Geometric difference between a homotopy equivalance and a homeomorphism

WWGD

Quote by homeomorphic

I guess they have to be homotopy equivalent to CW complexes, since they can be realized as CW complexes, and they are unique up to homotopy equivalence. The way you construct them is what I have been talking about. Make a wedge of spheres to get generators of the non-zero homotopy group. By the Hurewicz theorem, it's isomorphic to the homology in that dimension because all the lower homotopy groups vanish. Then, just keep attaching cells to kill off all the homotopy groups above that dimension. This doesn't give you a very concrete construction, so in the end, you don't really know what you've built. But some Eilenberg-Maclane spaces occur in nature, so to speak, like ℝP^∞, ℂP^∞, or S^1. As far as I know, those are the only naturally occurring ones. Some asked for examples of them when I first encountered them in my algebraic topology class, and the prof didn't seem to know of any other than those few examples and the abstract construction of them. Of course, maybe the term "natural occurring" doesn't have too much meaning.

Not to go too-far off-topic, but maybe a reasonable meaning for " x being naturally-occuring" is that one is somewhat likely to either run into x or hear about it while doing research that is not too wildly unusual.

homeomorphic

Not to go too-far off-topic, but maybe a reasonable meaning for " x being naturally-occuring" is that one is somewhat likely to either run into x or hear about it while doing research that is not too wildly unusual.

Yeah, and I forgot to say the closed (orientable?) surfaces are Eilenberg Maclane. And aspherical manifolds of all sorts, since those are manifolds with vanishing higher homotopy groups (which came up in my 3-manifold readings).

lavinia

The classifying spaces for flat bundles,bundles with discrete structure group, are all EMs. For finite groups these are all probably all infinite dimensional CW complexes. For instance the classifying space for Z2 bundles is the infinite real projective space.

An example of a Z2 bundle is the tangent bundle of the Klein bottle (itself an EM space).
It follows that the classifying map into the infinite Grassmann of 2 planes in Euclidean space can be factored through the infinite projective space. (I wonder though whether it can actually be factored through the two dimensional projective plane by following a ramified cover of the sphere by a torus with the antipodal map.)

In terms of group cohomology this corresponds to the projection map,

[itex]\pi_{1}[/itex](K) -> Z2 obtained by modding out the maximal two dimensional lattice.

Group cohomology is the same as the cohomology of the universal classifying space for vector bundles with that structure group - I think

In the case of the flat Klein bottle, this shows that its holonomy group is Z2.

lavinia

Quote by mathwonk

note that even an exotic sphere has a morse function with just two critical points. so when you attach that disc you cannot always attach it differentiably in the usual way. In fact this is apparently how one produces exotic spheres. you produce a manifold that is not an ordinary smooth sphere somehow, but that does have a morse function with only two critical points. then it is homeomorphic to a sphere.

reference?

homeomorphic

The classifying spaces for flat bundles,bundles with discrete structure group, are all EMs.

Well, what I was saying is that most of those aren't some familiar space that has a name, like S^1.

For finite groups these are all probably all infinite dimensional CW complexes. For instance the classifying space for Z2 bundles is the infinite real projective space.

Not always. For example, for surface groups, as we just mentioned. And there are some aspherical manifolds that are finite-dimensional. This includes, for example, all hyperbolic 3-manifolds.

http://en.wikipedia.org/wiki/Aspherical_space

Group cohomology is the same as the cohomology of the universal classifying space for vector bundles with that structure group - I think

I would say principal bundles, rather than vector bundles. I would prefer to say the cohomology of G is the cohomology of a K(G,1). That's my favorite definition, and of course, it's equivalent to other definitions of group cohomology, like the one as a derived functor or from the bar resolution of Z over ZG.

lavinia

Quote by homeomorphic

Not always. For example, for surface groups, as we just mentioned. And there are some aspherical manifolds that are finite-dimensional. This includes, for example, all hyperbolic 3-manifolds.

http://en.wikipedia.org/wiki/Aspherical_space

I think that the fundamental groups of ashperical manifolds are infinite e.g. tori. The fundamental groups of closed orientable surfaces are infinite except for the sphere.

I would say principal bundles, rather than vector bundles. I would prefer to say the cohomology of G is the cohomology of a K(G,1). That's my favorite definition, and of course, it's equivalent to other definitions of group cohomology, like the one as a derived functor or from the bar resolution of Z over ZG.

Same thing I think.

For aspherical manifolds the fundamental domain in the universal covering space generates a free resolution of the integers over the fundamental group. e.g. for a 2 dimensional torus one has four vertices and edjes and one rectange as a basis over ZxZ.

lavinia

Quote by mathwonk

Lavinia, have you read the classical reference, Milnor's Morse Theory? This is Thm. 3.5 proved in roughly the first 24 pages. He proves that the region on the manifold "below" c+e where c is a critical value, has the homotopy type of the region below c-e with the attachment of a single cell determined by the index of the critical point with value c.

The passage from local to global you ask about may be Lemma 3.7. In it he proves that a homotopy equivalence between two spaces extends to one between the spaces obtained from them by attaching a cell.

He makes use of deformation retractions and ultimately uses Whitehead's theorem that a map is a homotopy equivalence if it induces isomorphism on homotopy groups, at least for spaces dominated by CW complexes.

Thanks again Mathwonk. I just browsed through the first chapter. The lemmas you mention do the trick.

So what are two homotopy equivalent compact manifold without boundary that are not homeomorphic?

mathwonk

read the first page of this paper for some related results:

http://deepblue.lib.umich.edu/bitstr.../1/0000331.pdf

but perhaps these examples are not compact.

mathwonk

here you go: (lens spaces)

http://en.wikipedia.org/wiki/Spherical_3-manifold

mathwonk

to be explicit:

"In particular, the lens spaces L(7,1) and L(7,2) give examples of two 3-manifolds that are homotopy equivalent but not homeomorphic."

homeomorphic

I think that the fundamental groups of ashperical manifolds are infinite e.g. tori. The fundamental groups of closed orientable surfaces are infinite except for the sphere.

True, I sometimes confuse finite with finitely generated. That's why I was confused. Yeah, I think if you even have a subgroup of finite order, you have to have an infinite-dimensional complex. The proof was really cool, but I'll have to try and remember it. There was a covering-spaces proof.

lavinia

Quote by mathwonk

to be explicit:

"In particular, the lens spaces L(7,1) and L(7,2) give examples of two 3-manifolds that are homotopy equivalent but not homeomorphic."

pretty cool. So how do their Morse functions distinguish them?

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mathwonk Recognitions: Homework Help Science Advisor	read the first page of this paper for some related results: http://deepblue.lib.umich.edu/bitstr.../1/0000331.pdf but perhaps these examples are not compact.