text:
Talk:Juan Manuel Fangio

Pagani
This guy just had the Zonda F named after him

The Maestro
JMF has (still) the best win% in F1 history. Can somebody confirm he was called "The Maestro"? And add it at List of athletes by nickname? Trekphiler 11:18, 3 January 2006 (UTC)

Actually the statement is technically false. Argentines refer to JMF with the Latin language masculine the, which makes it "El Maestro". However, I suspect the references {14][15] likely did translate El to The. On another vector, I am a HUGE fan of Sterling Moss, who drove with JMF on both the Ferrari and MB teams, and I have seen several quotes, attributed to Sterling, referring to his teammate as either, The Maestro, or El Maestro, with great respect. As for me I have been using El Maestro for JMF since 1962. I drove Formula F with the Four Seasons Racing Team (North Texas) in the SCCA Summer season 1964. It was the year before Jackie's debut in F!, and for our biggest race that summer the track management managed to sponsor Jackie Stewart to race with us in the A Stock class to push up gate draw. The local Jag dealer provided a Race prepped E type that looked for all the world like a D type. I had the privilege of shooting the breeze with Sir Stewart (to be) between races and he knew exactly of whom I spoke when I referred to JMF as El Maestro. Even if you doubt my story, just Google "El Maestro" + Fangio and you will get over 24,000 hits from all over the racing world, including this Argentine URL www.cosasdeautos.com.ar/.../arte-se-inaugura-en-balcarce-la-muestra-fangio-el-maestro-el-mejor/
 * The Argentins actually say "El maestro, el mejor", which loosely translates into The Teacher, the best one. — Preceding unsigned comment added by Skylab72 (talk • contribs) 00:10, 5 February 2011 (UTC)

Introduction
I think the line "winning the world championship no less than five times..." is antiquated and out of place. It should be replaced with some more encyclopedic. Rtcpenguin

First paragraph
Just a note by way of explanation about the removal of the line in the first paragraph by user "Ernham". I reverted it once but he keeps changing it back. He's done it to several other articles as well, and engaged in personal attacks on Talk:Michael Schumacher. No doubt it'll be resolved soon enough, if it hasn't been already. Bretonbanquet 04:19, 12 September 2006 (UTC)
 * I hope the reference I brought calms down the editor. Mariano (t/c) 09:23, 12 September 2006 (UTC)
 * your atricle demonstrates absolutely nothing. first, it nowhere says Fangio was the greatest. Second, it rests its arguemnt that he may have something about him better than schumacher on the grounds that he won 5 WCorld championships in 8 years, which is supposedly better than Schumacher's. But wait-- it's not! Good old BBc logic. Schumacher won 5 WC in FIVE YEARS. Gone.
 * It's a pitty you didn't take the time to read the article, it's actually good. I really don't know why you are so blindly agressive. If he won 5 championships in 8 years, and 24 races of 51, then he does have a better average than Schummy, who has 7 championships in 15 years, and 90 wins in 247 races. The article actually does name Fangio as the best ("[a] genius against whom all others are measured"), even if it is only an oppinion, you can't deny a lot of people think like that.
 * Your edits to Fangio took me to take a look at Schumacher's article; you are seriously pushing your oppinion over other peoples, what goes against Wikipedia Policies. I don't even know exactly what bothers you from the previous version of Fangio's article, that a lot of people consider him the best driver?? It is a verifiable fact, you have to learn to live with other people's oppinions.
 * Finally, as the article of BBC says: "being the most successful in history does not necessarily mean you are the best". Mariano (t/c) 10:03, 12 September 2006 (UTC)

 * Uh, it's actually your opinion of someone else's opinion that the said quote establishes that one person's words actualy are trying to convey that Fangio was the best. And, good luck with that. As I demonstrated above, I totally destroyed the argument of his greater WC average. Plus, as most people know in racing, half of Schumacher's racing career he was racing with sub-par cars. He went from a winning team to a losing team and turned them into the "winningest" team ever.

 * Your use of words and phrases such as "demonstrated" and "totally destroyed" are misleading. Shumacher raced 15 years, not 5; same goes for wins/races. Then, first you talk about how I interpret people's oppinion, and right after that you talk about "people in racing knowing...". On that, Fangio won titles with 4 different constructors. What the BBC article significates is that all drivers in the article are considered, by some people, to be the best. But then again, if you are not listening, you'll not hear. Mariano (t/c) 10:37, 12 September 2006 (UTC)

 * Actually, the article seems to be infering that Schumacher was the best --but-- there are some other people that have claims to being great on one level or another. But such inference is only assumption, as the article never substantiates any driver being the "greatest". Ernham 11:12, 12 September 2006 (UTC)

 * Again, it's not about Being the greatest, but about Being considered the greatest. Mariano (t/c) 11:15, 12 September 2006 (UTC)
 * "considered the greatest" = POV. "The greatest" = fact, assuming you qualify and validly substantiate the given criterion. Is wiki about facts, then, or POV? Ernham 11:20, 12 September 2006 (UTC)

 * No, the other way around; To be considered the best can be sourced and verified, being the best can't!! I have already explained you that you can't compare two drivers and say that one of them is better, specially if from different times. Mariano (t/c) 11:25, 12 September 2006 (UTC)
 * regardless of your opinion on that matter, it's really moot. No poll ever done has shown that he is considered greater than Schumacher. In fact, all the polls ever done have shown Schumacher the greatest above him, with Senna trailing a bit behind. Fangio is always a distant third. He's not the people's greatest; he's not the experts' greatest; and he's not statistic's greatest. Calling him greatest in any capacity is intellectually dishonest. Further, you CAN prove someone is the best via data and statistics, which has already been done by a well known race car driver biographer. And popularity can be polled as well. Fangio loses on all accounts. Ernham 13:02, 12 September 2006 (UTC)
 * No offence, but your statement that "No poll ever done has shown that he is considered greater than Schumacher" is well - how do I put this? Silly. As I've come across several polls which suggest the oposite to your opinion. Sadly, I can't access those websites at the moment because of my college internet restrictions.--Skully Collins 14:51, 29 September 2006 (UTC)

Well, there's this:-- http://sputniknews.com/world/20160416/1038120655/best-ever-formula-one-driver.html -- a University of Sheffield statistical study, weighting drivers' performances according to team, which finds that Fangio was indeed (as everybody knows) the greatest. Of course you can design statistical exercises all sorts of ways, but Schumi, Prost, Senna and Clark all rank high, as they should, though Niki Lauda seems to have taken a massive hit. Khamba Tendal (talk) 18:08, 18 April 2016 (UTC)

Intro - Schumacher comments and Considered Greatest
For the attn of User:Ernham; rgding your reverts

Claims of quoted out of context is incorrect. Quote true to intention as indicated by BBC article's title. I've just changed the sentencing of Michael Schumacher overcoming the record. It was required to rephrase, because in this article, the subject is Fangio and not Schumacher. The information should be on Fangio, and correct. Whether Michael schumacher has seven records or nine, has to be present in Michael Schumacher's article. Here what matters is that one shows it was a 45 year (long standing) record, and the person who surpassed it. Since the quote was made after beating the record, and because it was about Fangio, becomes relevant to the article and the sentence.

As with the "considered greatest by many", Fangio is somebody who has a distinction of regarded so, by his peers and followers(something michael was doing). But still we dont say "considered greatest" as it says more that it might. For instance you seem not to share that. Many people do, and that is evident from Fangio's profile on the official site. So we say "considered greatest by many". POV is wrong so is dilution. Please check the profiles of other notable drivers - Senna or Schumacher in the same official site Champions' Profile. That would speak about their achievements, but you dont see a mention of greatest in the opening intro..

