Talk:Covering space

Surjectivity

Is there any reason why the word "surjective" does not appear in the informal definition, but it appears in the formal one a couple of lines further down. I would prefer adding it there, too. (I am fine with informal definitions, as long as they are really equivalent to the formal ones and do not "define" something else). — Preceding unsigned comment added by 2A00:5BA0:20:67:0:0:0:104 (talk) 14:26, 18 September 2018 (UTC)[reply]

Surjectivity is implicit in the informal statement "each point in X has ...", as in each point is covered. Anyway, it's a technical condition that matters only for disconnected base spaces, otherwise I think it can be proved from the remainder of the definition; I recall some authors do not even require surjectivity in general. Tokenzero (talk) 14:40, 18 September 2018 (UTC)[reply]

Overlap

User:Charles Matthews There is now some overlap between the content here and at local homeomorphism.

The way covering map has been defined allows it not to be surjective (the condition holds vacuously for points with empty pre-image); the usual definition has a covering map being surjective. I think surjective should be added to the definition since that's what is needed for most purposes.

Just noticed that one property that a covering map is supposed to have, according to whoever made the page, is being surjective. So I'll add 'surjective' to the definition.

I removed this paragraph of mine:

The composition of two covering maps need not be a covering map: consider the unit circle S¹ as a subset of the complex plane, and for any natural number n define p_n : S¹ → S¹ by p_n(z) = z⁻ⁿ. Consider the map p : S¹ × N → S¹ × N by p(z,n) = (p_n(z),n). If N is equipped with the discrete topology and S¹ × N carries the product topology, then p is a covering map. The natural projection q : S¹ × N → S¹ defined by q(z,n) = z is obviously a covering map. The composition qp : S¹ × N → S¹ is not: no matter how small an open set U you pick in S¹, there will always be an n large enough so that p_n⁻¹(U) = S¹ which cannot be isomorphic to U.

The last statement, p_n⁻¹(U) = S¹, is false, and that kills the whole argument. I don't know if the composition of two covering maps is always again a covering map. AxelBoldt 14:52, 23 Nov 2003 (UTC)

Huh? I'm often muddle-headed and confused, but ... the last statement is perfectly true. What's false is the statement that p is a covering map. The problem being that p restricted to to the inverse image S¹ does not produce a homeomorphism to U. That is, one can always find an n large enough so that p_n(S¹) is not equal to U; thus p was never a covering to begin with. Changing -n to +n in the definition would make p into a covering. Interesting example, though. Non-trival fundamental group. linas 15:47, 3 Apr 2005 (UTC)

The composition of two covering maps need not be a covering map. To devise a counterexample is an exercise in the Munkres book (which I don't have ready to hand, so I can't give a page citation). It is also an exercise (with a hint) in Allen Hatcher's book Algebraic Topology, which is available online at http://www.math.cornell.edu/~hatcher/AT/ATpage.html. (See exercise 6 on page 79.) The question comes up frequently in discussions at the "Topology Atlas" web site (http://at.yorku.ca/topology/); see e.g. http://at.yorku.ca/cgi-bin/bbqa?forum=ask_an_algebraic_topologist;task=show_msg;msg=0002.0001. If p and q are (composable) covering maps and every fiber q^-1(x) is finite, then the composition qp is a covering map; see http://at.yorku.ca/cgi-bin/bbqa?forum=ask_a_topologist_2002&task=show_msg&msg=0348.0001 for a proof. --Logician1989 19:41, 10 March 2006 (UTC)[reply]

Would someone who knows what is meant by the "opposite" of a group like to make a stub/redirection? I can't find anything on this. — Preceding unsigned comment added by 163.1.237.46 (talk) 01:52, 26 February 2004 (UTC)[reply]

Special case of dual (category theory); anyway like defining g*h = hg.

Charles Matthews 09:06, 26 Feb 2004 (UTC)

Ah yes, of course. I've added links - that ok? — Preceding unsigned comment added by 163.1.237.46 (talk) 12:51, 26 February 2004 (UTC)[reply]

Deck transformations and fixed points

In the first paragraph of "Deck transformation group, regular covers", it is claimes that a non-trivial deck transformation has no fixed points. I think that's false. Take C to be the disjoint union of X, X and X and p to be the "disjoint union" of id, id and id (in the obvious way). Then you could define f to swap the first to copies of X. This is a deck transformation with all the points of the third copy of X as fixed points.

