# 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.

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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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### Geometric Interpretation of the Lower Central Series for the Fundamental Group?

For any group G we can form the lower central series of normal subgroups by taking $G_0 = G$, $G_1 = [G,G]$, $G_{i+1} = [G,G_i]$. We can check this gives a normal chain
$G_0 > G_1 > ... > ...

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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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### Are fundamental groups of aspherical manifolds Hopfian?

A group $G$ is Hopfian if every epimorphism $G\to G$ is an isomorphism. A smooth manifold is aspherical if its universal cover is contractible. Are all fundamental groups of aspherical closed smooth ...

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### Homology theory constructed in a homotopy-invariant way

Singular homology sends homotopic morphisms on equal morphisms and weakly equivalent spaces on isomorphic objects. So singular homology is in fact defined on the homotopy category of topological ...

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### Which paths in a graph are orthogonal to all cycles?

Start with some standard stuff. Suppose we have a directed graph $\Gamma$. I'll write $e : v \to w \,$ when $e$ is an edge going from the vertex $v$ to the vertex $w$. We get a vector space of 0-...

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### Why localize spaces with respect to homology?

A basic construction in algebraic topology is the localization of spaces or spectra with respect to a homology theory: one formally inverts the $E$-homology isomorphisms, reflecting each space into ...

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### Is there a map of spectra implementing the Thom isomorphism?

A well known theorem in algebraic topology relates the (co)homology of the Thom space $X^\mu$ of a orientable vector bundle $\mu$ of dimension $n$ over a space $X$ to the (co)homology of $X$ itself: $...

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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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### A homology theory which satisfies Milnor's additivity axiom but not the direct limit axiom?

Let us agree on the following: a "homology theory" means a functor $h_*$ from the category of pointed CW complexes to the category of graded abelian groups, together with natural isomorphisms $h_{*+1}(...

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### Are the homology and cohomology Serre spectral sequences dual to each other?

If we use homology and cohomology over a field $k$, if a space has homology and cohomology groups of finite type in each degree, then $H_\ast(X;k)$ is dual to $H^\ast(X;k)$ using the universal ...

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### Simple curves on non-orientable surfaces.

Given an element in the (first) homology group of a surface, I would like to know if it can be represented as a simple closed curve. For orientable surfaces, this is well-known, but I wasn't able to ...

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### Does the bordism homology theory satisfy the weak equivalence axiom?

There is an interesting and important homology theory called bordism. Briefly speaking, a singular manifold in a space $X$ is a pair $(M, f)$ where $M$ is a closed smooth manifold and $f : M \to X$ is ...

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### What is the Hopf algebra structures in the homology of the based loop spaces of $E_7$ and $E_8$?

Since $\Omega X$ is a $H$-space, if it has homology of finite type, the homology acquires the structure of a Hopf algebra. Bott has shown that for $X=G$ a Lie group, in fact $H_*(\Omega X)$ is free ...

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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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### Homology classes of subvarieties of toric varieties

Let $X$ be a smooth proper toric variety, $Z\subseteq X$ a smooth subvariety.
Is the fundamental class $[Z] \in H_\ast(X) = A_\ast(X)$ nonzero?
Background
If $X$ is a Kaehler variety, this is of ...

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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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### What fraction of n x n invertible integer matrices contain at least one unit?

The question is simple:
What fraction of matrices in $G_n = \text{GL}_n(\mathbb{Z})$ have at least one unit entry (i.e., either $\lbrace\pm 1 \rbrace$)?
I'm not sure what the correct measure on $...

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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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### Is the singular homology of a real algebraic set always finitely generated?

Here is a precise statement of my question:
Let $p\in \mathbb{R}[x_1, \ldots, x_n]$ be a polynomial, and let $Z(p)\subset \mathbb{R}^n$ be the set of zeros of $p$. Must the singular homology $H_i (Z(...

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### Are totally degenerate chains null-homologous?

Let $X$ be a CW complex.
Suppose $\gamma\in C_n(X)$ is a cycle which is a sum of maps $\sigma:\Delta^n\to X$ which factor as $\Delta^n\to\mathbb R^{n-1}\to X$. Does it follow that $\gamma$ is null-...

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### About maps inducing bijections on homotopy classes

Let us assume that $f:X \to Y$ is a map of connected CW complexes, having the following property: if $K$ is a finite CW complex, then the induced map $f_{\ast}:[K,X] \to [K,Y]$ on \emph{free} homotopy ...

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### Request: intermediate-level proof: every 2-homology class of a 4-manifold is generated by a surface.

Hi, everyone:
For the sake of context, I am a graduate student, and I have taken classes in
algebraic topology and differential geometry. Still, the 2 proofs I have found
are a little too terse for ...

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### homology with compact supports

In one of the exercises in McDuff and Salamon, they mention homology with compact supports. I know how to define *co*homology with compact supports, but I can't picture the homology version. How do ...

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### What does this naive attempt at $S^1$-equivariant homology describe?

After reading Cohen and Voronov's notes on string topology, one can find the following construction: Suppose we have a topological space $X$ with continuous action of $S^1$. This means we have a map $\...

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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 ...

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### Examples of non-simply connected manifolds with trivial H^1

It is known that, if a topological space is simply connected,its first homology group vanishes. The converse is not true, since for every presentation of a (say, finite) perfect group G we can ...

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### What would be the ramifications of homotopy theory being as easy as homology theory?

Greg Muller, in a post called Rational Homotopy Theory on the blog "The Everything Seminar" wrote
"I tend to think of homotopy theory a little bit like ‘The One That Got Away’ from mathematics as a ...