# 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.

**61**

votes

**16**answers

5k 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 ...

**12**

votes

**0**answers

707 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

**0**answers

125 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 ...

**8**

votes

**1**answer

260 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

176 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 ...

**3**

votes

**2**answers

299 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 if ...

**15**

votes

**1**answer

1k 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

317 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

238 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

**0**answers

207 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

**1**answer

164 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

**0**answers

95 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 B_0(1-\...

**7**

votes

**1**answer

459 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 ...

**8**

votes

**6**answers

2k 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 ...

**1**

vote

**0**answers

300 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 ...

**2**

votes

**2**answers

960 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: \mathcal{...

**10**

votes

**1**answer

671 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 $...

**4**

votes

**1**answer

706 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 ...

**16**

votes

**0**answers

818 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 of ...

**15**

votes

**5**answers

852 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 on $...

**2**

votes

**2**answers

318 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: \...

**14**

votes

**3**answers

1k 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 subcomplex....

**1**

vote

**2**answers

687 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 $\mathcal{A}$...

**6**

votes

**1**answer

359 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 H^{k}(X,\mathbb{Z})\...

**9**

votes

**4**answers

2k 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 ...

**10**

votes

**1**answer

637 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 ...

**6**

votes

**3**answers

742 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 direct?...

**5**

votes

**1**answer

463 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

**1**answer

447 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

**2**answers

555 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 (Z(...

**1**

vote

**1**answer

386 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 non-...

**12**

votes

**1**answer

2k 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 ...

**5**

votes

**2**answers

2k 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 ...

**1**

vote

**1**answer

375 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 $\...

**0**

votes

**1**answer

276 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 ...

**21**

votes

**2**answers

1k 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 ...

**17**

votes

**1**answer

1k views

### 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 ...

**3**

votes

**1**answer

823 views

### Rational Homology of a Covering Space

I have heard that the rational homology of a covering space is easy to compute, compared with the ordinary homology. However, I don't know any details about that. Can anyone help me? Any reference ...

**2**

votes

**1**answer

694 views

### Computing the homology groups of spaces in a fibration

Let $F\rightarrow X\rightarrow B$ be a fibration. If we know very well the spaces $F$ and $B$ and wish to compute the homology of $X$. One possible tool is the Serre Spectral Sequence. However, it ...

**2**

votes

**1**answer

654 views

### Coboundary map on the cochain complex of abelian cosimplicial groups?

Maybe I'm looking at the wrong places, but I can't find a definition of
the coboundary map on the cochain complex of abelian cosimplicial groups.
What I have in mind is something similar to the "...

**14**

votes

**2**answers

820 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 $\pi_1(X)$ trivial?)...

**25**

votes

**4**answers

2k views

### 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 > ... > ...

**22**

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 ...

**10**

votes

**4**answers

3k views

### 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 ...

**9**

votes

**0**answers

381 views

### Singular chains as an HZ-module spectrum

For $R$ any ring and $H R$ its Eilenberg-MacLane spectrum -- a ring spectrum -- there is an equivalence between the $\infty$-categories of $H R$-module spectra and that of unbounded chain complexes of ...

**8**

votes

**1**answer

356 views

### Third bordism group of BG, where G is an arbitrary compact Lie group.

Is anything known about $\Omega_3(BG)$, where $G$ is an arbitrary compact Lie group; i.e., is it possible to describe the structure of $\Omega_3(BG)$ for any compact Lie group? I know that $H_3(BG)$ ...

**2**

votes

**1**answer

506 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
$$ \...

**1**

vote

**0**answers

322 views

### connection between non-orientable manifolds and torsion in 1D (co) homology

I'm interested in understanding the probability that given a prime $p$, $p$ divides the order of the torsional part of $H^k(X,Z)$, where $X$ is a finite simplicial complex.
Lets say you have a ...

**4**

votes

**0**answers

123 views

### Uniform distribution of special homology classes mod-p

Let $X_0(N)$ be the usual modular curve. For $\chi$ a quadratic Dirichlet character of conductor $D$, define the homology class $c(\chi)=\sum_{i=0}^{|D|-1}\chi(i)\{\frac{i}{|D|}{\infty} \}$; here $\{...

**1**

vote

**1**answer

876 views

### Is there a connection between the theory of motives and homotopy theory?

I have read that motives were designed to be the common part of the many homology theories, a way of unifying them. But as I understand it: homotopy is closely related to homology, there is only 1 ...