Maybe the statement is true if C is simply connected. -- Sven — Preceding unsigned comment added by 80.184.149.245 (talk) 08:59, 27 July 2005 (UTC)[reply]

If a topological space is path connected and locally path connected and its covering space Y is path connected, then the statement is true. The statement in Allen Hatcher's algebraic topology text is the following: each point in Y has a neighborhood U such that all the images g(U) are disjoint for varying deck transformations g. That is, the only element of the group of deck transformations that has a fixed point is the identity element. Orthografer 17:24, 26 January 2006 (UTC)[reply]

COMMENT ON ABOVE STATEMENT: a Deck transformation acts as a lift of the projection map from the universal covering space to itself. By the unique lifting property, a Deck transformation that fixes a point is the same as the identity map. The universal cover has to be simply connected, so the counter example above is invalid. — Preceding unsigned comment added by 128.36.156.224 (talk) 16:07, 28 April 2012 (UTC)[reply]

I disagree. I have not checked the proof in that book, but here is a counterexample. X is a bouquet of 2 circles. For the covering space Y, take a triangle ABC, attach a circle to one of the vertices, A, and add two extra segments between the two other vertices. The deck group can transpose B and C, and I think that's it, because the covering map then identifies uniquely which segments go to which loops in what orientation. Anyway, for sure A is a fixed point of the group. GaborPete (talk) 22:43, 16 November 2020 (UTC)[reply]

Oh, I think I have to take this back. Thinking of graphs, I thought that the transposition of B and C is for sure a deck transformation. But it is not: the covering map on the edges actually identifies uniquely who is B and who is C. So, the deck group seems trivial. But this raises the question for me: What is the group we are "seeing" here, transposing B and C, and maybe permuting some of the edges? It looks like the obvious group of symmetries of this covering. GaborPete (talk) 22:56, 16 November 2020 (UTC)[reply]

The Opposite of a Group

Don't you think it's overly pedantic to say the fundamental group of a space is isomorphic to the opposite of the group of deck transformations of its universal cover? After all, every group is canonically isomorphic to its opposite, via

$g\mapsto g^{-1}.$

In a detailed textbook, it might make sense to introduce the concept of the opposite of a group... but in an encyclopedia article, more people will be confused than helped - since most people will not know that every group is isomorphic to its opposite!

So, I urge that we delete the bit about the opposite of...

John Baez 02:52, 23 March 2006 (UTC)[reply]

I agree. In fact, it doesn't even show up in one textbook which I consider pretty detailed, Hatcher's algebraic topology book (searchable pdf). I'm deleting it. Why would something so trivial deserve an entire english word? Those things are getting ~~rare~~ scarce in math... Orthografer 14:19, 23 March 2006 (UTC)[reply]

Universal cover

The author or authors of this note have been cavalier with the hypotheses. The definition of universal cover given here is such that there is no universal cover. For instance let $X=\mathbb {R} \times D$ and let $B=\mathbb {R}$ , where D is any set with the discrete topology. The covering projection is $p:X\rightarrow \mathbb {R}$ is $p(x,n)=x$ . Now choose $D$ of greater cardinality than D and let $X'=\mathbb {R} \times D'$ with projection $p':X'\rightarrow \mathbb {R}$ given by $p'(x,n)=x$ . Notice that p lifts to $p'$ but it is not a covering map because no lift is onto. — Preceding unsigned comment added by 64.6.88.31 (talk) 22:36, 18 August 2006 (UTC)[reply]

Hi, I added the math tags to make it easier to read - revert if this annoys. It seems that the only definition of universal cover is at the beginning of the article, and it is the standard one: a covering space X→B is a universal cover if X is simply connected. In your example, p is not in general a covering map because it does not satisfy the path lifting property (for example when D is an interval, the preimage of an interval could be straight or curvy). In order to address your comment, I would need to know what hypotheses you are referring to. Further, in your example, which of X or

X'

is a universal cover? Orthografer 04:17, 19 August 2006 (UTC)[reply]

not locally path connected?