Thats the difference.. If say, I am a Senna fan, i might feel like going and editing in Senna is greatest.. But that defeats the whole purpose. And that is what you seem to be doing with Schumacher.. And when you are editing with such an intention, you are wasting your own and others time. It is evident that you dont care about the article you edit as much as your opinion. Shown from this edit; Besides manipulating the quotes, you've taken off Mercedes Benz and Maserati... and you probably noticed only now; So much for the concern of the article. You have the right of an opinion, but please dont be here just to impose that, esp when there are no grounds and you are in violation of even the basic guidelines. Just a reminder because honestly, one doesnt enjoy wasting time repeatedly with changing rash edits. Having to say coz, had tried discussing matter very politely with you previously to no avail. --Su30 15:16, 30 September 2006 (UTC)

 * As the saying goes, "by their fruits you shall know them". If Fangio is the greatest, show us that cornucopia of fruit he bears, and let people make their own decision. If you put direct quotes in your wiki, particularly when those quotes come from living persons, you better make sure that it is in context and conveys the whole meaning that was stated. if i just cut up peoples quotes, I could provide a page bigger than the entirety of Fangio's wiki of modern race drivers/experts/commentators all proclaiming Schumacher as "the greatest". Ernham 15:27, 30 September 2006 (UTC)

 * The argument on the quotes being misleading is not valid(see above). You may not have liked him say that, i can see from your attempts. But can it be taken off on that grounds alone? It can be discussed if you have a problem on the "considered greatest by many", coz there is some sense. But to take off/modifying the quotes has no grounds whatsoever.--Su30 15:35, 30 September 2006 (UTC)

 * btw, I didn't mean to clip out part of the four teams. no, i was actually going to change that to what someone had put in there before that was complimentary to Fangio, the statement about how he won with 4 different teams. But i got caught up elsewhere last night and forgot about it. in any event, you continue to take Schuamcher's quote out of context. He stated that as a RACING PERSONALITY *he* considers fangio greater than he sees himself, and that basically you can't compare things across eras like that. I've stated my argument on this like 5 times now and no one has bothered coming up with a valid counter-argument outside of he "well he said these words".Ernham 15:40, 30 September 2006 (UTC)

 * I've pointed that out just to show your concern about the article. I am not very happy to criticise, but give it a thought. I see no point in an edit war. Just consider one thing. Most sports in any period has heroes, and fans for each of them and there is so much emotional content in the support and it is not out of the ordinary that a fan would want to see their hero on the ultimate summit; yeah yeah.. he/she would have lot of reasons to say that.. But when that is carried out to an encyclopedia, there is trouble. Dead sportsmen with dead fans would go out of the window, however good they were.. And each era, would redefine the summit. Fangio's best years were done before he arrived in europe, the 5 titles dont measure the driver completely. Besides knowledge of the equipment, doesnt know any other driver who peers unanimously talk with reverence. Not to pull down what recent drivers like Schumacher or Senna have done. Will be back on this one--Su30 16:08, 30 September 2006 (UTC)

 * Look, I'm not going to rag on Fangio too much, because he's probably the nicest guy ever to race in F-1; however, you simply cannot call him the greatest when it comes to driving. The reality is, that in Fangio's Era the sport was dominated by ultra-rich overweight men. It was more a country club than it was a competition. I wrote that it was "feat" that fangio won with 4 different teams, but in reality it wasn't really a feat because Fangio was well-known for absolutely having to have the best car. He ALWAYS went to the team that had the best car, hence his "feat", both of winning 5 WDC and of winning with 4 different teams.(as a side note, Senna did the exact same thing) I think I've been more than flattering to Fangio in light of reality.Ernham 16:19, 30 September 2006 (UTC)

Since when did you connect with reality? In Fangio's era, drivers got killed at the rate of one a month. Your comments show an astonishing lack of knowledge and respect. Your continuing persistence in twisting articles to reflect your feelings about Michael Schumacher are not only non-encyclopedic, but insulting. Please know that your opinions are not shared by many of the more knowledgable followers of the sport. *Stands back and waits for insults / childish retort* Bretonbanquet 22:47, 30 September 2006 (UTC)

 * I don't even need to answer with words: http://members.atlasf1.com/williams/collisions.jpg Ernham 23:10, 30 September 2006 (UTC)

 * Apparently you don't need to answer it at all, you just have to post some irrelevant stats. Bretonbanquet 23:12, 30 September 2006 (UTC)
 * and meaningless as well - what do those stats actually mean? Bretonbanquet 23:15, 30 September 2006 (UTC)

 * ahh, ok. To the unbiased observer: at least you can see what kind of ridiculousness I have to put up with around here.Ernham 23:16, 30 September 2006 (UTC)

 * What are you talking about? Don't pretend that table of stats is self-explanatory, because it isn't. No wonder people have stopped arguing with you, it's like talking to a pile of bricks. Bretonbanquet 23:20, 30 September 2006 (UTC)

 * I would like to point out, for the sake of sanity, that you cannot compare what Fangio achieved with what Schumacher achieved statistically. There were less races, different rules and different machinery. Both drivers were absolutely the best throughout their careers in F1 and both can be "considered the greatest" depending on your point of view so can the Schumacher worshipper please get down off his high horse!86.1.38.43 21:00, 8 December 2006 (UTC)

Schumacher quote on Fangio
I've just edited the Schumacher quote on Fangio from the lead. The previous version read:

"[in regards to personality], Fangio is on a level much higher than I see myself. [But], you can't take a personality like Fangio and compare him with what has happened today. There is not even the slightest comparison."

The quote as given on the BBC website is:

"Fangio is on a level much higher than I see myself. What he did stands alone and what we have achieved is also unique. I have such respect for what he achieved. You can't take a personality like Fangio and compare him with what has happened today. There is not even the slightest comparison."

I have, after going through a couple of version, summarised the BBC quote as:

"Fangio is on a level much higher than I see myself. What he did stands alone and what we have achieved is also unique...There is not even the slightest comparison.""

The square bracketed comments in the previous version - particularly 'in regards to personality' placed at the beginning of the sentence imply that Schumacher directly and clearly qualified his statement that he saw Fangio as being on a higher level than himself as relating only to their respective personalities. The original words from the BBC website give no such clear qualification.

You could read Schumacher's comment that 'there is not even the slightest comparision' as meaning that he feels his achievements are greater than Fangio's, although it seems unlikely because it would be a public relations disaster of a monumental proportions to make such a comment. (Note: If that were the meaning it would also break with the tone of the immediately preceding comment: 'I have such respect for what he achieved') Nevertheless, the edited version of the BBC quote that I have given leaves the decision on whether that is what Schumacher meant to the reader. Cheers. 4u1e 18 October. Lunchtime.

 * What you have just made is very biased... you have turned the citation the direction YOU want, and you don't even try to hide it! Who are you to decide, what he really meant with this? Probably Michael meant that you cant compare what happened then with what happens today, because there is a higher and tighter competition today... so in reality he means that he respected Fangio because of his pioneer work and achievement, BUT you cant compare it with higher competitive races today...but you cut out this very relevant information!
 * Only a complete quote is valid, your shortened quotation is very "constructed" and obviously misleading.--84.58.34.138 23:05, 22 October 2006 (UTC)

 * Well, I don't think it is very biased, I certainly had no intention of making it so. I hope you don't feel that hiding my changes and reasoning would have been better! ;-) Anyway - none of us are perfect. I'm quite positive my version's very much closer to an accurate 'compression' of the quote than the previous version which distorted the full quote quite seriously to give a certain point of view. What I have left out is "I have such respect for what he achieved. You can't take a personality like Fangio and compare him with what has happened today." The bit about not comparing across eras ('You can't take.....') I consider to be covered by "What he did stands alone and what we have achieved is also unique.", which seems to me to say the same thing quite clearly. So, to the best of my understanding, I have only left out bits that effectively repeat or aren't directly relevant ('I have such respect....').
 * Given the history of arguing over this point I'm very happy to have the full quote - without editorialising - it's just a bit long for the lead, so it would have been nice to compact it.
 * By the way, please try to be polite when commenting on others work - it's easy for editors to get drawn into editing wars when we're not all nice to each other. Cheers --4u1e 00:24, 23 October 2006 (UTC)

 * Full quote in place. Ta. --4u1e 00:29, 23 October 2006 (UTC)

Juan Manuel Fangio II
Fangio's nephew also was a race car driver -- Juan Manuel Fangio II I am unsure about placing a "see also" or "for this driver, see this article" marker on the top, but I have acted boldly and done so. I would say that the link is up for discussion, though. -- Guroadrunner 10:27, 6 May 2007 (UTC)

Abduction
I am not exactly sure where this should go, but I am shocked that there is no reference to his abduction by Fidel Castro's Rebels during the 1958 Cuban Grand Prix (released unharmed after the race had concluded)... This is extremely well documented in history MikeinBlack (talk) 03:57, 30 October 2008 (UTC)
 * The information is already present in the "Later life and death" section of the article:
 * "Cuban rebels kidnapped him on February 23, 1958, but he was later released, and remained a good friend of his captors afterwards."
 * DH85868993 (talk) 01:50, 12 November 2008 (UTC)

Pre-F1 Career
Considering Fangio's record in domestic racing prior to going to Europe, this article could probably do with more than a single sentence on the matter. Mr Larrington (talk) 14:44, 26 July 2010 (UTC)

