The homology tag has no wiki summary.

**14**

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**5**answers

538 views

### 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} ...

**1**

vote

**0**answers

83 views

### Role of determinant of the matrix corresponding to $i$-th Homology group.

I was thinking about the proof of the Lefschetz's Fixed point theorem and the ingeniuty of the Hopf's Trace formula, i.e. associating the trace of the matrix for deciding about the fixed points. Now ...

**1**

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**0**answers

59 views

### Homology and Exterior Square

Let $G$ be a finite group and $G \wedge G$ denote the exterior square of $G$. It is well known that the second integral homology $H_2(G,\mathbb{Z})$ is the kernel of homomorphism $x \wedge y \mapsto ...

**1**

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**0**answers

71 views

### Tubular neighborhoods in the proof of the Morse homology theorem

I have a question regarding the proof of the Morse homology theorem given by D. Salamon in "Morse theory, the Conley index and Floer homology". The full text can be found here:
...

**1**

vote

**2**answers

184 views

### The cohomology groups of $\Omega U(n)$

Let $\Omega U(n)$ be the loop space of $U(n)$. Is it true that the cohomology groups $H^*(\Omega U(n); \mathbb{Z})$ are torsion-free? How can one calculate these groups?

**8**

votes

**1**answer

264 views

### Software for computing Thurston's unit ball

Is there any software which can be used for computing Thurston's unit ball(for second homology of 3-manifolds) of link complements? In particular can I do that with SnapPy?
PS: even a table for ...

**8**

votes

**1**answer

333 views

### Naturality of a Kunneth formula for cohomology

Let $X,Y$ be CW complexes. By Kunneth formula, we have a group isomorphim
$$ H^n(X\times Y;G) \cong \oplus_{p+q=n} H^p(X;H^q(Y;G))$$
Is there a natural map realizing this isomorphism?

**-1**

votes

**2**answers

179 views

### Interpretation of $H_1(A_\mathbb{C}^{top},\mathbb{Q})$

In the paper "Sato-Tate Distributions and Galois Endomorphism Modules in Genus 2" (arxiv: http://arxiv.org/abs/1110.6638), the authors use the singular homology $H_1(A_\mathbb{C}^{top},\mathbb{Q})$ ...

**8**

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**0**answers

358 views

### 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 ...

**3**

votes

**1**answer

225 views

### Second betti number of compact analytic spaces

Let $V$ be a proper singular complex algebraic variety, possibly nonprojective ($dim(V)=n>0$). I would like to know:
1) if its second Betti number is non zero,
2) same question but now $V$ is a ...

**21**

votes

**9**answers

2k views

### 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 ...

**26**

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**4**answers

2k views

### 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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**0**answers

211 views

### Divisibility in homology/homotopy

I have a simply-connected CW-complex $F$ of finite-type, and I know that the imprimitivity of its particular integral homology is divisible by an odd prime $p$; that is,
$$ \forall n,\exists \delta, ...

**1**

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**0**answers

54 views

### some intuition about the degree of a map

Consider a map
$$ f: \Sigma \to X/\sigma,$$
where $\Sigma=\Sigma_g/\Omega$ is a quotient of a Riemann surface by an antiholomorphic involution,
$\sigma:X\to X$ is an antiholomorphic involution
of some ...

**2**

votes

**1**answer

395 views

### Homology of a region of the plane

This is related to this MO question, I don't know if it's really "research-level". As in that question, let $U$ be a domain of the complex plane $\mathbb{C}$, i.e. an open connected subset. Let
$$ ...

**2**

votes

**0**answers

268 views

### Cech homology (!) of the Warsaw Circle

Can anyone can give me a reference to the fact that first Cech homology (not cohomology!) group of the Warsaw Circle is $\mathbb{R}$? Thank you in advance :)

**11**

votes

**3**answers

821 views

### 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 ...

**18**

votes

**3**answers

2k views

### 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 ...

**1**

vote

**1**answer

150 views

### What are finite groups $H$ such that $H^n(H,\mathbb{Q/Z}) \cong H_n(G,\mathbb{Z})$?

Let $G$ be a finite group and $G^{\prime}$ be its commutator subgroup. Let $\mathbb{Z}$ and $\mathbb{Q}$ denote the integers and rationals. $\mathbb{Z}$ and $\mathbb{Q/Z}$ treated as trivial ...

**1**

vote

**1**answer

144 views

### Universal coefficient theorem for local ring

Let $R$ be a commutative local artin $k$-algebra,where $k$ is a field with characteristic $0$.I wonder whether universal coefficient theorem holds in this case.Namely,if $C$ is a chain of flat ...

**8**

votes

**1**answer

221 views

### worked out examples in borel-moore homology

I'm trying to learn about BM homology. I've found a few references and they are all quite abstract and high-powered. I am actually familiar with derived categories and sheaves so I don't mind thinking ...

**4**

votes

**1**answer

117 views

### 0-homologous surface bounds

Given a map $f : S \to M^4$ from a compact closed not necessarily connected oriented surface to a compact oriented 4-manifold, such that $f_*([S])$ is zero in $H_2(M)$, is there a compact oriented ...

**11**

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**0**answers

374 views

### 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 ...

**12**

votes

**1**answer

314 views

### 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 ...

**47**

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**15**answers

3k views

### 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 ...

**2**

votes

**2**answers

872 views

### Proving homotopy invariance of cellular homology by constructing a chain homotopy

I'm trying to follow an argument in Lück's "Algebraische Topologie: Homologie und Mannigfaltigkeiten" (to which there apparently doesn't exist an english translation). The aim is to check homotopy ...

