α) ¡ β + (-1) i+1 α ¡ ∂ j (β). □ Corollary 2.3. For α ∈ M i (L) and an element v ∈ L we have α -(v ¡ (∂ i α)) = ∂ i+1 (v ¡ α).
Proof. We can consider v as an element of M 0 (L). Then by Corollary 2.2

∂ i+1 (v ¡ α) = ∂ 0 (v) ¡ α + (-1) 1 v ⊗ ∂ i (α) = () ¡ α -v ¡ ∂ i (α) = α -v ¡ ∂ i (α). □

Synors and relaxed chains

For this chapter we return to the situation where P is a poset, and we consider an ecient way to generate their homology. We do so using the notion of synors that is naturally inspired by the main theorem. An i-synor is an element x in P such that P <x has nontrivial (i -1)-st homology, and a synor is an element that is an i-synor for some i. It is not hard to see that P synors , the subposet of synors, and P are homologically equivalent, that is, the inclusion induces an isomorphism of homology groups (this is a trivial consequence, for instance, of Quillen's Theorem A [START_REF] Quillen | Homotopy properties of the poset of nontrivial p-subgroups of a group[END_REF].)

Let us be more explicit, and construct a chain complex, synor complex S(P) and associated synor homology using only these synors. We will construct this as a subspace of the simplicial chain complex, so that the boundary operator need not be specied; it is simply the simplicial boundary operator. We should note that this construction leaves us some freedom, and is not entirely unique in general.

As for the subspace, we construct it inductively, and with the property that the restriction of a synor complex to any ideal is a synor complex of that ideal.

On the empty poset {∅}, not to be confused with the empty set, we have the natural chain complex 0 -→ K -→ 0.

Given P any nonempty poset, we may assume we have constructed, for x any maximum of P, the chain complexes S(P -x) and S(P <x ) by induction.

We consider the nontrivial synor homology cycles of P <x , called principal (synor) cycles, and the cones over them: given a simplicial i-chain γ, we denote by x * γ the (i + 1)-chain obtained by appending x to every simplex 3 .

We call the chains of the form x * ζ, where ζ is a principal cycle, the principal (synor) chains. Now, we consider a principal cycle basis Z of the synor homology of P <x . Then

S(P) = S(P -x) ⊕ K[Z * x].
where K[A] is the free K group over a set A. Naturally, we should think of K[x * Z] as S(P, P -x).

3

Of course, this is but a more ecient way to write x ¡γ if we consider x as a 0-chain.

For instance, we obtain that S 0 (P) is the free group over minimal elements of P.

We obtain immediately:

Proposition 3.1. The natural inclusion of chain complexes S • (P) into C • (P) induces an isomorphism between synor and simplicial homology.

Proof. The restriction of a synor chain complex to an ideal is a synor chain complex again. Hence, by inductive assumption, the statement is true for P -x and P <x . The rest follows as a quasiisomorphism of the long exact sequence in homology induced by 0 -→ S(P -x) -→ S(P) -→ S(P, P -

x) ∼ = K[x * Z] -→ 0. □
We have a useful corollary: We can extend this notion to chains: if γ is a chain, a weak relaxation is a chain γ in the simplicial chain complex of the poset such that the boundary ∂γ is a relaxation of ∂γ and the deformations are realized in the respective ideals; it is also a relaxation if γ is a relative cycle in (S • (P), S • (P ≤∂γ )).

Formally: A (weak) relaxation of a chain γ is a (weak) relaxation of the relative cycle γ in (P, P ≤∂γ ) whose boundary is also a (weak) relaxation of ∂γ. This relaxation always exists as well, simply by introducing relative homology and repeating the arguments of Proposition 3.1 with pairs (P, a), where a is any ideal in P. Again, formally: Proposition 3.3. Given a poset P and an ideal a within, the natural inclusion of chain complexes S • (P) into C • (P) induces an isomorphism between synor and simplicial homology of the pair (P, a).