Meeting Senna
Can anyone clarify the sentence "In 1992, Fangio met the three-time world champion, Ayrton Senna, who had genuinely felt the encounter had reflected the mutual affection for both drivers." in the Later life and death section. Without access to the source it's not clear who's saying what about whom or even what it is they're saying. Mighty Antar (talk) 23:49, 31 May 2013 (UTC)

External links modified
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Grand Prix Race record
This list seems to leave off at least some of his lower place finishes, at least in the later Fifties. Any comments before I look into updating the record? Proscriptus (talk) 18:24, 22 October 2015 (UTC)

External links modified
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External links modified
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Shared drives and entries
Shared drives 1950 Italian Grand Prix, 1951 French Grand Prix, 1953 Belgian Grand Prix, 1953 Swiss Grand Prix, 1956 Argentine Grand Prix, 1956 Monaco Grand Prix and 1956 Italian Grand Prix. That is 7 shared drives. He has started 51 races and has entered 53 races. His 2 non starts are the 1952 Swiss Grand Prix where his car did not arrive and the Indy 500 where he did not qualify. Mobile mundo (talk) 00:52, 7 September 2017 (UTC)

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title: Talk:Juan Manuel Fangio


text:
Healthy city

Healthy city is a term used in public health and urban design to stress the impact of policy on human health. It is a municipality that continually improves on a physical and a social level until environmental and pathological conditions are reached establishing an acceptable morbidity rate for the population. Its modern form derives from a World Health Organization (WHO) initiative on Healthy Cities and Villages in 1986, but has a history dating back to the mid 19th century. The term was developed in conjunction with the European Union, but rapidly became international as a way of establishing healthy public policy at the local level through health promotion. It emphasises the multi-dimensionality of health as laid out in WHO's constitution and, more recently, the Ottawa Charter for Health Promotion. An alternative term is Healthy Communities, or Municipios saludables in parts of Latin America.

Approaches
Many jurisdictions which have healthy community programmes and cities can apply to become a WHO-designated "Healthy City". WHO defines the Healthy City as: "one that is continually creating and improving those physical and social environments and expanding those community resources which enable people to mutually support each other in performing all the functions of life and in developing to their maximum potential."

Measuring the indices required, establishing standards and determining the impact of each component on health is difficult. In some regions such as Europe, a health impact assessment is a required piece of public policy development.

There are many networks of healthy cities, including in Europe and internationally, such as the Alliance for Healthy Cities. A key feature is ensuring that the social determinants of health are taken into consideration in urban design and urban governance. For example, "urbanization and health" was the theme of the 2010 World Health Day. One tool in developing healthy cities is social entrepreneurship.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Healthy city


text:
Talk:Lidsville

The formatting of the text of this article would seem to indicate it was copied verbatim from a web site. This could be copyright infringement. Can anyone verify or disprove this? —Frecklefoot 18:19 20 Jun 2003 (UTC)

Lidsville/ Hatty Town
This article says that Hatty Town was a much later show, starting in 1980. It wasn’t - it was virtually simultaneous with Lidsville, being made in 1969, and shown in 1970. The article that is linked to from the article page confirms this… Jock123 (talk) 10:58, 15 October 2009 (UTC)

Theme Song
Near the end, the opening theme song suddendly changes from telling the orgin story to a happy dancing tune. Was there an extended version that may have only aired on the first broadcast? — Preceding unsigned comment added by 98.233.59.72 (talk) 23:56, 17 November 2013 (UTC)

Film is now becoming a television reboot
Hi, so I visited Comic Con in 2015 and in 2016 and I asked about the Lidsville movie and the Kroffts told me{ there are some articles confirming this on the internet} they aren't going to do a lidsville film, instead they are looking into updating lidsville as a tv show — Preceding unsigned comment added by 2601:642:4280:8EE0:6D9D:AC00:A790:CDF4 (talk) 18:45, 27 August 2016 (UTC)

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Covering space

In topology, a covering or covering projection is a map between topological spaces that, intuitively, locally acts like a projection of multiple copies of a space onto itself. In particular, coverings are special types of local homeomorphisms. If $$ p : \tilde X \to X $$ is a covering, $$(\tilde X, p)$$ is said to be a covering space or cover of $$X$$, and $$X$$ is said to be the base of the covering, or simply the base. By abuse of terminology, $$\tilde X$$ and $$p$$ may sometimes be called covering spaces as well. Since coverings are local homeomorphisms, a covering space is a special kind of étale space.

Covering spaces first arose in the context of complex analysis (specifically, the technique of analytic continuation), where they were introduced by Riemann as domains on which naturally multivalued complex functions become single-valued. These spaces are now called Riemann surfaces.

Covering spaces are an important tool in several areas of mathematics. In modern geometry, covering spaces (or branched coverings, which have slightly weaker conditions) are used in the construction of manifolds, orbifolds, and the morphisms between them. In algebraic topology, covering spaces are closely related to the fundamental group: for one, since all coverings have the homotopy lifting property, covering spaces are an important tool in the calculation of homotopy groups. A standard example in this vein is the calculation of the fundamental group of the circle by means of the covering of $S^1$ by $\mathbb{R}$ (see below). Under certain conditions, covering spaces also exhibit a Galois correspondence with the subgroups of the fundamental group.

Definition
Let $$X$$ be a topological space. A covering of $$X$$ is a continuous map
 * $$\pi : \tilde X \rightarrow X$$

such that for every $$x \in X$$ there exists an open neighborhood $$U_x$$ of $$x$$ and a discrete space $$D_x$$ such that $$\pi^{-1}(U_x)= \displaystyle \bigsqcup_{d \in D_x} V_d $$ and $$\pi|_{V_d}:V_d \rightarrow U_x $$ is a homeomorphism for every $$d \in D_x $$. The open sets $$V_{d}$$ are called sheets, which are uniquely determined up to homeomorphism if $$U_x$$ is connected. For each $$x \in X$$ the discrete set $$\pi^{-1}(x)$$ is called the fiber of $$x$$. If $$X$$ is connected, it can be shown that $$\pi$$ is surjective, and the cardinality of $$D_x$$ is the same for all $$x \in X$$; this value is called the degree of the covering. If $$\tilde X$$ is path-connected, then the covering $$ \pi : \tilde X \rightarrow X$$ is called a path-connected covering. This definition is equivalent to the statement that $$\pi$$ is a locally trivial Fiber bundle.

Some authors also require that $$\pi$$ be surjective in the case that $$X$$ is not connected.

Examples

 * For every topological space $$X$$, the identity map $$\operatorname{id}:X \rightarrow X$$ is a covering. Likewise for any discrete space $$D$$ the projection $$\pi:X \times D \rightarrow X$$ taking $$(x, i) \mapsto x$$ is a covering. Coverings of this type are called trivial coverings; if $$D$$ has finitely many (say $$k$$) elements, the covering is called the trivial $$k$$-sheeted covering of $$X$$.
 * The map $$r : \mathbb{R} \to S^1$$ with $$r(t)=(\cos(2 \pi t), \sin(2 \pi t))$$ is a covering of the unit circle $$S^1$$. The base of the covering is $$S^1$$ and the covering space is $$\mathbb{R}$$. For any point $$x = (x_1, x_2) \in S^1$$ such that $$x_1 > 0$$, the set $$U := \{(x_1, x_2) \in S^1 \mid x_1 > 0 \}$$ is an open neighborhood of $$x$$. The preimage of $$U$$ under $$r$$ is
 * $$r^{-1}(U)=\displaystyle\bigsqcup_{n \in \mathbb{Z}} \left( n - \frac 1 4, n + \frac 1 4\right)$$
 * and the sheets of the covering are $$V_n = (n - 1/4, n+1/4)$$ for $$n \in \mathbb{Z}.$$ The fiber of $$x$$ is
 * $$r^{-1}(x) = \{t \in \mathbb{R} \mid (\cos(2 \pi t), \sin(2 \pi t)) = x\}.$$

 * Another covering of the unit circle is the map $$q : S^1 \to S^1$$ with $$q(z)=z^{n}$$ for some $$n \in \mathbb{N}.$$ For an open neighborhood $$U$$ of an $$x \in S^1$$, one has:
 * $$q^{-1}(U)=\displaystyle\bigsqcup_{i=1}^{n} U$$.

 * A map which is a local homeomorphism but not a covering of the unit circle is $$p : \mathbb{R_{+}} \to S^1$$ with $$p(t)=(\cos(2 \pi t), \sin(2 \pi t))$$. There is a sheet of an open neighborhood of $$(1,0)$$, which is not mapped homeomorphically onto $$U$$.