I have a technical question. What if a space is path connected and simply connected, but not locally path connected? (You can usually make up an example from something that's connected but not path connected by adding an appropriate path going around). Then technically this space would be the universal cover of itself, yet it is not locally path connected. It contradicts the conditions of having a universal cover, listed here... 146.186.132.190 (talk) 19:23, 29 February 2008 (UTC)[reply]

I'm not sure who put the "if and only if" statement for existence of universal covers. The comb space is an example of what you're describing - good catch! Orthografer (talk) 16:46, 10 August 2008 (UTC)[reply]

Lead now perfect

I have rewrote the lede. It now gives applications of covering spaces to mathematics, as well as a cite note and an image. It also carefully describes their importance in mathematics. PST — Preceding unsigned comment added by Point-set topologist (talk • contribs) 16:34, 15 January 2009 (UTC)[reply]

I have trimmed the introduction by fixing the grammar and removing material which I feel is too advanced. I kept the picture -- that is very nice and perhaps should be repeated later in the article. I also kept the links to ramification and homotopy groups. I think the introduction is now easier to read. best, Sam nead (talk) 23:17, 24 January 2009 (UTC)[reply]

Hi Sam,

Per WP:LEAD, the lead should contain a description of the concept and its importance in mathematics (for this particular case). The applications of covering maps in homotopy theory are crucial (note that fiber bundles are also important) and I think that this should be mentioned in the lead. Despite the technical terms there, I think that the lede should be like before. Of course, if you think differently, feel free to comment here again and we will try to even out the content. --PS T 12:20, 26 January 2009 (UTC)[reply]

Dear PST - I believe that you may be misunderstanding the point of WP:LEAD. The introduction to an article cannot be both concise, clear, and mention every possible application of covering spaces. I find the addition of G-covers, isomorphisms of homotopy groups, forward references, and technical definitions to be an imposing obstacle to readability. My version wasn't great, but it was better than the current version, which I find to be unreadable. To be clear -- the current version includes material that is too advanced; it will serve no purpose other than to block understanding. That is my opinion. Best, Sam nead (talk) 21:23, 31 January 2009 (UTC)[reply]

Dear PST -- Unless I hear back from you, I will rewrite the intro in a few days. all the best, Sam nead (talk) 21:37, 31 January 2009 (UTC)[reply]

I have now done so. If you'd like to rewrite the lead again, I urge you to keep in mind an important class of reader -- that is, the reader who does not know what a covering space is. all the best, Sam nead (talk) 23:44, 2 February 2009 (UTC)[reply]

Rewrite

Except for the lead, most of the article has to be rewritten in order to go for FA. PS T 09:27, 16 January 2009 (UTC)[reply]

Uniqueness

If an open set U in X is even covered, is the collection of pairwise disjoint "open lifts" of U unique? Also, is it true that infinite product of covering maps is not necessarily a covering map? Take the product of the real numbers to circle map with itself infinitely many times, since a covering map must be injective on some open set but the product map is not even injective on any basis element. Money is tight (talk) 16:13, 27 December 2009 (UTC)[reply]

In general, if you don't assume that U is connected and locally path connected, there is no uniqueness : "In general, the representation of an evenly covered open set as a disjoint union of open sets, mapped homeomorphically, is not unique (consider the case of an evenly covered discrete set)..." (Spanier, Algebraic topology, Mc GrawHill, page 64)--Cbigorgne (talk) 16:33, 27 December 2009 (UTC)[reply]

Typesetting

Per WP:MSM, special math symbols should be done in LaTeX. As is, many of them aren't rendering correctly in multiple browsers, like the circle for function composition. There seems to be extreme avoidance of LaTeX formatting in this article, and I don't really know why, but it's a problem in many cases. — Preceding unsigned comment added by 24.144.122.240 (talk) 21:55, 11 September 2015 (UTC)[reply]

In the 2006-2009 timeframe, when this was written, it was highly fashionable to avoid LaTeX, because the size of the LaTeX images did not match the user's native font size, and so inline formulas looked our of proportion. Less of an issue now, now that everyone has high-resolution screens and are thus using large fonts. Anyway, there are still some people running around, doing this conversion, away from LaTeX; I guess they don't agree with the math style manual.