Moreover, this process of relaxation behaves well with respect to shue products if we ignore cancellations. Given two subsets A, B in a lattice, we denote by A ∨ B the ideal generated by elements a ∨ b, where a ∈ A and b ∈ B.

As we only move downwards in the poset, and the shue is supported in joins, the shue with a (weak) relaxation is a weak relaxation in the following sense. Proposition 3.4. Consider a lattice L and a cycle of the form α ¡β := ∂α ¡ β + (-1) i+1 α ¡ ∂β within Ľ (with sign from Corollary 2.2, that is, i being the length of order chains in α resp. its dimension). If β ′ is a weak relaxation of β then α ¡β ′ is homologous to α ¡β in the ideal α ∨ ∂β ∪ β ∨ ∂α. The analogous statement applies to the chain α¡β, which is homologous to

α¡β ′ in (α ∨ β, α ∨ ∂β ∪ β ∨ ∂α).
Proof. This is essentially just an application of associativity of the shue product. The chain β, seen as a relative cycle in (P, P ≤∂β ), is homologous to its relaxation β; their dierence is boundary to a chain E. Then the cycles α ¡β and α ¡β ′ are homologous via

∂α ¡ E + (-1) i+1 α ¡ (∂E -β + β ′ ).
Similarly, the homology equivalence of α¡β to α¡β ′ , as relative cycles, is realized via the chain α¡E.

□

Shuffling elements into cycles

Let us note another useful fact that the synor chain complex provides: Consider a simplicial d-dimensional cycle ζ in a simplicial chain complex, and let us consider any (d -1)-dimensional face τ . Then where λ σ,τ is the oriented coecient of ζ at σ with respect to τ .

In other words, if we look at τ , and all the coecients of ζ added by shuing in one element. To understand why this observation is true is a trivial exercise: simply apply the boundary operator and consider the terms that support τ.

What about cycles in posets P? Of course, the same holds, but it is a mess. Indeed,

ζ = λ σ σ
where λ σ is a scalar as before and σ is a order chain

σ 0 > σ 1 > • • • > σ d . Consider now τ = (τ 0 > τ 1 > • • • > τ d-1 ) another order chain. Of course σ d-simplex containing τ λ σ,τ = 0,
but it is chaos, as the elements of σ \ τ are shued into the order chain all over the place; in rst, last or any other place. Let us consider the subsum

ζ τ,i := σ d-simplex containing τ σ i =τ λ σ,τ = 0,
where σ i is the removal of the i-th entry of σ. Of course

σ d-simplex containing τ λ σ,τ = d+1 i=1 ζ τ,i = 0.
But the individual ζ τ,i may not vanish in the simplicial chain complex of P.

The synor chain complex xes that. Proof. The proof is by induction on d. The case d = 0 is trivial. Then

σ d-simplex containing τ λ σ,τ = d+1 i=1 ζ τ,i = 0.
Moreover, for i ̸ = 1, that is, when shuing an element into τ not in the rst place, then, of course, we obtain only contributions from the principal d-synor chains at τ 0 , which is of course the cone over a cycle. Hence

ζ τ,i = 0
for all i ̸ = 1 by induction applied to the principle synor d-chain at τ 0 obtained as restriction of ζ to

K[Z * τ 0 ] = S(P ≤τ 0 , P <τ 0 ).
Hence ζ τ,1 = 0 as well.

□

Synor representations and final steps

For this section, let us consider for a lattice L the subset L obtained by removing the minimal element 0. Assume that 1 is a m-synor, that is there is a principal synor (m -1)-cycle ζ in Ľ, and the associated principal relative synor cycle γ = 1 * ζ in S(L, Ľ).

Consider any element v in L; think of it as a 0-chain. The following observation is trivial, and our starting point.

Proposition 5.1. v ¡ ζ is homologous to 1 * ζ in S(L, Ľ).
This is obvious: Both chains have the same boundary, and L is contractible so both chains are homologous.

Let us rethink what was done here: We presented the m-chain as a shue product of an synor (m -1)-cycle and a 0-chain; canonically, 1 ¡ ζ.