Local homeomorphism
Since a covering $$\pi:E \rightarrow X$$ maps each of the disjoint open sets of $$\pi^{-1}(U)$$ homeomorphically onto $$U$$ it is a local homeomorphism, i.e. $$\pi$$ is a continuous map and for every $$e \in E$$ there exists an open neighborhood $$V \subset E$$ of $$e$$, such that $$\pi|_V : V \rightarrow \pi(V)$$ is a homeomorphism.

It follows that the covering space $$E$$ and the base space $$X$$ locally share the same properties.

 * If $$X$$ is a connected and non-orientable manifold, then there is a covering $$\pi:\tilde X \rightarrow X$$ of degree $$2$$, whereby $$\tilde X$$ is a connected and orientable manifold.
 * If $$X$$ is a connected Lie group, then there is a covering $$\pi:\tilde X \rightarrow X$$ which is also a Lie group homomorphism and $$\tilde X := \{\gamma:\gamma \text{ is a path in X with }\gamma(0)= \boldsymbol{1_X} \text{ modulo homotopy with fixed ends}\}$$ is a Lie group.
 * If $$X$$ is a graph, then it follows for a covering $$\pi:E \rightarrow X$$ that $$E$$ is also a graph.
 * If $$X$$ is a connected manifold, then there is a covering $$\pi:\tilde X \rightarrow X$$, whereby $$\tilde X$$ is a connected and simply connected manifold.
 * If $$X$$ is a connected Riemann surface, then there is a covering $$\pi:\tilde X \rightarrow X$$ which is also a holomorphic map and $$\tilde X$$ is a connected and simply connected Riemann surface.

Factorisation
Let $$ X, Y$$ and $$E$$ be path-connected, locally path-connected spaces, and $$p,q$$ and $$r$$ be continuous maps, such that the diagram commutes.

 * If $$p$$ and $$q$$ are coverings, so is $$r$$.
 * If $$p$$ and $$r$$ are coverings, so is $$q$$.

Product of coverings
Let $$X$$ and $$X'$$ be topological spaces and $$p:E \rightarrow X$$ and $$p':E' \rightarrow X'$$ be coverings, then $$p \times p':E \times E' \rightarrow X \times X'$$ with $$(p \times p')(e, e') = (p(e), p'(e'))$$ is a covering. However, coverings of $$X\times X'$$ are not all of this form in general.

Equivalence of coverings
Let $$X$$ be a topological space and $$p:E \rightarrow X$$ and $$p':E' \rightarrow X$$ be coverings. Both coverings are called equivalent, if there exists a homeomorphism $$h:E \rightarrow E'$$, such that the diagram commutes. If such a homeomorphism exists, then one calls the covering spaces $$E$$ and $$E'$$ isomorphic.

Lifting property
All coverings satisfy the lifting property, i.e.:

Let $$I$$ be the unit interval and $$p:E \rightarrow X$$ be a covering. Let $$F:Y \times I \rightarrow X$$ be a continuous map and $$\tilde F_0:Y \times \{0\} \rightarrow E$$ be a lift of $$F|_{Y \times \{0\}}$$, i.e. a continuous map such that $$p \circ \tilde F_0 = F|_{Y \times \{0\}}$$. Then there is a uniquely determined, continuous map $$\tilde F:Y \times I \rightarrow E$$ for which $$\tilde F(y,0) = \tilde F_0$$ and which is a lift of $$F$$, i.e. $$p \circ \tilde F = F$$.

If $$X$$ is a path-connected space, then for $$Y=\{0\}$$ it follows that the map $$\tilde F$$ is a lift of a path in $$X$$ and for $$Y=I$$ it is a lift of a homotopy of paths in $$X$$.

As a consequence, one can show that the fundamental group $$\pi_{1}(S^1)$$ of the unit circle is an infinite cyclic group, which is generated by the homotopy classes of the loop $$\gamma: I \rightarrow S^1$$ with $$\gamma (t) = (\cos(2 \pi t), \sin(2 \pi t))$$.

Let $$X$$ be a path-connected space and $$p:E \rightarrow X$$ be a connected covering. Let $$x,y \in X$$ be any two points, which are connected by a path $$\gamma$$, i.e. $$\gamma(0)= x$$ and $$\gamma(1)= y$$. Let $$\tilde \gamma$$ be the unique lift of $$\gamma$$, then the map
 * $$L_{\gamma}:p^{-1}(x) \rightarrow p^{-1}(y)$$ with $$L_{\gamma}(\tilde \gamma (0))=\tilde \gamma (1)$$

is bijective.

If $$X$$ is a path-connected space and $$p: E \rightarrow X$$ a connected covering, then the induced group homomorphism
 * $$ p_{\#}: \pi_{1}(E) \rightarrow \pi_{1}(X)$$ with $$ p_{\#}([\gamma])=[p \circ \gamma]$$,

is injective and the subgroup $$p_{\#}(\pi_1(E))$$ of $$\pi_1(X)$$ consists of the homotopy classes of loops in $$X$$, whose lifts are loops in $$E$$.

Holomorphic maps between Riemann surfaces
Let $$X$$ and $$Y$$ be Riemann surfaces, i.e. one dimensional complex manifolds, and let $$f: X \rightarrow Y$$ be a continuous map. $$f$$ is holomorphic in a point $$x \in X$$, if for any charts $$\phi _x:U_1 \rightarrow V_1$$ of $$x$$ and $$\phi_{f(x)}:U_2 \rightarrow V_2$$ of $$f(x)$$, with $$\phi_x(U_1) \subset U_2$$, the map $$\phi _{f(x)} \circ f \circ \phi^{-1} _x: \mathbb{C} \rightarrow \mathbb{C}$$ is holomorphic.

If $$f$$ is holomorphic at all $$x \in X$$, we say $$f$$ is holomorphic.

The map $$F =\phi _{f(x)} \circ f \circ \phi^{-1} _x$$ is called the local expression of $$f$$ in $$x \in X$$.

If $$f: X \rightarrow Y$$ is a non-constant, holomorphic map between compact Riemann surfaces, then $$f$$ is surjective and an open map, i.e. for every open set $$U \subset X$$ the image $$f(U) \subset Y$$ is also open.

Ramification point and branch point
Let $$f: X \rightarrow Y$$ be a non-constant, holomorphic map between compact Riemann surfaces. For every $$x \in X$$ there exist charts for $$x$$ and $$f(x)$$ and there exists a uniquely determined $$k_x \in \mathbb{N_{>0}}$$, such that the local expression $$F$$ of $$f$$ in $$x$$ is of the form $$z \mapsto z^{k_{x}}$$. The number $$k_x$$ is called the ramification index of $$f$$ in $$x$$ and the point $$x \in X$$ is called a ramification point if $$k_x \geq 2$$. If $$k_x =1$$ for an $$x \in X$$, then $$x$$ is unramified. The image point $$y=f(x) \in Y$$ of a ramification point is called a branch point.

Degree of a holomorphic map
Let $$f: X \rightarrow Y$$ be a non-constant, holomorphic map between compact Riemann surfaces. The degree $$\operatorname{deg}(f)$$ of $$f$$ is the cardinality of the fiber of an unramified point $$y=f(x) \in Y$$, i.e. $$\operatorname{deg}(f):=|f^{-1}(y)|$$.

This number is well-defined, since for every $$y \in Y$$ the fiber $$f^{-1}(y)$$ is discrete and for any two unramified points $$y_1,y_2 \in Y$$, it is: $$|f^{-1}(y_1)|=|f^{-1}(y_2)|.$$

It can be calculated by:
 * $$\sum_{x \in f^{-1}(y)} k_x = \operatorname{deg}(f)$$

Definition
A continuous map $$f: X \rightarrow Y$$ is called a branched covering, if there exists a closed set with dense complement $$E \subset Y$$, such that $$f_{|X \smallsetminus f^{-1}(E)}:X \smallsetminus f^{-1}(E) \rightarrow Y \smallsetminus E$$ is a covering.

Examples

 * Let $$n \in \mathbb{N}$$ and $$n \geq 2$$, then $$f:\mathbb{C} \rightarrow \mathbb{C}$$ with $$f(z)=z^n$$ is branched covering of degree $$n$$, where by $$z=0$$ is a branch point.
 * Every non-constant, holomorphic map between compact Riemann surfaces $$f: X \rightarrow Y$$ of degree $$d$$ is a branched covering of degree $$d$$.