Ideally, a browser would tell you what font size it wants to use, and the inline LaTeX images would be rendered so that they math that font size. No clue if this is doable in practice. 67.198.37.16 (talk) 17:55, 16 August 2016 (UTC)[reply]

Implication of covers that are not regular (or normal or Galois)?

Less than entirely clear is what the implications are of not having a regular cover. For what I can tell, the orbit space is homeomorphic to the base space iff the cover is regular. A worked example of a non-regular cover, and how that wrecks things, would be nice (e.g. with a non-regular cover, then the automorphism group of a fiber does not act transitively. Right?) thus, somehow the orbit space gets messed up too, but I'm having trouble visualizing this just right now. (I'm having trouble finding a simple example). 67.198.37.16 (talk) 17:20, 16 August 2016 (UTC)[reply]

Riemann surfaces

Skimming through this article, can't help noticing that Riemann surfaces are never mentioned ... yet the classic example is of a torus covered by R^2 is a Riemann surface. It seems to me that this might be worth mentioning... another example would be the Lens spaces. Then there's e.g. the toric varieties and so on. Another factoid: the fundamental group of any topological group has to be abelian, or else there's no way to define multiplication.

Also, more attention to things that don't work: e.g. the wedge product of infinite number of circles does not have a universal cover because it is not locally one-connected because the open sets are given by the cylinder sets of the initial topology on the cartesian product of the circles, and so no open set is simply connected and thus not contractible. 67.198.37.16 (talk) 18:09, 16 August 2016 (UTC)[reply]

External links modified

Hello fellow Wikipedians,

I have just modified one external link on Covering space. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:

Added archive http://arquivo.pt/wayback/20160514115224/http://www.bangor.ac.uk/r.brown/topgpds.html to http://www.bangor.ac.uk/r.brown/topgpds.html

When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.

This message was posted before February 2018. After February 2018, "External links modified" talk page sections are no longer generated or monitored by InternetArchiveBot. No special action is required regarding these talk page notices, other than regular verification using the archive tool instructions below. Editors have permission to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the RfC before doing mass systematic removals. This message is updated dynamically through the template {{source check}} (last update: 5 June 2024).

If you have discovered URLs which were erroneously considered dead by the bot, you can report them with this tool.
If you found an error with any archives or the URLs themselves, you can fix them with this tool.

Cheers.—InternetArchiveBot (Report bug) 22:46, 13 August 2017 (UTC)[reply]

Misleading section

The section Alternative definitions includes this passage:

" many authors require both spaces to be path-connected and locally path-connected. This can prove helpful because many theorems hold only if the spaces in question have these properties."

It is true that the conditions of path-connected and locally path-connected — and also semi-locally simply connected — can prove helpful in proving certain theorems about covering spaces.

However: It is utterly ridiculous to suggest that the definition of covering spaces must be restricted to path-connected or locally path-connected (or semi-locally simply connected) spaces. Any space can have a covering space!

For this reason I strongly suggest that this passage be removed.

Even restricting to connected spaces is strictly unnecessary. For instance, a discrete subgroup of the rotation group SO(3) (such as the tetrahedral group T of order 12) should be seen as covered by its inverse image π^-1(T) by the covering map π : S³ → SO(3) (which for T is the binary tetrahedral group 2T ⊂ S³).71.37.182.254 (talk) 19:15, 15 November 2020 (UTC)[reply]

Incomprehensible definition

In this sweeping revision the definition was made incomprehensible. Concrete gripes:

a space $E$ is introduced as the codomain of $\pi$ (" $\pi :E\rightarrow X$ "), but does not play any further role;
the notation $V_{d}$ is used, but remains undefined.