We can go further and write

γ = χ X χ ¡ ζ χ
where the sum is over χ described as follows:

• χ is an order chain in L of the form

1 = x m > x m-1 > • • • > x m-ℓ+1
where x i is an i-synor,

• X χ = ¡(x i ) χ , where we think of the x i as simplicial 0-chains, and

• ζ χ are principal synor (m -ℓ)-cycles in L <x m-ℓ+1 .
Consider now an index j between 1 and ℓ, just for a second. We x one χ as above and consider the sum χ ′ = j χ ζ χ ′ where χ ′ = j χ simply denotes the fact that χ and χ ′ coincide except possibly in the j-th, that is, the entry x m-j+1 .

Lemma 5.2. χ ′ = j χ ζ χ ′ = 0 unless j = 1. Proof. We might as well nish the process of writing γ. Then γ = χ λ χ X χ , where the χ are now chains of length m + 1, starting with 1, and λ χ are scalars. Then the sum transforms to

χ ′ = j,m-ℓ,... χ λ χ ′ X χ ′ \χ ′ = χ ′ = j χ ζ χ ′
where of course χ ′ = j,m-ℓ,... χ denotes that both order chains coincide in all but the indexed entries.

As all χ ′ contain 1, we may remove that element without aecting the vanishing; then we have a representation of the cycle

ζ = χ (X χ ) 1 . Because ζ is a cycle, Lemma 4.2 gives χ ′ = j,m-ℓ,... χ λ χ ′ (X χ ′ ) 1,j = 0
Now, removing common elements from the top of these order chains does not aect vanishing, so we can also remove the remaining indices of χ. We obtain

χ ′ = j,m-ℓ,... χ λ χ ′ (X χ ′ ) 1,j = χ ′ = j,m-ℓ,... χ λ χ ′ X χ ′ \χ ′ = 0
as desired.

□

Let us nish: We dene a map

ρ k : K[L k ] -→ S k-1 (L)
inductively as follows:

• ρ 0 sends {∅} to the unique -1-chain.

• ρ k (V ) is a relaxation of the chain ¡V with boundary ρ(∂V ).

Such a map exists by the existence of a relaxed state Proposition 3.3; that is all that is necessary for us to know.

Hence, we can write, similar to Proposition 3.4, the homologous equivalence

Lemma 5.3. γ ∼ χ X χ ¡ ζ χ ∼ χ ρ(X χ ) ¡ ζ χ in (L, Ľ).
Proof. To see this, it suces to prove that

∂γ = ∂ χ X χ ¡ ζ χ and ∂ χ ρ(X χ ) ¡ ζ χ
are homologous in Ľ; the claimed homology equivalence then follows from contractibility of L. Now, we can assume that we have inductively proven the equivalence for smaller ℓ. Then

∂ χ ρ(X χ ) ¡ ζ χ = χ (∂ρ(X χ )) ¡ ζ χ + (-1) ℓ χ ρ(X χ ) ¡ ∂ζ χ Now, χ ρ(X χ ) ¡ ∂ζ χ = 0
by associativity and because ∂ζ χ = 0. Hence, we are left with

χ (∂ρ(X χ )) ¡ ζ χ = χ   ℓ j=1 (-1) j ρ((X χ ) j )   ¡ ζ χ ,
where (X χ ) j denotes the omission of the j-th element in the chain

x m > x m-1 > • • • > x m-ℓ+1 . Now, we have χ ρ((X χ ) 1 ) ¡ ζ χ ∼ χ (X χ ) 1 ¡ ζ χ ∼ ∂γ by induction.
For j ̸ = 1, declare χ = j χ ′ if they coincide in all but the j-th entry as in Lemma 5.2. Consider the equivalence classes Λ. Then

χ ρ(X χ ) j ¡ ζ χ = Λ ρ(X χ ) j ¡ χ∈Λ ζ χ ,
which is well-dened because ρ(X χ ) j only depends on the equivalence class Λ. But the last sum is 0 by Lemma 5.2. Hence χ ρ((X χ ) j ) ¡ ζ χ = 0 for j ̸ = 1 which nishes the proof.