Definition
Let $$p: \tilde X \rightarrow X$$ be a simply connected covering. If $$\beta : E \rightarrow X$$ is another simply connected covering, then there exists a uniquely determined homeomorphism $$\alpha : \tilde X \rightarrow E$$, such that the diagram commutes.

This means that $$p$$ is, up to equivalence, uniquely determined and because of that universal property denoted as the universal covering of the space $$X$$.

Existence
A universal covering does not always exist, but the following properties guarantee its existence:

Let $$X$$ be a connected, locally simply connected topological space; then, there exists a universal covering $$p:\tilde X \rightarrow X$$.

$$\tilde X$$ is defined as $$\tilde X := \{\gamma:\gamma \text{ is a path in }X \text{ with }\gamma(0) = x_0 \}/\text{ homotopy with fixed ends} $$ and $$p:\tilde X \rightarrow X$$ by $$p([\gamma]):=\gamma(1)$$.

The topology on $$\tilde X$$ is constructed as follows: Let $$\gamma:I \rightarrow X$$ be a path with $$\gamma(0)=x_0$$. Let $$U$$ be a simply connected neighborhood of the endpoint $$x=\gamma(1)$$, then for every $$y \in U$$ the paths $$\sigma_y$$ inside $$U $$ from $$x$$ to $$y$$ are uniquely determined up to homotopy. Now consider $$\tilde U:=\{\gamma.\sigma_y:y \in U \}/\text{ homotopy with fixed ends}$$, then $$p_{|\tilde U}: \tilde U \rightarrow U$$ with $$p([\gamma.\sigma_y])=\gamma.\sigma_y(1)=y$$ is a bijection and $$\tilde U$$ can be equipped with the final topology of $$p_{|\tilde U}$$.

The fundamental group $$\pi_{1}(X,x_0) = \Gamma$$ acts freely through $$([\gamma],[\tilde x]) \mapsto [\gamma.\tilde x]$$ on $$\tilde X$$ and $$\psi:\Gamma \backslash \tilde X \rightarrow X$$ with $$\psi([\Gamma \tilde x])=\tilde x(1)$$ is a homeomorphism, i.e. $$\Gamma \backslash \tilde X \cong X $$.

Examples

\exp(2 \pi i t) & 0\\ 0 & I_{n-1} \end{bmatrix}_\vphantom{x} A $$ is the universal covering of the unitary group $$U(n)$$. X = \bigcup_{n\in \N}\left\{(x_1,x_2)\in\R^{2} : \Bigl(x_1-\frac{1}{n}\Bigr)^2+x_2^2=\frac{1}{n^2}\right\} $$ One can show that no neighborhood of the origin $$(0,0)$$ is simply connected.
 * $$r : \mathbb{R} \to S^1$$ with $$r(t)=(\cos(2 \pi t), \sin(2 \pi t))$$ is the universal covering of the unit circle $$S^1$$.
 * $$p : S^n \to \mathbb{R}P^n \cong \{+1,-1\}\backslash S^n$$ with $$p(x)=[x]$$ is the universal covering of the projective space $$\mathbb{R}P^n$$ for $$n>1$$.
 * $$q : \mathrm{SU}(n) \ltimes \mathbb{R} \to U(n)$$ with $$q(A,t)= \begin{bmatrix}
 * Since $$\mathrm{SU}(2) \cong S^3$$, it follows that the quotient map $$f : \mathrm{SU}(2) \rightarrow \mathrm{SU}(2) \backslash \mathbb{Z_2} \cong \mathrm{SO}(3)$$ is the universal covering of the $$\mathrm{SO}(3)$$.
 * A topological space which has no universal covering is the Hawaiian earring: $$

G-coverings
Let G be a discrete group acting on the topological space X. This means that each element g of G is associated to a homeomorphism Hg of X onto itself, in such a way that Hg h is always equal to Hg ∘ Hh for any two elements g and h of G. (Or in other words, a group action of the group G on the space X is just a group homomorphism of the group G into the group Homeo(X) of self-homeomorphisms of X.) It is natural to ask under what conditions the projection from X to the orbit space X/G is a covering map. This is not always true since the action may have fixed points. An example for this is the cyclic group of order 2 acting on a product X × X by the twist action where the non-identity element acts by (x, y) ↦ (y, x). Thus the study of the relation between the fundamental groups of X and X/G is not so straightforward.

However the group G does act on the fundamental groupoid of X, and so the study is best handled by considering groups acting on groupoids, and the corresponding orbit groupoids. The theory for this is set down in Chapter 11 of the book Topology and groupoids referred to below. The main result is that for discontinuous actions of a group G on a Hausdorff space X which admits a universal cover, then the fundamental groupoid of the orbit space X/G is isomorphic to the orbit groupoid of the fundamental groupoid of X, i.e. the quotient of that groupoid by the action of the group G. This leads to explicit computations, for example of the fundamental group of the symmetric square of a space.

Definition
Let $$p:E \rightarrow X$$ be a covering. A deck transformation is a homeomorphism $$d:E \rightarrow E$$, such that the diagram of continuous maps commutes. Together with the composition of maps, the set of deck transformation forms a group $$\operatorname{Deck}(p)$$, which is the same as $$\operatorname{Aut}(p)$$.

Now suppose $$p:C \to X$$ is a covering map and $$C$$ (and therefore also $$X$$) is connected and locally path connected. The action of $$\operatorname{Aut}(p)$$ on each fiber is transitive. If this action is free on some fiber, then it is free on all fibers, and we call the cover regular (or normal or Galois). Every such regular cover is a principal $G$-bundle, where $$G = \operatorname{Aut}(p)$$ is considered as a discrete topological group.

Every universal cover $$p:D \to X $$ is regular, with deck transformation group being isomorphic to the fundamental group $\pi_1(X)$.

Examples

 * Let $$q : S^1 \to S^1$$ be the covering $$q(z)=z^{n}$$ for some $$n \in \mathbb{N} $$, then the map $$d_k:S^1 \rightarrow S^1 : z \mapsto z \, e^{2\pi ik/n} $$ for $$k \in \mathbb{Z}$$ is a deck transformation and $$\operatorname{Deck}(q)\cong \mathbb{Z}/ n\mathbb{Z}$$.
 * Let $$r : \mathbb{R} \to S^1$$ be the covering $$r(t)=(\cos(2 \pi t), \sin(2 \pi t))$$, then the map $$d_k:\mathbb{R} \rightarrow \mathbb{R} : t \mapsto t + k$$ for $$k \in \mathbb{Z}$$ is a deck transformation and $$\operatorname{Deck}(r)\cong \mathbb{Z}$$.
 * As another important example, consider $$\Complex$$ the complex plane and $$\Complex^{\times}$$ the complex plane minus the origin. Then the map $$p: \Complex^{\times} \to \Complex^{\times}$$ with $$ p(z) = z^{n} $$ is a regular cover. The deck transformations are multiplications with $$n$$-th roots of unity and the deck transformation group is therefore isomorphic to the cyclic group $$\Z/n\Z$$. Likewise, the map $$\exp : \Complex \to \Complex^{\times}$$ with $$\exp(z) = e^{z}$$ is the universal cover.

Properties
Let $$X$$ be a path-connected space and $$p:E \rightarrow X$$ be a connected covering. Since a deck transformation $$d:E \rightarrow E$$ is bijective, it permutes the elements of a fiber $$p^{-1}(x)$$ with $$x \in X$$ and is uniquely determined by where it sends a single point. In particular, only the identity map fixes a point in the fiber. Because of this property every deck transformation defines a group action on $$E$$, i.e. let $$U \subset X$$ be an open neighborhood of a $$x \in X$$ and $$\tilde U \subset E$$ an open neighborhood of an $$e \in p^{-1}(x)$$, then $$\operatorname{Deck}(p) \times E \rightarrow E: (d,\tilde U)\mapsto d(\tilde U)$$ is a group action.

Definition
A covering $$p:E \rightarrow X$$ is called normal, if $$\operatorname{Deck}(p) \backslash E \cong X$$. This means, that for every $$x \in X$$ and any two $$e_0,e_1 \in p^{-1}(x)$$ there exists a deck transformation $$d:E \rightarrow E$$, such that $$d(e_0)=e_1$$.

Properties
Let $$X$$ be a path-connected space and $$p:E \rightarrow X$$ be a connected covering. Let $$H=p_{\#}(\pi_1(E))$$ be a subgroup of $$\pi_1(X)$$, then $$p$$ is a normal covering iff $$H$$ is a normal subgroup of $$\pi_1(X)$$.

