Talk:Simply connected space

Mathworld bug

The article said:

A subset X of Euclidean 2-space R² is simply connected if and only if both X and R² - X are connected.

I tried to correct this yesterday by changing "subset" to "bounded subset" and "connected" to "path-connected", but it's still wrong. It is possible to construct a bounded simply connected subset of the plane with disconnected complement. --Zundark, 2002 Feb 19

Thanks for finding the bug. I mindlessly copied it from Mathworld. Maybe we need X to be open? AxelBoldt

Never trust Mathworld. I think

A bounded open subset X of Euclidean 2-space R² is simply connected if and only if both X and R² - X are connected.

may be correct, but we need to check it before adding it to the article. --Zundark, 2002 Feb 19

SU(N) is not simply connected?

There was the statement in the article (before I deleted it) that SU(N) is simply connected. But in general it is not! Since for some N it has nontrivial fundamental group. The only case is N=2 (when it is 3-sphere). Am I right? —Preceding unsigned comment added by 193.124.167.14 (talk) 13:37, 30 April 2009 (UTC)

No. A 3-sphere is certainly simply connected, although that is not to say that its other homotopy groups are non-trivial (for example, the third homotopy group of this space is isomorphic to the additive group of the integers). I should also point out that formally, SU(2) is not the 3-sphere (as a subspace of $R^{8}$) but rather diffeomorphic to it. But as far as simple connectedness goes, this is irrelevant. The reason for making this point is that SU(2) has applications in other branches of mathematics - not solely topology. --PST 11:53, 1 May 2009 (UTC)

Ok, thanks for the answer. It is realy the fact that $\pi _{3}(SU(N))$ is isomorphic to Z - additive group of the integers (for N equal to or greater than 2). But where can I find the elementary proof of the SU(N) simple connectedness (i.e. $\pi _{1}(SU(N))$ triviality)? The matter for my hesitations is the reminiscence for $\pi _{1}(U(N))$ to be isomorphic to Z. —Preceding unsigned comment added by 195.46.33.25 (talk) 16:35, 2 May 2009 (UTC)

I do not know it to be too hard to prove it yourself - at least if you have an intuitive understanding of SU(N). If you want the proof, and similar properties about these matrix groups, any good textbook on differential geometry will do (or differential topology). By good textbook, I mean a graduate level book (such as a book in the series "Graduate texts in mathematics" or one that has a thorough treatment of algebraic topology also). --PST 01:33, 13 May 2009 (UTC)

Assuming n ≥ 2, SU(n) is a fibre bundle over the sphere S^2n-1, with fibre = SU(n-1). This implies there is an exact sequence of homotopy groups:

(*)        π₁(SU(n-1)) → π₁(SU(n)) → π₁(S^2n-1)) = {1}

. Since the last group is the trivial group, it follows from exactness that the first map π₁(SU(n-1)) → π₁(SU(n)) must be onto. But since we know that for n=2, SU(2) = S³ is indeed simply-connected, (*) implies that the map {1} = π₁(SU(2)) → π₁(SU(3)) is onto, and hence π₁(SU(3)) = {1}, too. Continuing this way, we see that (*) tells us that π₁(SU(n-1)) = {1} always implies π₁(SU(n)) = {1} (for n ≥ 3), and hence SU(n) is simply-connected for all n ≥ 2.

On the other hand, SU(1) is the trivial group {1} which is simply connected as well.Daqu (talk) 18:30, 1 June 2014 (UTC)

a hollow ball is?

I smell inconsistency: a hollow ball (in 3D) can't be simply connected any more than a circle (in 2D). Whether or not they have a hole depends on your definition of hole.

Yes, it depends on your definition of a hole, which is why we give a formal of definition of simply connected which is unambiguous. A hollow ball (a sphere) is simply connected - it's intuitively obvious that a circle on a sphere can be contracted to a point. --Zundark 15:52 Nov 28, 2002 (UTC)

(I'm here to look up these concepts because the descriptions are infinitely better than those in the books of our math library's geometry/topology section, but there seems to be an imperfection here.)

The definition given is over my head: why involve the circle? If "simply connected" is meant to express the notion "completely filled up", isn't it better to define it as a connected subspace of which the complement's interior is also connected, or something like that?

