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1
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1answer
123 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
1answer
121 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 ...
7
votes
1answer
187 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 ...
3
votes
1answer
99 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
votes
0answers
334 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 ...
11
votes
1answer
288 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 ...
46
votes
15answers
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
2answers
865 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
votes
0answers
419 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
vote
0answers
100 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
vote
0answers
37 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
1answer
206 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
1answer
104 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
2answers
209 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
votes
2answers
569 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
1answer
432 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
0answers
102 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
1answer
105 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
votes
0answers
84 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
votes
1answer
133 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 ...
0
votes
0answers
73 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
1answer
276 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
3answers
610 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 ...
14
votes
0answers
543 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
6answers
996 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
3answers
504 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
3answers
831 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
0answers
173 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
2answers
426 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: ...
13
votes
3answers
1k 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 ...
7
votes
1answer
301 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
2answers
272 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: ...
6
votes
2answers
519 views

Computer aided homology computations

Background I am currently working on the homology of some modulispace and there exists a much simpler chaincomplex with the same homology. It is a quotient of a bisimplicial complex by a subcomplex. ...
3
votes
1answer
339 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 ...
15
votes
5answers
718 views

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 ...
1
vote
2answers
352 views

Hyper(co)homology of exact (acyclic) complexes

Let $\mathcal{A}$ be an abelican category with enough injectives, let $K^\bullet \in Kom^+(\mathcal{A})$ be a complex, where $Kom^+(\mathcal{A})$ is the category of cochain complexes over ...
6
votes
1answer
273 views

Torsion in $H^2(X,\mathbb{Z})$ induced by torsion in $H_1(X,\mathbb{Z})$

Let $X$ be a topological space. The universal coefficient theorem says that there is a short exact sequence $$ 0\rightarrow Ext(H_{k-1}(X,\mathbb{Z}),\mathbb{Z})\rightarrow ...
9
votes
4answers
937 views

A map inducing isomorphisms on homology but not on homotopy

As a consequence of the Whitehead theorem, Spanier's Algebraic Topology book has on 7.6.25 the following theorem: A weak homotopy equivalence induces isomorphisms of the corresponding integral ...
18
votes
12answers
5k views

“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 ...
10
votes
1answer
308 views

finite complex with non-finitely generated homology with local coefficients

I am looking for an explicit example, if one exists, of a (pointed) finite connected CW-complex $X$ such that some homology group with local coefficients $H_n(X,{\mathbb Z}[\pi_1 X])$ is not a ...
3
votes
1answer
330 views

Bitopological spaces and algebraic topology

Is it possible to introduce the concept of bitopological spaces such as $(X,T_1,T_2)$ (introduced by J.C.Kelly see Proc. London Math. Soc. (3) 13 (1963) 71–89 MR0143169, J.C. Kelly) in the homotopy ...
0
votes
1answer
312 views

The query concerning the Euler-Poincare formula’s generalizations

Euler's equation for polyhedral, Euler's polyhedral formula, V – E + F = 2, where V, E, and F, are the number of points, edges and faces, was discovered by Leonhard Euler in 1752. However, the basic ...
12
votes
2answers
428 views

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 ...
0
votes
1answer
265 views

What is a quasi-isomorphism of two crossed modules

Could you tell me how are two crossed modules quasi-isomorphic. And I have known a result: Let $\mu: M \rightarrow G$ and $\mu': M' \rightarrow G'$ are isomorphic, then the integral homology of them ...
1
vote
1answer
354 views

Can a function from R^n to R^n fail to be one-to-one, and also lack critical points? [closed]

I feel like there should either be a straightforward example of such a thing, or a homological reason why it can't exist. The reason why I ask is that if the Hessian matrix of a Lagrangian is ...
11
votes
1answer
803 views

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 ...
4
votes
2answers
520 views

Homology of Covering Spaces

Let $A$ be a subgroup of a group $G$. Then since $A$ is a subgroup of the fundamental group $\pi_1(K(G,1))=G$, there is a covering space $p\colon Y\to K(G,1)$ with $p_*(\pi_1(Y))=A$. So the homology ...
15
votes
2answers
912 views

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 ...
0
votes
1answer
259 views

Simple Equivariant homology [no borel-Moore]

Hey. I'm working with Bredon's equivariant cohomology. At some point I need to compute the $4$th equivariant cohomology group of $S^1 \x D^3$ relatively to its boundary for the antipodal action of ...
14
votes
2answers
546 views

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 ...