If $$p:E \rightarrow X$$ is a normal covering and $$H=p_{\#}(\pi_1(E))$$, then $$\operatorname{Deck}(p) \cong \pi_1(X)/H$$.

If $$p:E \rightarrow X$$ is a path-connected covering and $$H=p_{\#}(\pi_1(E))$$, then $$\operatorname{Deck}(p) \cong N(H)/H$$, whereby $$N(H)$$ is the normaliser of $$H$$.

Let $$E$$ be a topological space. A group $$\Gamma$$ acts discontinuously on $$E$$, if every $$e \in E$$ has an open neighborhood $$V \subset E$$ with $$V \neq \empty$$, such that for every $$d_1, d_2 \in \Gamma $$ with $$d_1 V \cap d_2 V \neq \empty $$ one has $$d_1 = d_2$$.

If a group $$\Gamma$$ acts discontinuously on a topological space $$E$$, then the quotient map $$q: E \rightarrow \Gamma \backslash E $$ with $$q(e)=\Gamma e$$ is a normal covering. Hereby $$\Gamma \backslash E = \{\Gamma e: e \in E\}$$ is the quotient space and $$\Gamma e = \{\gamma(e):\gamma \in \Gamma\}$$ is the orbit of the group action.

Examples

 * The covering $$q : S^1 \to S^1 $$ with $$q(z)=z^{n}$$ is a normal coverings for every $$n \in \mathbb{N}$$.
 * Every simply connected covering is a normal covering.

Calculation
Let $$\Gamma$$ be a group, which acts discontinuously on a topological space $$E$$ and let $$q: E \rightarrow \Gamma \backslash E $$ be the normal covering.

 * If $$E$$ is path-connected, then $$\operatorname{Deck}(q) \cong \Gamma$$.

 * If $$E$$ is simply connected, then $$\operatorname{Deck}(q)\cong \pi_1(\Gamma \backslash E)$$.

Examples

 * Let $$n \in \mathbb{N}$$. The antipodal map $$g:S^n \rightarrow S^n$$ with $$g(x)=-x$$ generates, together with the composition of maps, a group $$D(g) \cong \mathbb{Z/2Z}$$ and induces a group action $$D(g) \times S^n \rightarrow S^n, (g,x)\mapsto g(x)$$, which acts discontinuously on $$S^n$$. Because of $$\mathbb{Z_2} \backslash S^n \cong \mathbb{R}P^n$$ it follows, that the quotient map $$q : S^n \rightarrow \mathbb{Z_2}\backslash S^n \cong \mathbb{R}P^n$$ is a normal covering and for $$n > 1$$ a universal covering, hence $$\operatorname{Deck}(q)\cong \mathbb{Z/2Z}\cong \pi_1({\mathbb{R}P^n})$$ for $$n > 1$$.
 * Let $$\mathrm{SO}(3)$$ be the special orthogonal group, then the map $$f : \mathrm{SU}(2) \rightarrow \mathrm{SO}(3) \cong \mathbb{Z_2} \backslash \mathrm{SU}(2)$$ is a normal covering and because of $$\mathrm{SU}(2) \cong S^3$$, it is the universal covering, hence $$\operatorname{Deck}(f) \cong \mathbb{Z/2Z} \cong \pi_1(\mathrm{SO}(3))$$.
 * With the group action $$(z_1,z_2)*(x,y)=(z_1+(-1)^{z_2}x,z_2+y)$$ of $$\mathbb{Z^2}$$ on $$\mathbb{R^2}$$, whereby $$(\mathbb{Z^2},*)$$ is the semidirect product $$\mathbb{Z} \rtimes \mathbb{Z} $$, one gets the universal covering $$f: \mathbb{R^2} \rightarrow (\mathbb{Z} \rtimes \mathbb{Z}) \backslash \mathbb{R^2} \cong K $$ of the klein bottle $$K$$, hence $$\operatorname{Deck}(f) \cong \mathbb{Z} \rtimes \mathbb{Z} \cong \pi_1(K)$$.
 * Let $$T = S^1 \times S^1$$ be the torus which is embedded in the $$\mathbb{C^2}$$. Then one gets a homeomorphism $$\alpha: T \rightarrow T: (e^{ix},e^{iy}) \mapsto (e^{i(x+\pi)},e^{-iy})$$, which induces a discontinuous group action $$G_{\alpha} \times T \rightarrow T$$, whereby $$G_{\alpha} \cong \mathbb{Z/2Z}$$. It follows, that the map $$f: T \rightarrow G_{\alpha} \backslash T \cong K$$ is a normal covering of the klein bottle, hence $$\operatorname{Deck}(f) \cong \mathbb{Z/2Z}$$.
 * Let $$S^3$$ be embedded in the $$\mathbb{C^2}$$. Since the group action $$S^3 \times \mathbb{Z/pZ} \rightarrow S^3: ((z_1,z_2),[k]) \mapsto (e^{2 \pi i k/p}z_1,e^{2 \pi i k q/p}z_2)$$ is discontinuously, whereby $$p,q \in \mathbb{N}$$ are coprime, the map $$f:S^3 \rightarrow \mathbb{Z_p} \backslash S^3 =: L_{p,q}$$ is the universal covering of the lens space $$L_{p,q}$$, hence $$\operatorname{Deck}(f) \cong \mathbb{Z/pZ} \cong \pi_1(L_{p,q})$$.

Galois correspondence
Let $$X$$ be a connected and locally simply connected space, then for every subgroup $$H\subseteq \pi_1(X)$$ there exists a path-connected covering $$\alpha:X_H \rightarrow X$$ with $$\alpha_{\#}(\pi_1(X_H))=H$$.

Let $$p_1:E \rightarrow X$$ and $$p_2: E' \rightarrow X$$ be two path-connected coverings, then they are equivalent iff the subgroups $$H = p_{1\#}(\pi_1(E))$$ and $$H'=p_{2\#}(\pi_1(E'))$$ are conjugate to each other.

Let $$X$$ be a connected and locally simply connected space, then, up to equivalence between coverings, there is a bijection:

$$ \begin{matrix} \qquad \displaystyle \{\text{Subgroup of }\pi_1(X)\} & \longleftrightarrow & \displaystyle \{\text{path-connected covering } p:E \rightarrow X\} \\ H & \longrightarrow & \alpha:X_H \rightarrow X \\ p_\#(\pi_1(E))&\longleftarrow & p \\ \displaystyle \{\text{normal subgroup of }\pi_1(X)\} & \longleftrightarrow & \displaystyle \{\text{normal covering } p:E \rightarrow X\} \end{matrix} $$

For a sequence of subgroups $$ \displaystyle \{\text{e}\} \subset H \subset G \subset \pi_1(X) $$ one gets a sequence of coverings $$ \tilde X \longrightarrow X_H \cong H \backslash \tilde X \longrightarrow X_G \cong G \backslash \tilde X \longrightarrow X\cong \pi_1(X) \backslash \tilde X $$. For a subgroup $$ H \subset \pi_1(X) $$ with index $$ \displaystyle[\pi_1(X):H] = d $$, the covering $$ \alpha:X_H \rightarrow X $$ has degree $$d$$.

Category of coverings
Let $$X$$ be a topological space. The objects of the category $$\boldsymbol{Cov(X)}$$ are the coverings $$p:E \rightarrow X$$ of $$X$$ and the morphisms between two coverings $$p:E \rightarrow X$$ and $$q:F\rightarrow X$$ are continuous maps $$f:E \rightarrow F$$, such that the diagram commutes.

G-Set
Let $$G$$ be a topological group. The category $$\boldsymbol{G-Set}$$ is the category of sets which are G-sets. The morphisms are G-maps $$\phi:X \rightarrow Y$$ between G-sets. They satisfy the condition $$\phi(gx)=g \, \phi(x)$$ for every $$g \in G$$.

Equivalence
Let $$X$$ be a connected and locally simply connected space, $$x \in X$$ and $$G = \pi_1(X,x)$$ be the fundamental group of $$X$$. Since $$G$$ defines, by lifting of paths and evaluating at the endpoint of the lift, a group action on the fiber of a covering, the functor $$F:\boldsymbol{Cov(X)} \longrightarrow \boldsymbol{G-Set}: p \mapsto p^{-1}(x)$$ is an equivalence of categories.

Applications
[[Image:Rotating gimbal-xyz.gif|thumb|300px|[[Gimbal lock]] occurs because any map T3 → RP3 is not a covering map. In particular, the relevant map carries any element of T3, that is, an ordered triple (a,b,c) of angles (real numbers mod 2$\pi$), to the composition of the three coordinate axis rotations Rx(a)∘Ry(b)∘Rz(c) by those angles, respectively. Each of these rotations, and their composition, is an element of the rotation group SO(3), which is topologically RP3.