Your definition does not give an intrinsic property of the space, it gives a property of the embedding of the space in some larger space. The idea of "simply connected" is that there is a path from every point to every other point (this is path-connectedness) and that there is essentially only one such path (thus "simply" connected), in the sense that any two paths from A to B can be deformed into one another. --Zundark 15:52 Nov 28, 2002 (UTC)

---

Zundark, if you're right about the 3-sphere, then the bit on the Poincaré Conjecture at Clay Mathematics Institute needs to be corrected. It says, "In topology, a sphere with a two-dimensional surface is essentially characterized by the fact it is simply connected. The Poincaré conjecture is that this is also true for spheres with three-dimensional surfaces. The question has been solved for all dimensions above three. Solving it for three is central to the problem of classifying 3-manifolds." I don't know enough about the topic to correct it. -FM (FunnyMan3595 22:20, 2 Mar 2004 (UTC))

The point here is not that we don't know the fundamental group of the 3-sphere (we certainly do). It is the characterisation of the 3-sphere by topological invariants - which goes the other way.

Charles Matthews 22:23, 2 Mar 2004 (UTC)

(Charles beat me to it, but I'll post my reply anyway, as I've typed it out.) What the Clay Mathematics Institute says is correct. It's easy to prove that the 3-sphere is a simply connected compact 3-manifold without boundary, but the Poincaré Conjecture says that these properties characterize the 3-sphere - that is, any space with all these properties is (homeomorphic to) a 3-sphere. --Zundark 22:31, 2 Mar 2004 (UTC)

Ahh, you answered the question I was about to ask while I was typing it. Someone should probably (and I hope I'm right on this) put up a page at characterize/characterization explaining the term, it would have helped my confusion. Every time Charles opens his mouth I feel like a babbling idiot... guess now I know how the non-math majors feel when I open mine.  :) -FM (FunnyMan3595 22:40, 2 Mar 2004 (UTC))

multiply connected???

"Multiply Connected" seemed to be in general use up until about 1980: the term is definitely used but definitely "old fashioned" RodVance (talk) 02:36, 25 April 2014 (UTC)

Is this term really in use? it is usually not simply conneced At least it should say "rearly", but it is safe to remove it.

Tosha 18:40, 8 May 2004 (UTC)

I knew the term - perhaps it is just old-fashioned.

Charles Matthews 08:20, 9 May 2004 (UTC)

Probably old-fashioned. I've seen the term in some old books. BTW it would be nice if the article explained the rationale or etymology behind the term "simply connected". My guess is that the connection between any two points is "simple" since the connecting path is unique up to homotopy, and in "multiply" connected spaces there exists several essentially different paths. Can anyone confirm this?Punainen Nörtti 13:37, 27 March 2007 (UTC)

I think that's right. The article already says this at the end of the second paragraph in the Formal definition and equivalent formulations section. --Zundark 15:04, 27 March 2007 (UTC)

The explanation in the introduction is confusing

I don't like the explanation of simple connectedness given in the introduction. It attempts to explain the concept without being formal, but is imprecise, long and confusing.

• Why give an explanation of connected space at all? It doesn't belong here. What's more, it is not rigourous. Imagine the object {M \in R^3, d(O,M) <= 1 and d(O,M) \ne 0.5}, a ball with a sphere missing inside. The inside ball can't move. It is true it can still spin, but this could be prevented easily by changing a little the shape of the object.
• The illustration with a string is too long and imprecise. It could be misleading, because people will think they can only insert the string at the boundary of the object (imagine a 3D object in 3D space). "Feeding some extra string" is also imprecise.

I think we should give a more mathematical definition in the introduction to begin with. To aid non mathematically inclined people, we could give a few diagrams of simply connected and non-simply connected spaces, and show how a path can be deformed in an example. --Bernard Helmstetter 16:16, 25 Dec 2004 (UTC)

I don't understand that section either. It seems to me that it tries to examine whether the complement of the space is simply connected. But again, my English is not perfect. \Mike(z) 17:33, 15 May 2005 (UTC)

No link with the previous discussion: is the Mandelbrot space simply connected? It would be a nice example in the appropriate section. --Bernard Helmstetter 16:16, 25 Dec 2004 (UTC)

New intro

I just rewrote the intro (had been "insert a string in the object", now is "imagine object as aquarium with diver, magic loop of string"). I hope it's better. I'm not quite sure how to convey the idea that the string will never get hung up on obstacles (for example, think of a 3D donut glued to a flat piece of plywood; the union is simply connected), without making too big a deal of it and making the reader wonder, "So when are obstacles important?"