Is it really required that the preimages are disjoint unions? Why is that important? --Lambiam 21:16, 26 June 2022 (UTC)[reply]

Also, is what is being defined here not usually called a "covering map"?^[1][2][3] --Lambiam 21:52, 26 June 2022 (UTC)[reply]

I'm not sure why these edits were made. The article now seems completely impenetrable for someone who doesn't already know about covering maps/spaces. 122.161.93.162 (talk) 05:33, 13 July 2022 (UTC)[reply]

I agree, I found the previous revision to be much more agreeable to read. The article is now way too technical. DeathOfBalance (talk) 15:59, 11 March 2023 (UTC)[reply]

section headings cannot repeat

The heading "Examples" is repeated several times. As are "Definition", "Properties", etc. This breaks anchor links and runs contra MOS:HEADINGS. Can someone here try to invent unique headings for each of these? –jacobolus (t) 23:30, 5 November 2022 (UTC)[reply]

Very short lead

Maybe an old version of the lead can be used? The longer version of the lead was removed after this edit: [4] Christian75 (talk) 20:58, 8 June 2023 (UTC)[reply]

@Christian75 I took a stab at it, please do let me know what you think. (I figured if we have a definition section no need to spell out disjoint-union-homeomorphism explicitly; best to give some intuition & use-cases) Winthrop23 (talk) 17:54, 28 October 2023 (UTC)[reply]

@Winthrop23 I think it looks better, but I can not fact check. The lead should be more or less understandably by laymen but still correct. Christian75 (talk) 12:03, 4 November 2023 (UTC)[reply]

Definition restrictive compared to some references

Here is current's version of def:

Let $X$ be a topological space. A covering of $X$ is a continuous map

\pi :E\rightarrow X

such that there exists a discrete space $D$ and for every $x\in X$ an open neighborhood $U\subset X$ , such that $\pi ^{-1}(U)=\displaystyle \bigsqcup _{d\in D}V_{d}$ and $\pi |_{V_{d}}:V_{d}\rightarrow U$ is a homeomorphism for every $d\in D$ .

In Allen Hatcher's "Algebraic Topology" and "Godbillon's Éléments de topologie algébrique" in both cases the definition allows examples where D depends on x (the cardinality of D be locally constant, hence constant if X is connected) but in the definition above D is fixed. Similarly, the notion of degree of a covering is only defined when the base X is connected. Arnaud Chéritat (talk) 07:55, 4 October 2023 (UTC)[reply]

Well spotted; made some changes to show the possible dependence of

D

on

x

. Winthrop23 (talk) 18:06, 28 October 2023 (UTC)[reply]

The non-existence of a covering map T^3 to RP^3 is not really the fundamental reason for gimbal lock

Every proper (surjective) local homeomorphism between (connected) manifolds is a (surjective) covering map. A version of this appears as Exercise 11-9 in Lee's Topological manifolds book, and this is discussed in many places. So, by compactness of T^3, the existence of a covering map T^3 to RP^3 is equivalent to the existence of a local homeomorphism T^3 to RP^3. Hence there is perhaps no formal issue being raised by this topic.

However, the gimbal lock phenomenon, as explained on the page, is more fundamentally due to the map not being a local homeomorphism than it is to the map not being a covering map. The explanation on the page naturally suggests it is because on every neighborhood of the triplet of angles corresponding to the total alignment of the gimbals, the map to SO(3) ~= RP^3 does not restrict to a homeomorphism to its image.

As mentioned, in this setting there is not much difference between local homeomorphisms and covering maps. In other settings there is a difference however, and the difference has to do with non-local behavior of the map. The gimbal locking phenomenon teaches something about continuous maps versus local homeomorphisms (or smooth maps versus local diffeomorphisms).

To prove that Euler angles cannot determine elements of SO(3) "without some singularity issue", it does seem useful to invoke the covering map technology. The universal cover of T^3 is not homeomorphic to the universal cover of RP^3, so there can exist no covering map T^3 to RP^3, hence by the point mentioned at the beginning there can exist no local homeomorphism T^3 to RP^3. I still contend that the actual gimbal locking phenomenon (according to the gimbals in the animation) is most directly explained just with the language of local homeomorphisms, however.

73.39.58.52 (talk) 16:15, 23 May 2024 (UTC)[reply]

Sorry, but "the existence of a covering map T^3 to RP^3 is equivalent to the existence of a local homeomorphism T^3 to RP^3" makes no sense.

For any two (nonempty) 3-manifolds M, N there is always a "local homeomorphism" between them.

Perhaps you mean something else.

Aha, I suspect that what you mean is a locally one-to-one, onto mapping T^3 —> P^3.