This animation shows a set of three gimbals mounted together to allow three degrees of freedom. When all three gimbals are lined up (in the same plane), the system can only move in two dimensions from this configuration, not three, and is in gimbal lock. In this case it can pitch or yaw, but not roll (rotate in the plane that the axes all lie in).]]

An important practical application of covering spaces occurs in charts on SO(3), the rotation group. This group occurs widely in engineering, due to 3-dimensional rotations being heavily used in navigation, nautical engineering, and aerospace engineering, among many other uses. Topologically, SO(3) is the real projective space RP3, with fundamental group Z/2, and only (non-trivial) covering space the hypersphere S3, which is the group Spin(3), and represented by the unit quaternions. Thus quaternions are a preferred method for representing spatial rotations – see quaternions and spatial rotation.

However, it is often desirable to represent rotations by a set of three numbers, known as Euler angles (in numerous variants), both because this is conceptually simpler for someone familiar with planar rotation, and because one can build a combination of three gimbals to produce rotations in three dimensions. Topologically this corresponds to a map from the 3-torus T3 of three angles to the real projective space RP3 of rotations, and the resulting map has imperfections due to this map being unable to be a covering map. Specifically, the failure of the map to be a local homeomorphism at certain points is referred to as gimbal lock, and is demonstrated in the animation at the right – at some points (when the axes are coplanar) the rank of the map is 2, rather than 3, meaning that only 2 dimensions of rotations can be realized from that point by changing the angles. This causes problems in applications, and is formalized by the notion of a covering space.

title: Covering space


text:
George Bowering

George Harry Bowering, (born December 1, 1935) is a prolific Canadian novelist, poet, historian, and biographer. He was the first Canadian Parliamentary Poet Laureate.

Life and career
He was born in Penticton, British Columbia, and raised in the nearby town of Oliver, where his father was a high-school chemistry teacher. Bowering is author of more than 100 books.

Bowering is the best-known of a group of young poets including Lionel Kearns, Frank Davey, Fred Wah, Jamie Reid, and David Dawson, who studied together at the University of British Columbia in the 1960s. There they founded the journal TISH. In the late 1960s Bowering's work was published in 0 to 9 magazine, an avant-garde journal which experimented with language and meaning-making.

Bowering lives in Vancouver, British Columbia, and is Professor Emeritus at Simon Fraser University, where he worked for 30 years. Never having written as an adherent of organized religion, he has in the past wryly described himself as a Baptist agnostic. In 2002, Bowering was appointed the first ever Canadian Parliamentary Poet Laureate. That same year, he was made an Officer of the Order of Canada. He was awarded the Order of British Columbia in 2004.

When the Indian Hungryalist, also known as Hungry generation, poet Malay Roy Choudhury, was arrested at Kolkata, India, Bowering brought out a special issue of Imago for helping the Indian poet in his trial.

Bowering was one of the judges for the 2008 Griffin Poetry Prize.

Novels

 * 1967: Mirror on the Floor, Toronto: McClelland & Stewart; Vancouver: Anvil Press, 2014
 * 1973: Fiddler's Night, Vancouver: Dos Equis
 * 1977: A Short Sad Book, Vancouver: Talonbooks; Vancouver: New Star Books, 2017
 * 1980: Burning Water, Toronto: Musson Book Company; Vancouver: New Star Books, 2007
 * French edition: En eaux troubles, translated by L.-Philippe Hébert, Montreal: Quinze, 1982
 * 1988: Caprice, Toronto/New York: Penguin; Vancouver: New Star Books, 2010
 * 1990: Harry's Fragments, Vancouver: Talonbooks
 * 1994: Shoot!, Toronto: Key Porter Books; Vancouver: New Star Books, 2008
 * 1998: Piccolo Mondo (with Angela Bowering, Michael Matthews & David Bromige), Toronto: Couch House Books
 * 2012: Pinboy, Toronto: Cormorant Books
 * 2015: Writing the Okanagan, Vancouver: Talonbooks
 * 2018: No One, Toronto: ECW Press

Short fiction

 * 1974: Flycatcher and Other Stories, Ottawa: Oberon Press
 * 1977: Concentric Circles, Windsor: Black Moss Press
 * 1978: Protective Footwear, Toronto: McClelland & Stewart
 * 1983: A Place to Die, Ottawa: Oberon Press
 * 1994: The Rain Barrel, Vancouver: Talonbooks
 * 2004: Standing on Richards, Toronto: Viking
 * 2009: The Box, Vancouver: New Star Books
 * 2015: Ten Women, Vancouver: Anvil Press

Young adult fiction

 * 1994: Parents from Space, Montreal: Roussan
 * 1998: Diamondback Dog, Montreal: Roussan
 * 2015: Attack of the Toga Gang, Toronto: Dancing Cat Books

Book-length poems

 * 1967: Baseball, Toronto: Coach House Books
 * 1970: George,Vancouver, Kitchener, Weed/Flower
 * 1971: Geneve, Toronto: Coach House Books
 * 1972: Autobiology, Vancouver: New Star Books, 1972; Vancouver, Pooka Press, 2006
 * 1972: Curious, Toronto: Coach House Books
 * 1974: At War With the U.S., Vancouver: Talonbooks
 * 1976: Allophanes, Toronto: Coach House Books
 * 1982: Ear Reach, Vancouver: Alcuin
 * 1984: Kerrisdale Elegies, Toronto: Coach House Books; Vancouver, Pooka Press, 2008; Vancouver: Talonbooks, 2008
 * Italian edition: Elegie di Kerrisdale, translated by Annalisa Goldoni, Rome: Edizioni Empiria, 1996
 * 2000: His Life: a poem, Toronto: ECW Press
 * 2010: My Darling Nellie Grey, Vancouver: Talonbooks
 * 2019: Taking Measures, Vancouver: Talonbooks

Collections of poems (including gathered long poems)

 * 1963: Sticks & Stones, Vancouver: Tishbooks; Vancouver: Talonbooks, 1989
 * 1964: Points on the Grid, Toronto: Contact Press
 * 1965: The Man in Yellow Boots/ El hombre de las botas amarillas, Mexico: Ediciones El Corno
 * 1966: The Silver Wire, Kingston, Quarry Press
 * 1969: Rocky Mountain Foot, Toronto: McClelland & Stewart
 * 1969: The Gangs of Kosmos, Toronto: House of Anansi Press
 * 1971: Touch: selected poems 1960-1969, Toronto: McClelland & Stewart
 * 1974: In the Flesh, Toronto: McClelland & Stewart
 * 1976: The Catch, Toronto: McClelland & Stewart
 * 1976: Poem & Other Baseballs, Windsor: Black Moss Press
 * 1977: The Concrete Island, Montreal: Véhicule Press
 * 1979: Another Mouth, Toronto: McClelland & Stewart
 * 1981: Particular Accidents: selected poems, Vancouver: Talonbooks
 * 1982: West Window: selected poetry, Toronto: General
 * 1982: Smoking Mirror, Edmonton: Longspoon
 * 1985: Seventy-One Poems for People, Red Deer: RDC Press
 * 1986: Delayed Mercy & other poems, Toronto: Coach House Books
 * 1992: Urban Snow, Vancouver: Talonbooks
 * 1993: George Bowering Selected: Poems 1961-1992, Toronto: McClelland & Stewart
 * 1997: Blonds on Bikes, Vancouver: Talonbooks
 * 2004: Changing on the Fly, Vancouver, Polestar Press
 * 2006: Vermeer's Light: Poems 1996-2006, Vancouver, Talonbooks
 * 2013: Teeth: Poems 2006-2011, Toronto: Mansfield Press
 * 2015: The World, I Guess, Vancouver: New Star Books
 * 2018: Some End, Vancouver: New Star Books

Criticism

 * 1970: Al Purdy, Toronto: Copp Clark
 * 1971: Robert Duncan: an Interview, Toronto: Coach House/Beaver Kosmos
 * 1979: Three Vancouver Writers, Toronto: Open Letter/Coach House
 * 1982: A Way With Words, Ottawa: Oberon Press
 * 1983: The Mask in Place, Winnipeg: Turnstone Press
 * 1985: Craft Slices, Ottawa: Oberon Press
 * 1988: Errata, Red Deer: RDC Press
 * 1988: Imaginary Hand, Edmonton: NeWest Press
 * 2005: Left Hook, Vancouver: Raincoast Books
 * 2010: Horizontal Surfaces, Toronto: BookThug
 * 2012: Words, Words, Words, Vancouver: New Star Books
 * 2019: Writing and Reading, Vancouver: New Star Books

