Re: Hurewicz homomorphism
- From: "W. Dale Hall" <wdunderscorehallatpacbelldotnet@last>
- Date: Sat, 24 Nov 2007 21:35:46 GMT
sanchopancho80@xxxxxx wrote:
Hello,
the Hurewicz theorem of algebraic topology asserts, that there is a relation between the homotopy and the singular homology of a space X.
If the first m \pi_n(X) are trivial (n>0), then H_n(X) is trivial
too. In m+1 it is an epimorphism but my question is the following:
Does a space X exists, such that there is a m with H_n(X) trivial for
all n>m but \pi_n eventually nonzero?
Visualized I am looking for an easy space X with something like 0 0 0 0 x x x 0 x 0 0 0 0 Homotopy and 0 0 0 0 x x x 0 0 0 0 0 0 Homology.
Does it exists?
Thanks Sancho
Sure. Try the 2-sphere S^2. There is a fibre bundle (the
Hopf bundle), with fibre S^1, total space S^3, and base S^2:
S^1 --> S^3 --> S^2
The long exact sequence in homotopy for this bundle gives
isomorphisms in homotopy between pi_n(S^3) and pi_n(S^2)
for all n >= 3, leading to many nonzero higher homotopy
groups for S^2:
H_*(S^2) = Z, 0, Z, 0, 0, ... listed according to dimension
pi_*(S^2) = {*}, 0, Z, Z, Z/2Z, Z/2Z, Z/12Z, ...
This list was obtained from the table here:
http://en.wikipedia.org/wiki/Homotopy_groups_of_spheres#Table_of_homotopy_groups
Dale
.
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