The homology tag has no wiki summary.
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1answer
124 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
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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
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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
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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
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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
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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
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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 ...
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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 ...
11
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1answer
433 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
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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
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1answer
106 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 ...
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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.
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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: ...
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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 ...
3
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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 ...
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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 ...
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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 ...
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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
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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. ...
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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
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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 ...
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4answers
940 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
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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 ...
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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 ...
3
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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 ...
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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 ...
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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 ...
1
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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
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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 ...
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2answers
521 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 ...
0
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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 ...
0
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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 ...
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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 ...
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1answer
579 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 ...
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1answer
352 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 ...
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1answer
376 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 ...
1
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1answer
240 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 ...
12
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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 $\pi_1(X)$ ...
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3answers
877 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 > ... > ...
13
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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 ...