I removed the "Naming" section, which had been

• In topology, a connection is often referred to as a handle. This is probably a reference to the way that a (singly-connected) beaker can be topologically turned into a (doubly-connected) teacup by the addition of a handle.
• In theoretical physics, an additional connection is known as a wormhole.

The article never defines connection, although I assume a connection is something like an equivalence class of paths between two fixed points that can deform into each other. Except I'm fairly certain that a "wormhole" is really another name for a handle-shaped hole, and not for the connection itself. A wormhole would be a connection in the complementary space, right? Anyway, the section seems unnecessary and doesn't make sense as written, so, I commented it out. --Quuxplusone 23:13, 8 May 2006 (UTC)

What about spaces that aren't path-connected?

The introduction seems to include those, while in the example section, they seem to be excluded, ipso facto.

So, what's the deal? Are both definitions in use? Should we say that the term is usually reserved for CW complexes, or complex domains?

RandomP 16:42, 3 June 2006 (UTC)

Path-connectedness is always required for simply connected spaces, and this is how the article defines the term. (The introduction isn't meant to be rigorous, but feel free to change it if you think it's unclear or misleading.) The term isn't usually reserved for CW complexes, or complex domains. --Zundark 18:49, 3 June 2006 (UTC)

Hrm. The introduction isn't just not rigorous, it's incorrect, then. I don't think that's acceptable, even in this case (my impression is that non-path-connected connected spaces are thought of as less of a pathology nowadays than they used to be).

RandomP 19:53, 3 June 2006 (UTC)

I think the assumption that the space is path-connected should be removed. A space is simply-connected if any closed curve in it is null-homotopic. Thus for instance a union of two disjoint closed discs in the plane is simply-connected, but not connected. Standard literature, especially on Lie groups, makes this distinction, often talking about a "connected, simply-connected" Lie group.

user:AMAN I think even if we remove the " path connected" constrain from the definition, then also the way rest of the definition goes itself will imply that it is path connected. So if simply connected region itself will be path connected. A union of two disjoint discs also cant be simply connected as by definition " if every simple closed curve in D encloses only points that are in D, it is simply connected". — Preceding unsigned comment added by 202.78.172.162 (talk) 12:33, 30 November 2012 (UTC)

Connected space

Shouldn't Connected space be mentioned at least once in this article? --Abdull 15:08, 5 June 2006 (UTC)

R² not, but R³

It is not obvious to me why R² minus (0,0) is not simply connected, but R³ minus (0,0,0) is. Maybe someone can prove it? --Abdull 11:04, 6 June 2006 (UTC)

The easiest proof proceeds via showing that the first is homotopy equivalent to the circle, while the second is homotopy equivalent to the 2-sphere; this implies that the fundamental group of the first is infinite, while the fundamental group of the second is trivial. A path-connected space is simply connected iff its fundamental group is trivial.

Intuitively: if you have a loop in either space, you can "shrink" it so that it will actually be on the unit sphere of the respective vector space - it's clear where to map each point, since there is exactly one ray through every point, and every ray intersects the sphere in exactly one point. It's furthermore easy to see that every loop in the sphere arises in such a manner. The difficult bit is showing that the two-sphere is simply connected, since you need to show that you can deform every loop on it not to be surjective (see Peano curve for why this is necessary); thankfully, such a proof succeeds, and the result stands.

Hope this helps!

RandomP 11:34, 6 June 2006 (UTC)

Is a cylinder simply connected?

I think it is. If so, it ought to be listed as such in the "Examples" section of this article.

I answered my own question. The answer is no, as there are definitely two different paths along a cylinder between any two points on that cylinder that can't be deformed into each other. Kevin Lamoreau 02:23, 14 September 2006 (UTC)

The cylinder is homeomorphic to S¹ X R so can't be simply connected (it is a simple result that the fundamental group of X X Y (where X and Y are topological spaces), is isomorphic to the direct product of the fundamental group of X with that of Y). The circle is not simply connected (well known); its fundamental group is the integers.

