The homology tag has no usage guidance.

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### CW-structures and Morse functions: a reference request

The following is probably well known, but I wasn't able to locate a reference in the literature.
Let $f$ be a Morse function on a smooth compact manifold $M$ without boundary and let $\rho$ be a ...

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### Does homology detect chain homotopy equivalence?

Is the following true: If two chain complexes of free abelian groups have isomorphic homology modules then they are chain homotopy equivalent.

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### Examples of the varying strengths of topological invariants

In my first algebraic topology class, I remember being told that the simplest reason for homology was to distinguish spaces. For example, if is X=circle and a Y= wedge of a circle and a 2-sphere then ...

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### Realizing complexes with bases as cellular complexes

This is a question a friend of mine asked me some time ago. I suspect the answer is "no" but can't prove it.
Every free complex of abelian groups is isomorphic to the reduced cellular complex of some ...

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### disagreement between two definitions of the singular boundary map

Hi everyone, I have a little problem with the definition of singular boundary map in singular homology theory. It appears to be some disagreement between two authors. The first one is Hatcher in his '...

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### Poincaré quasi-isomorphism

Suppose we have a simplicial combinatorial manifold (just a triangulated manifold) and its Poincaré dual cell complex.
Corresponding homology simplicial and homology cell complexes are quasi-...

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### Cohomology of associative algebras

Let $A$ be an associative algebra over a commutative ring $k$. I've read statements saying that Hochschild (co)homology is the "right" notion of (co)homology for associative algebras. When $A$ is ...

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### What is a cohomology theory (seriously)?

This question has bugged me for a long time. Is there a unifying concept behind everything that is called a "cohomology theory"?
I know that there exist generalized cohomology theories, Weil ...

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### Torsion in homology or fundamental group of subsets of Euclidean 3-space

Here's a problem I've found entertaining.
Is it possible to find a subset of 3-dimensional Euclidean space such that its homology groups (integer coefficients) or one of its fundamental groups ...

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### Are there two non-diffeomorphic smooth manifolds with the same homology groups?

I know that there definitely are two topological spaces with the same homology groups, but which are not homeomorphic. For example, one could take T^2 and S^{1}\vee S^{1}\vee S^{2} (or maybe S^{1}\...

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### For which spaces is homology (or cohomology) determined by the Eilenberg-Steenrod axioms

This is a spinoff of Can anyone give me a good example of two interestingly different ordinary cohomology theories? . By an ordinary homology theory, I mean a functor on topological spaces which ...

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### Can one calculate the (co)homology of the loopspace of a Lie group from its Lie algebra?

Compact connected simply-connected Lie groups have so much structure that you can calculate their cohomology from their Lie algebras using Lie algebra cohomology (certain Ext-groups) and similarly ...

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### Why is homology not (co)representable?

This is in the same vein as my previous question on the representability of the cohomology ring. Why are the homology groups not corepresentable in the homotopy category of spaces?

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### Smooth classifying spaces?

Take G to be a group. I care about discrete groups, but the answer in general would be welcome too. There are the various ways to construct the classifying space of G, bar construction, cellular ...

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### What is (co)homology, and how does a beginner gain intuition about it?

This question comes along with a lot of associated sub-questions, most of which would probably be answered by a sufficiently good introductory text. So a perfectly acceptable answer to this question ...

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### Differentials in the Lyndon-Hochschild spectral sequence

The Lyndon-Hochschild(-Serre) spectral sequence applies to group extensions in a manner analogous to the Serre-Leray spectral sequence applied to a fibration.
Does anyone know of a good description (...

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### Does homology have a coproduct?

Standard algebraic topology defines the cup product which defines a ring structure on the cohomology of a topological space. This ring structure arises because cohomology is a contravariant functor ...