□

We obtained:

Theorem 5.4. Consider a lattice L, and assume that 1 is an m-synor. Let ℓ ≤ m be a nonnegative integer.

Then any principal m-synor γ at 1 in L is homologous to a chain of the form λα ¡ β where λ are scalars, α are principal (m-ℓ)-chains and β are principal (ℓ-1)chains.

We obtain the proof of the Main Theorem: Assume that that L is a lattice such that Ľ has nontrivial (m -1)-homology. Then the associated principal synor chain is of the form as above. Hence, one of the summands must involve 1, which means that one of the (m -ℓ)-synors and one of the (ℓ -1)-synors has join 1. This nishes the proof.

Further applications

It is likely that we can achieve many more results regarding the Betti table just from this result. Let us give two more applications of the proven vanishing results: Corollary 6.1. Suppose z ∈ L is an (a + b -k)-synor for k ≤ a, b. Then z = x ∨ y for some a, b-synors x, y respectively, and hence t a+b-k ≤ t a + t b .

Proof. Suppose 1 is an (a + b -k)-synor, and let ζ be a principal synor (a + b -k -1) cycle in Ľ. By Lemma 5.3, we can express γ = 1 * ζ up to homologous equivalence in (L, Ľ) as γ ∼ χ ρ(X χ ) ¡ζ χ with ρ(X χ ) (a -1)synor chains and ζ χ (b -k) synor cycles, so for some order chain χ 0 we get 1 appearing as a join of synors in ρ(X χ 0 ) ¡ ζ χ 0 . We can similarly write γ ∼ χ ′ ρ(X χ ′ ) ¡ζ χ ′ with ρ(X χ ′ ) (a -k)-synors and ζ χ ′ (b -1)-synor cycles, with the order chains χ ′ containing b-synors in the (a -k + 1) th position. As χ and χ ′ agree in the rst (a -k + 1) positions, all order chains χ will contain a b-synor. Hence all the (b -k)-synors appearing in ζ χ 0 will have some b-synor above them in the lattice. As some synors from ζ χ 0 and ρ(X χ 0 ) have 1 as their join, so will the b-synors above ζ χ 0 and ρ(X χ 0 ). □ Corollary 6.2. Let a k denote the number of distinct shifts in homological degree k. If s ≤ t, then a s+t ≤ a s a t .

Proof. By the main theorem we know every (s + t)-synor is a join of an ssynor and a t-synor, and there are a s a t distinct pairs of s-and t-synors in L. Hence we get at most a s a t possibilities for distinct (s + t)-synors.

□

We note that subadditivity is a sharp inequality: for example the ideals I a = (x a 1 , . . . , x a n ) with a ≥ 1 in k[x 1 , . . . , x n ] satisfy t k = ak for all 0 ≤ k ≤ pd(I a ).

  Lemma 4.1. σ d-simplex containing τ λ σ,τ = 0.

Lemma 4. 2 .

 2 Consider ζ a d-cycle in S(P), and τ an order chain of length d. Then ζ τ,i = 0 for all i.

  Corollary 3.2. Consider a poset P and a simplicial homology cycle ζ within it. Then the ideal generated by ζ contains a synor cycle ζ in S(P) homologous to ζ, and the homology equivalence is realized within the ideal, that is, ζ and

ζ are homologous in P ≤ζ .

We say ζ is the relaxation of ζ. A weak relaxation is any simplicial homology cycle in the ideal P ≤ζ homologous to ζ in P ≤ζ ; intuitively it corresponds to nding a representative of a cycle somewhere lower in the poset. Let us note though that this does not induce a poset order on cycles as cycles can be weak relaxations of each other without being equal. Formally, a weak relaxation is any cycle ζ ′ homologous to ζ in P ≤ζ .

Acknowledgements: We thank David Eisenbud and Craig Huneke for