**11**

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**0**answers

458 views

### 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 ...

**1**

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**0**answers

107 views

### Does compactly supported cohomology make sense for cosimplicial spaces?

As far as I understand, whenever one has something (co)simplicial in spaces, one should take a sort of diagonal to reduce the study to a single space. I'm never sure though whether one preserves only ...

**1**

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**0**answers

45 views

### Natural isomorphism between locally finite homology and homology of one-point compactifcation of a forward tame ANR

Let $X,Y$ be a locally compact, separable, metric ANR that is forward tame, which means that for some closed subset $V\subseteq X$ such that $\overline{X\smallsetminus V}$ is compact a proper map ...

**8**

votes

**1**answer

217 views

### The Image of the Mod 2 Homology of BSp in the Homology of BSO

I'm essentially trying to figure out exactly what the title asks for. I've been scouring old Seminaires Henri Cartan and books by Stong to try to see exactly how to do this, but the combination of ...

**2**

votes

**1**answer

112 views

### Spectral sequence in Lie algebra

The exact sequence $0\rightarrow sl(A)\rightarrow gl(A)\rightarrow A/[A,A]\rightarrow 0$ gives rise to a spectral sequence in homology, I want to know the details for this spectral sequence at the ...

**1**

vote

**2**answers

215 views

### Dimension of the homology group with coefficients in $\mathbb{Z}/2\mathbb{Z}$

I asked this on math.stackexchange.com, but didn't get a single answer.
Charles Weibel writes in his survey of homological algebra
Riemann defined a surface $S$ to be $(n + 1)$-fold
connected ...

**12**

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**2**answers

603 views

### 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 $\pi_1(X)$ ...

**11**

votes

**1**answer

564 views

### 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 ...

**2**

votes

**0**answers

114 views

### Twisted product structure on product of Eilenberg-MacLane spaces

The product of Eilenberg-MacLane spaces $K(\mathbb{Z}/2,1) \times K(\mathbb{Z}/2,2)$ is an $H$-space with respect to a "twisted product structure", where you add the product of the first entries to ...

**2**

votes

**1**answer

134 views

### Finding null-homologous curves via the matrix equation $AB^iC^jx=0$

Motivation: I have a family of curves obtained from a single curve by repeatedly applying two automorphisms of the surface (Dehn twists to be specific). I am interested in the images of these curves ...

**0**

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**0**answers

93 views

### loop space homology and lens spaces

Is the homology of free loop space of lens spaces known?
Thanks in advance for your help.

**0**

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**1**answer

137 views

### Subvarieties with different topology representing the same cycle

Let $X$ be a topological space, and $Y,Z$ be subspaces. My question is a bit vague and open-ended: when is it the case that, if $Y$ and $Z$ represent the same (nonzero) cycle in homology, then $Y$ and ...

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**0**answers

83 views

### Relative homology and a map with degree 1

Let M be a manifold and $B_p(1,M)$ a ball of radius 1 and center p in M. Let $F:B_p(1,M)\to \mathbb{R}^n$ a map such that $F(\partial B_p(1,M))\subset B_0(1+\epsilon,\mathbb{R}^n)\setminus ...

**7**

votes

**1**answer

311 views

### A description of cellular boundary maps in terms of a Morse function

I'm writing a paper on classical Morse Theory and I'm interested in applying Morse functions to the computation of homology groups of a compact manifold $M$. The standard way in which this is done is ...

**6**

votes

**3**answers

634 views

### 2-cycle of K3 surface

Hi there,
I want to ask about the 2-cycle of K3 surface.
As we know, its betti number $b_2$=22, so there will be 22 2-cycle generators.
Is there any topological way to figure out such cycles ...

**15**

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**0**answers

579 views

### 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 ...

**6**

votes

**6**answers

1k views

### reference for (co)homology theories

Hi everyone,
Every now and then, I find myself dealing with such or such (co)homology theory, and I'm frustrated I don't feel more comfortable around it.
I was wondering if someone could recommend a ...

**6**

votes

**3**answers

535 views

### Projective dimension of zero module

Is there any consensus on what the projective dimension of the zero module should be? Here are three statements one commonly encounters in textbooks, sometimes with or without the condition $M\neq 0$:
...

**7**

votes

**3**answers

887 views

### 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 ...

**0**

votes

**0**answers

188 views

### Question on Steenrod realizability problem

René Thom proved that for any topological space $X$, any integral homology class of $X$ (of any degree) has an odd multiple that is the pushforward of the fundamental class of a smooth compact ...

**1**

vote

**2**answers

481 views

### Theorem on composition of derived functors, question about proof

I got a question about a proof I found in Gelfand-Manin's "Methods of homological algebra" (Page 200):
Theorem 1. Let $\mathcal{A}, \mathcal{B}, \mathcal{C}$ be three abelian categories, $F: ...

**7**

votes

**1**answer

338 views

### What are the applications of Dowker's theorem?

Let $R \subset X \times Y$ be any relation between sets $X$ and $Y$. CH Dowker constructed two simplicial complexes $K$ and $L$ associated to $R$:
a simplex in $K$ consists of finitely many elements ...

**2**

votes

**2**answers

277 views

### Obtaining derived functors from derived functors of similar complexes or “bluntly truncated” unbounded complexes (without adding 0's to the left)

I don't know if I'm actually using the right terminology here, to be clear I'm going to state explicitly what I'm trying to figure out to see if I can be pointed in the right direction:
Let $F: ...

**3**

votes

**1**answer

387 views

### Unbounded complexes, resolutions and computation of derived functors

Hey guys, let $F: \mathcal{A} \rightarrow \mathcal{B}$ be a left exact functor between abelian categories with enough injectives, let $K \in Kom(\mathcal{A})$ be an unbounded complex, I've heard that ...