# Questions tagged [homology]

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.

212 questions
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### Why torsion is important in (co)homology ?

I've once been told that "torsion in homology and cohomology is regarded by topologists as a very deep and important phenomenon". I presume an analogous statement could be said in the context of ...
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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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### Teaching homology via everyday examples

What stories, puzzles, games, paradoxes, toys, etc from everyday life are better understood after learning homology theory? To be more precise, I am teaching a short course on homology, from ...
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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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### Spaces with same homotopy and homology groups that are not homotopy equivalent?

A common caution about Whitehead's theorem is that you need the map between the spaces; it's easy to give examples of spaces with isomorphic homotopy groups that are not homotopy equivalent. (See Are ...
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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 ...
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### Maps which induce the same homomorphism on homotopy and homology groups are homotopic

I am interested in the following question. Are maps which induce the same homomorphism on homotopy and homology groups homotopic? I am sure the answer is no, however I cannot imagine how to construct ...
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### “Homotopy-first” courses in algebraic topology

A first course in algebraic topology, at least the ones I'm familiar with, generally gets students to a point where they can calculate homology right away. Building the theory behind it is generally ...
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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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### Does this approach for the Poincaré conjecture work?

Several months ago a paper was posted at http://arxiv.org/abs/1001.4164 called "Another way of answering Henri Poincaré's fundamental question." The author gave a talk on it today at my institution. ...
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### Massey Products vs. $A_\infty$-Structures

Does anyone know a good reference for a proof of the fact that given a dga $A$, an $A_\infty$-structure on $HA$ is ''the same'' as coherent choices for all of the higher Massey products of $HA$? More ...
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### On the wikipedia entry for Borel-Moore homology

The wikipedia page on Borel-Moore homology claims to give several definitions of it, all of which are supposed to coincide for those spaces $X$ which are homotopy equivalent to a finite CW complex and ...
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### Ordinal-indexed homology theory?

Back when I was a grad student sitting in on Mike Freedman's topology seminar at UCSD, he posed the following question. Does there exist a good homology theory $H_{\alpha}(X)$ where $\alpha$ is an ...
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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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### Computer-aided homology computations

Background I am currently working on the homology of some moduli space and there exists a much simpler chain complex with the same homology. It is a quotient of a bisimplicial complex by a subcomplex....
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### Take contraction wrt a vector field twice and define kernel mod image. Does that give anything interesting?

First, we make the following observation: let $X: M \rightarrow TM$ be a vector field on a smooth manifold. Taking the contraction with respect to $X$ twice gives zero, i.e. $$i_X \circ i_{X} =0.$$...
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### Are acyclic subcomplexes of finite contractible 2-complexes contractible?

Let $Y$ be a contractible finite simplicial 2-complex. Let $X$ be an acyclic subcomplex of $Y$ (i.e. $X$ connected, $H_1(X)=0$, $H_2(X)=0$). Is $X$ contractible? (Equivalently, is $\pi_1(X)$ trivial?)...
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### Spectra with “finite” homology and homotopy

As known, any non-trivial finite spectrum $X$ can not have non-zero homotopy groups $\pi_i(X)$ only for finite number of $i$. As I understand, the same is true for any spectrum $X$ with finitely ...
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### When does homology represent an embedded sphere?

If we have a triangulation of a manifold $M$ of dimension $i$ and we have simplicial homology $H_i(M)=\mathbb{Z}$, what is the condition than there exists an embedded sphere $S^i$ that generates the ...
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### Is this “Homology” useful to study?

In the usual singular homology of a topological space $X$, one consider the free abelian group generated by all continuous maps from the standard simplex $\Delta^{n}$ to $X$. Now we can ...
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### Singular chains generated by manifolds with corners — does it really work?

Usually we define singular homology via the complex of singular chains: $$C_\ast(X)=\bigoplus_{n\geq 0}\mathbb Z\langle\sigma:\Delta^n\to X\rangle$$ where the right hand side denotes the free abelian ...
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### Is it true that all real projective space $RP^n$ can not be smoothly embedded in $R^{n+1}$ for n >1

So first for n even, $RP^n$ is not orientable, hence can not be embedded in $\mathbb{R}^{n+1}$. For odd n, $RP^{n}$ is orientable, hence the normal bundle is trivial. Now using stiefel-Whitney ...
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### Functorial description of mod-2 homology of an abelian group $A$ in terms of $A/2$ and ${}_2A.$

Let $A$ be an abelian group and $p$ be a prime. If $p\ne 2,$ there is a very nice functorial description of the homology algebra $H_*(A,\mathbb Z/p):$ H_*(A,\mathbb Z/p)\cong \Lambda^*(A/p)\otimes \...
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### Homology of Lie groups

Let $G$ be a Lie group and $G^{\delta}$ the underlying group (with discrete topology). Obviously, we have a continuous map of groups $i:G^{\delta}\rightarrow G$ which induces a map between classifying ...