Topology Expert (talk) 06:57, 8 November 2008 (UTC)

introduction

While the introduction is not incorrect, the standard definition of simply-connected space is given in terms of contractibility of loops in a pointed space (rather than paths between two points). Thus a space is simply connected if all loops (passing through a fixed point in the space) can be continuously deformed and contracted to a point. More formally, the first homotopy group of the space is trivial. - Subh83 (talk | contribs) 17:50, 11 December 2013 (UTC)

Lie Theory

The statement "In Lie theory, simple connectedness is prerequisite for working of important Baker–Campbell–Hausdorff formula." is made. This is either misleading or vacuous: the Campbell Baker Hausdorff formula is defined for small enough neighbourhoods of the identity (small enough that the matrix log series coverges in the adjoint representation of the group and a few other requirements), and the neighbourhoods one talks about are "automatically" simply connected by dint of the fundamental local homeomorphism of a Lie group with open neighbourhoods of the origin ℝ^N that the group gets through its charts. I mean, one could use open, non-simply connected neighbourhoods if one really wanted to as long as they were small enough in the sense above, and of course one can always carve out a simply connected open neighbourhood of the origin from the non-simply connected one and restrict discussion to the latter neighbourhood. This is what I mean by "the statement is vacuous": it is not needed to prove the CBH formula. What the comment's poster might be thinking of is that simple connectedness is prerequisite for the lifting of a homomorphism between two Lie groups' Lie algebras to a homomorphism between the Lie groups themselves: the preimage of the homomorphism must be a simply connected Lie group: see chapter 9 of Stillwell, "Naive Lie Theory" for a readable explanation. --RodVance (talk) 02:49, 25 April 2014 (UTC)

Agreed. I deleted the line. Sean Eberhard (talk) 12:55, 14 April 2015 (UTC)

Gives a characterisation of simply connected subsets of $\mathbb {C}$ and then contradicts it

The following sentences are contradictory:

> In complex analysis: an open subset $X\subseteq \mathbb {C}$ is simply connected if and only if both X and its complement in the Riemann sphere are connected. The set of complex numbers with imaginary part strictly greater than zero and less than one, furnishes a nice example of an unbounded, connected, open subset of the plane whose complement is not connected.

-- Svennik (talk) 13:03, 13 September 2019 (UTC)

Actually, I see now that the example given is simply connected in the Riemann Sphere (it does not include the point at $\infty$). Pictured on a sphere, the "width" of the band approaches zero as the band approaches the point at infinity. As a result, the complement is (just about) connected. --Svennik (talk) 13:10, 13 September 2019 (UTC)

as a layperson, this article is unreadable

Most importantly, what does "path" mean?

How come a path can't change what its endpoints are but also the definition is something about shrinking a circle down to a point? nhinchey (talk) 00:33, 28 May 2021 (UTC)

I have added a missing link to Path (topology) in the lead section and another missing link to Loop (topology). This doesn't really help much with the general level of this article, I'm afraid. The article isn't really helping by using different definitions in the lead section and in the first paragraph of Simply connected space § Definition and equivalent formulations. However, that section does mention the definition using homotopic paths given in the lead section as an equivalent definition. The article used to be more approachable, I think. See for example this revision from 2015. – Tea2min (talk) 06:56, 28 May 2021 (UTC)

image ambiguity, especially in search results

The image at the forefront of this article probably shouldn't be a counterexample, even if it almost certainly has a place elsewhere in the article. Not only does this lead to a visual conflict and possible confusion while reading, but it also causes problems when this page shows up in search results, as is often the case when someone looks for information on this topic. The problem is exemplified by this google search result as the image shows up by itself, sans clarifying caption, with the erroneous title of "simply connected space," which it explicitly is not. I have given similar feedback to google on this issue of the image being an incorrect search result, but it seems like the problem is mostly on this end. The article should probably be mostly about what the subject is and only parenthetically what it isn't. ZephyrCubic (talk) 18:56, 29 March 2023 (UTC)