Chapbooks

 * 1967: How I Hear Howl, Montreal: Beaver Kosmos
 * 1969: Two Police Poems, Vancouver: Talonbooks
 * 1972: The Sensible, Toronto: Massasauga
 * 1973: Layers 1-13, Kitchener: Weed/Flower
 * 1977: In Answer, Vancouver: William Hoffer
 * 1980: Uncle Louis, Toronto: Coach House Books
 * 1985: Spencer & Groulx, Vancouver: William Hoffer
 * 1991: Quarters, Prince George: Gorse Press (Winner, bp Nichol chapbook award 1991)
 * 1992: Do Sink, Vancouver: Pomflit (Winner, bp Nichol chapbook award, 1992).
 * 1992: Sweetly, Vancouver: Wuz
 * 1997: Blondes on Bikes, Ottawa: Above Ground
 * 1998: A, You're Adorable, Ottawa: Above Ground
 * 2000: 6 Little Poems in Alphabetical Order, Calgary: House Press
 * 2001: Some Writers, Calgary: House Press
 * 2002: Joining the Lost Generation, Calgary: House Press
 * 2004: Lost in the Library, Ellsworth, ME: Backwoods Broadsides
 * 2005: Rewriting my Grandfather, Vancouver: Nomados
 * 2006: Crows in the Wind, Toronto: BookThug
 * 2006: A Knot of Light, Calgary: No Press
 * 2007: Montenegro 1966, Calgary: No Press
 * 2007: U.S. Sonnets, Vancouver: Pooka Press
 * 2007: Eggs in There, Edmonton: Rubicon
 * 2007: Some Answers, Mt. Pleasant, ON: LaurelReed Books
 * 2007: Horizontal Surfaces, Edmonton: Olive Collective
 * 2007: Tocking Heads, Edmonton: above/ground
 * 2008: There Then, Prince George: Gorse Press
 * 2008: Animals, Beasts, Critters, Vancouver: JB Objects
 * 2008: Valley, Calgary: No Press
 * 2008: Fulgencio, Vancouver: Nomados
 * 2008: According to Brueghel, North Vancouver: Capilano
 * 2008: Shall I Compare, Penticton: Beaver Kosmos
 * 2009: A Little Black Strap, St. Paul: Unarmed
 * 2013: Los Pájaros de Tenacatita, Ootischenia: Nose-in-Book
 * 2016: Sitting in Jalisco, Ootischenia: Nose-in-Book
 * 2016: That Toddlin Town, Ottawa: above/ground books
 * 2019: David in Byzantium, Cobourg: Proper Tales Press

Memoirs

 * 1993: The Moustache: Memories of Greg Curnoe, Toronto: Coach House Books
 * 2001: A Magpie Life, Toronto: Key Porter
 * 2002: Cars, Toronto: Coach House Books
 * 2006: Baseball Love, Vancouver: Talonbooks
 * 2011: How I Wrote Certain of my Books, Toronto: Mansfield Press
 * 2016: The Hockey Scribbler, Toronto: ECW Press
 * 2016: The Dad Dialogues (with Charles Demers), Vancouver: Arsenal Pulp Press

History

 * 1996: Bowering's B.C., Toronto: Viking
 * 1999: Also published in 2000 by Penguin. ISBN 978-0-14-027550-6..
 * 2003: Stone Country, Toronto: Viking

Plays

 * 1962: "The Home for Heroes", Vancouver: Prism
 * 1966: "What Does Eddie Williams Want?", Montreal: CBC-TV
 * 1972: "George Vancouver", Vancouver: CBC radio network
 * 1973: "Sitting in Mexico", Vancouver: CBC radio network
 * 1986: "Music in the Park", Vancouver: CBC radio network
 * 1989: "The Great Grandchildren of Bill Bissett's Mice", Vancouver: CBC radio network

Editions

 * The 1962 Poems of R.S. Lane, Toronto: Ganglia Press, 1965
 * Vibrations: poems of youth, Toronto, Gage, 1970
 * The Story so Far, Toronto, Coach House, 1972
 * The City in her Eyes by David Cull, Vancouver: Vancouver Community Press, 1972
 * Imago Twenty, Vancouver, Talonbooks, 1974
 * Cityflowers by Artie Gold, Montreal: Delta Canada, 1974
 * Letters from Geeksville: letters from Red Lane 1960-64, Prince George: Caledonia Writing Series, 1976
 * Great Canadian Sports Stories, Ottawa: Oberon Press, 1979
 * Fiction of Contemporary Canada, Toronto: Coach House Books, 1980
 * Loki is Buried at Smoky Creek: selected poems of Fred Wah, Vancouver: Talonbooks, 1981
 * My Body was Eaten by Dogs: selected poems of David McFadden, Toronto: McClelland & Stewart, New York: CrossCountry, 1981
 * "1945-1980," in Introduction to Poetry: British, American, Canadian, David and Lecker, Toronto: Holt, Rinehart, and Winston, 1981
 * The Contemporary Canadian Poem Anthology, Toronto: Coach House Books, 1983
 * Sheila Watson and The Double Hook: the artist and her critics, Ottawa: Golden Dog Press, 1984
 * Taking the Field:the best of baseball fiction, Red Deer: RDC Press, 1990
 * Likely Stories: a postmodern sampler (with Linda Hutcheon), Toronto: Coach House Books, 1992
 * An H in the Heart: Selected works of bpNichol (with Michael Ondaatje), Toronto: McClelland & Stewart, 1994
 * And Other Stories, Vancouver: Talonbooks, 2001
 * The 2008 Griffin Poetry Prize Anthology, Toronto: House of Anansi, 2008
 * The Heart Does Break (with Jean Baird), Toronto: Random House, 2009

Editor

 * Tish, Vancouver, 1961–63
 * Imago, Calgary, London, Montreal, Vancouver, 1964-1974
 * Beaver Kosmos Folios, Calgary, London, Montreal, Vancouver, 1966–75.

About

 * A Record of Writing: an annotated and Illustrated Bibliography of George Bowering by Roy Miki, Vancouver: Talonbooks, 1989
 * Essays on Canadian Writing, George Bowering issue ed. Ken Norris, 1989
 * George Bowering: Bright Circles of Colour by Eva-Marie Kroller, Vancouver: Talonbooks, 1992
 * George Bowering and His Works by John Harris, Toronto: ECW Press, 1992
 * 71 (+) for GB ed. Jean Baird, David McFadden and George Stanley, Vancouver/Toronto: (printed at) Coach House Books, 2005
 * He Speaks Volumes: A Biography of George Bowering by Rebecca Wigod, Vancouver: Talonbooks, 2018

title: George Bowering


text:
Talk:Maureen Connolly

Tragedy struck again?
There is a paragraph starting "Tragedy struck again" without the article having said what the first tragedy was. I guess it is her horse accident in 1955, when she was only 19, leading to her never being able to play tennis again. Gmackematix (talk) 02:29, 26 June 2009 (UTC)

 * I would certainly called her leg being crushed and not being able to play tennis a tragedy, so "tragedy struck again" sounds ok to me. Fyunck(click) (talk) 02:52, 26 June 2009 (UTC)

According to Nancy Brinker,head of the Susan B Kommen foundation, Little Mo had ovarian cancer,not stomach cancer. —Preceding unsigned comment added by 67.173.124.178 (talk) 03:26, 17 April 2011 (UTC)

 * Unless you have a source for that, it's WP:OR. News articles at the time of Connolly's death specify no specific cancer. Just that she had a operation for removal of a stomach tumor. That might mean stomach cancer except we all know cancer metastasizes and that could have been the case with Connolly....William 15:56, 7 June 2012 (UTC)

Lew Hoad
is shown in the mixed-doubles table with an American flag against his name. However, he was Australian. 78.147.225.71 (talk) 18:28, 27 March 2013 (UTC)
 * ✅ - yep, an error that's been there a long time. Thanks. Fyunck(click) (talk) 18:35, 27 March 2013 (UTC)

Assessment comment
Substituted at 23:28, 29 April 2016 (UTC)

Update to better compare with other greats
It would be useful if this entry also had the timelines for doubles and mixed doubles and a records section to better appreciate the short but impressive career and achievements of Connolly. Antipodenz (talk) 04:05, 1 January 2024 (UTC)

title: Talk:Maureen Connolly