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.
34 questions
60
votes
6
answers
7k
views
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 is not ...
- 44.3k
51
votes
3
answers
12k
views
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 ...
- 10.1k
29
votes
4
answers
3k
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 \ge G_1 \ge ... \ge G_i ...
- 573
107
votes
10
answers
38k
views
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 ...
14
votes
1
answer
1k
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 $K$ is empty or consists of finitely ...
- 213
11
votes
2
answers
2k
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 ...
- 361
86
votes
16
answers
9k
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 ...
80
votes
15
answers
15k
views
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 ...
19
votes
7
answers
6k
views
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 ...
- 23.5k
17
votes
8
answers
3k
views
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 ...
- 1,422
83
votes
6
answers
16k
views
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 ...
- 5,233
66
votes
5
answers
8k
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 ...
- 661
24
votes
3
answers
4k
views
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: $...
- 8,167
24
votes
10
answers
4k
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 ...
- 66.7k
23
votes
3
answers
3k
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 ...
- 3,033
23
votes
3
answers
2k
views
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}(...
- 3,004
20
votes
2
answers
900
views
Integral homology classes that can be represented by immersed submanifolds but not embedded submanifolds
Let $M$ be an $m$-dimensional compact closed smooth manifold and $z\in H_n(M,\mathbb{Z})$ an $n$-dimensional integral homology class, with $m>n.$ Does there exist a pair of $M$ and $z$ so that $z$ ...
- 587
17
votes
2
answers
2k
views
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 ...
- 8,167
16
votes
1
answer
2k
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 ...
- 18.8k
16
votes
1
answer
5k
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 ...
- 1,598
13
votes
3
answers
2k
views
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 ...
- 1,465
11
votes
1
answer
1k
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 ...
- 1,280
11
votes
1
answer
670
views
Is it always possible to compute the Betti numbers of a nice space with a well-chosen Lefschetz zeta function?
Let $X$ be a smooth projective variety. If I've understood correctly, the Weil conjectures imply that it is possible to compute the Betti numbers of $X(\mathbb{C})$ by computing the local zeta ...
- 118k
11
votes
1
answer
580
views
Smallest volume representatives of homology
Given a Riemannian manifold, I have a notion of volume for each of my chains, so it makes sense to ask for a representative of a homology class with the smallest volume. Are there conditions for when ...
- 353
10
votes
0
answers
295
views
Hochschild homology of a Hopf algebra
Let $A$ be a Hopf algebra over the complex numbers.
Denote by $\mathcal{M}$ the dg-category of dg-$A$-modules.
The Hochschild homology of $\mathcal{M}$ is not going to be the Hochschild homology of $A$...
- 1,382
8
votes
3
answers
1k
views
Reference needed: Isomorphism on pi_1 and homology gives weak equivalence
Let $f : X \to Y$ be a map between a connected space $X$ and a space $Y$. If $\pi(f) : \pi_1(X) \to \pi_1(Y)$ is an isomorphism, and $H_n(f) : H_n(X, G) \to H_n(Y, G)$ is an isomorphism for all $n \ge ...
- 435
8
votes
2
answers
1k
views
Differentials in the Lyndon-Hochschild spectral sequence
The Lyndon-Hochschild(-Serre) spectral sequence applies to group extensions in a manner analogous to the Serre-Leray spectral sequence applied to a fibration.
Does anyone know of a good description (...
- 1,422
7
votes
0
answers
433
views
spectral sequence for a complex with two filtrations
Suppose $(C,d)$ is a chain complex: an abelian group with a map $d:C \to C$ such that $d^2 = 0$ (people like to assume $C$ is graded; if that helps - feel free to do so). A filtration is an ascending ...
- 353
7
votes
1
answer
557
views
Continuous maps $f:S^n \to \mathbb{C}P^m$ with $f(x)\perp f(-x) $
Question 1: What is a complete classification of all positive integers $m,n$ with the following property:
There is a continuous map $f:S^n \to \mathbb{C}P^m$ such that $f$ maps antipodal ...
- 356
7
votes
2
answers
1k
views
Proof of the ''trangression theorem''
Here is what I would call the transgression theorem. Let $X$ be a pointed space and $\Omega X$ its loop space. There are two maps $H_{p}(\Omega X) \to H_{p+1}(X)$ which should be the same. I am ...
- 20.9k
6
votes
1
answer
1k
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 ...
- 456
5
votes
1
answer
411
views
Homology of singular chain complex modulo subdivision
Let $S_p(X)$ be the $p$-th singular chain group and $\mathcal S(X)$ be the singular chain complex of a topological space $X$. There is a barycentric subdivision operator (which is also a chain map) $\...
- 807
4
votes
1
answer
264
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 3-...
- 2,283
3
votes
1
answer
537
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 ...
- 9,870