Or in other words, a local homeomorphism at each point. — Preceding unsigned comment added by 98.36.148.11 (talk) 17:58, 9 November 2024 (UTC)[reply]

Request/suggestion for more robust section featuring examples of non-covering map local homeomorphisms

Speaking from experience, seeking the difference between a local homeomorphism and a covering map is a very natural inquiry. I find it confusing that very little mention of this seems to appear on Wikipedia, although it is among the first things a person learning about this topic would try to address. The section should indicate common settings where the two types of objects are equivalent. Lee's Topological Manifolds exercise 11-9 says "Show that a proper local homeomorphism between connected, locally path connected, and compactly generated Hausdorff spaces is a covering map". I don't know of a source with a proof of this fact, but such a source would be preferable.

The page http://en.wiki.x.io/wiki/Local_homeomorphism says every covering map is a local homeomorphism, but gives no counterexample to the converse. This page (http://en.wiki.x.io/wiki/Covering_space) has only one counterexample to the converse. I think it would be helpful to indicate that no surjective non-trivial restriction of the map $p:\mathbb {R} \to S^{1},\ t\mapsto (\cos(2\pi t),\sin(2\pi t))$ is a covering map. (The counterexample listed on the page being a special case of this.) Particular attention to the restrictions having finite fibers also seems useful for building intuition; these can be helpfully visualized using a modification of the helix picture which already exists on the page. Ryan Budney's answer here https://math.stackexchange.com/a/100942/444923 gives good intuition for why one should expect non-covering map local homeomorphisms (actually local diffeomorphisms) to be abundant in general. That intuition should be achievable from reading this page. This page should indicate that in many settings a covering map is a very special object compared to a local homeomorphism, more akin to a homeomorphism (a "multi-sheeted homeomorphism") than to an arbitrary local homeomorphism. 73.39.58.52 (talk) 17:33, 23 May 2024 (UTC)[reply]

Composition properties of covering maps

Lemma 80.2. and Exercise 1 of Section 80 of Munkres's point-set topology book (Topology, 2nd edition) discuss the composition properties of covering maps. These should be available on this page since they are basic facts which are not easily found. Evidence for the demand for these facts is given already on this Talk page. Logician1989 states a special case of Munkres's Exercise 80.1. The special case of composition of covering maps where the second covering map is finite-sheeted appears in Munkres's Exercise 54.4 (section 54, exercise 4).

It is frequently stated "The composition of two covering maps need not be a covering map" (this direct quote even appears in this Talk page). This is true but somewhat misleading, because in very general settings a composition of covering maps actually is a covering map. Munkres's Exercise 80.1 says this is true if the codomain of the second map has a universal cover, which to me means a better working default is to assume the composition of whatever covering maps I'm thinking about is a covering map, and then be surprised in the unusual case (since I think about manifolds) when the fact doesn't hold. 73.39.58.52 (talk) 18:07, 23 May 2024 (UTC)[reply]

Covering with empty domain

Let $\pi :{\tilde {X}}\to X$ be a covering. The article currently claims that if $X$ is connected, then $\pi$ is surjective. However, consider the case where ${\tilde {X}}$ is empty and $X$ is non-empty and connected. I believe that the empty map $\pi :{\tilde {X}}\to X$ is a covering, which is non-surjective even though $X$ is connected.

To see this, note that the inverse image under $\pi$ of any open subset $U$ of $X$ is empty, and hence is given by an empty coproduct of spaces. And, for any $x\in X$ , the condition on $\pi |_{V_{d}}$ for each $d\in D_{x}$ is satisfied vacuously because $D_{x}$ is empty. 2001:569:7DE5:4900:64B6:F620:8F83:3766 (talk) 17:49, 25 September 2024 (UTC)[reply]

Good catch.

X

connected implies subjectivity if and only if

{\tilde {X}}

is non-empty (see e.g. this stackexchange). I've edited the article to include that condition. Winthrop23 (talk) 18:49, 25 September 2024 (UTC)[reply]

You mean "surjectivity". — Preceding unsigned comment added by 98.36.148.11 (talk) 17:58, 9 November 2024 (UTC)[reply]

Wrong domain

The section Holomorphic maps between Riemann surfaces contains this passage:

"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 $\phi _{f(x)}\circ f\circ \phi _{x}^{-1}:\mathbb {C} \rightarrow \mathbb {C}$ is holomorphic."