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Results tagged with homology
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user 23571
Homology is a general way of associating a sequence of algebraic objects such as abelian groups or modules to other mathematical objects such as topological spaces.
79
votes
Accepted
Maps which induce the same homomorphism on homotopy and homology groups are homotopic
Thus $gf$ induces the same maps on homology and homotopy groups as a constant map, but it isn't homotopic to a constant map. (I forget where I first saw this example, maybe in something of Arnold.) …
- 20k
22
votes
CW-structures and Morse functions: a reference request
The result you are looking for is Theorem 4.18 in "An Introduction to Morse Theory" by Yukio Matsumoto, published by AMS in 2002 (translated from Japanese). The connections between Morse functions, ha …
- 20k
14
votes
Accepted
Geometric intuition behind this chain homotopy
I have seen this method of subdividing $\Delta^p\times I$ in several books when they are developing homology theory, but it is more complicated than necessary. …
- 20k
7
votes
When are the homology and cohomology Hopf algebras of topological groups equal?
Taking the union over all $n$, the infinite-dimensional group $\mathrm{SO}$ has mod $2$ cohomology a polynomial ring with one generator in each odd degree and mod $2$ homology an exterior algebra with … Restricting to a finite dimensional $\mathrm{SO}_n$ has the effect of restricting the homology and cohomology algebras to a finite number of generators and truncating the polynomial algebra by relations …
- 20k
4
votes
Accepted
Realizing complexes with bases as cellular complexes
Here is a sketch of an argument to show that all based chain complexes are realizable. (This might end up being pretty similar to Tyler's argument.)
First one gives an algebraic argument that by a ch …
- 20k