Geometric difference between a homotopy equivalance and a homeomorphismby Flying_Goat Tags: difference, equivalance, geometric, homeomorphism, homotopy 

#37
Jul2412, 11:19 AM

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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. 



#38
Jul2412, 11:33 AM

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#39
Jul2412, 12:31 PM

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



#40
Jul2412, 10:18 PM

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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. 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. 



#41
Jul2412, 10:21 PM

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



#42
Jul2412, 11:02 PM

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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. 



#43
Jul2412, 11:19 PM

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#44
Jul2512, 05:52 PM

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



#45
Jul2512, 07:53 PM

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#46
Jul2512, 07:54 PM

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