Use this tag if your question involves some type of (co)homology, including (but not limited to) simplicial, singular or group (co)homology. Consider the tag (homological-algebra) for more abstract aspects of (co)homology theory.

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2
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1answer
35 views

Which maps on the zeroth homology maps do actually come from continuous maps?

Let $X, Y$ be topological spaces. What are the possible maps $H_0(X) → H_0(Y)$ on homology coming from continuous maps $X → Y$? For example, can a map $X → X$ on a connected space induce a ...
0
votes
0answers
13 views

Definitions of the group of cycles/group of boundaries

first I want to clarify that the class I am referring to is not really about homological algebra, rather about Galoistheory. Still we defined the group of cycles/ the group of boundaries (first ...
1
vote
1answer
46 views

Homology group $H_1(G;\mathbb{R})$ is a vector space?

I am reading a paper which is asking me to view the homology group $H_1(G;\mathbb{R})$ of a (presentation of a) group as a vector space. Now, my knowledge of homology is basically non-existent, but I ...
4
votes
3answers
94 views

Is reduced homology a full functor on connected spaces?

Let $X$ and $Y$ be connected topological spaces. Can we then realize each map between their homologies as coming from a continuous map between their spaces? For any arrow $φ\colon \tilde H_•(X) → ...
2
votes
2answers
29 views
+50

Homology groups of orientable surfaces.

Edit: I have a proof here but when I spoke last with my professor, she told me something was close, but not quite. Can someone help me patch this proof? I've been trying to get this down for quite a ...
2
votes
1answer
19 views

Boundary maps of the projective plane as a $\Delta$-complex (homology)

Hi, very simple question here. In Hatcher's 'Algebraic Topology' the diagram above is used to describe the projective plane as a $\Delta$-complex(see p.102). Later the 2-boundary maps are given by ...
0
votes
0answers
22 views

Prerequisites to study cohomology?

Work related I have to deal with cohomology theory fairly soon. Unfortunately, I never had any classes on this, so I'd like to study it on my own. Before I dive into a book or two, I'd like to make ...
1
vote
1answer
51 views

Closed non-exact $2$-form on $T = S^1 \times S^1$

I would like to find a closed non-exact $2$-form on $T = S^1 \times S^1$. Here are my thoughts: Since $d\theta$ and $d\varphi$ are closed non-exact $1$-forms an obvious candidate is $d\theta \wedge ...
0
votes
1answer
30 views

Finding two inequivalent closed, non-exact $1$-forms on $T = S^1 \times S^1$: second check

This is a follow up on my previous question. I would like to test whether I understand the first part of the answer given to me there by rewriting it in my own words. Please could someone tell me ...
3
votes
0answers
33 views

Non-split chain complex which is chain-homotopy equivalent to its homology sequence

This is exercise 1.4.4 from Weibel. Consider the homology $H_*(C)$ of chain complex $C$ as a chain complex with zero differentials. It is easy to show that if C is split, then there is a chain ...
1
vote
1answer
33 views

cohomology of semi-direct product of groups

Let $G, H$ be groups. Let $G\rtimes _\phi H$ be a semidirect product. The product is twisted. Let $BG$, $BH$, and $B(G\rtimes_\phi H)$ be the classifying spaces of $G$, $H$, and $G\rtimes _\phi H$. ...
1
vote
1answer
34 views

cohomology of permutation group with mod 2 coefficient

Let $S_n$ be the permutation group of order $n$. Let $\mathbb{Z}_2=\mathbb{Z}/2\mathbb{Z}$. What is the cohomology algebra $$H^*(S_n;\mathbb{Z}_2)?$$ For $n=2$, $BS_2=\mathbb{R}P^\infty$ hence I ...
6
votes
1answer
75 views

Finding two inequivalent closed, non-exact $1$-forms on $T = S^1 \times S^1$

I've been studying the torus and the first cohomology group $H^1_{dR}(T)$ for a couple of weeks now. I finally had a breakthrough of understanding and would like to kindly request the community to ...
1
vote
0answers
15 views

use homology groups to obtain minimal cell structure

On Hatcher's book Algebraic Topology, Section 4.C Prop. 4C.1, for a simply-connected CW-complex $X$, if $H_*(X;\mathbb{Z})$ is known as a graded module over $\mathbb{Z}$, then the minimal cell ...
2
votes
1answer
50 views

Relationship between differential forms and coordinates

For the purpose of this question let us restrict our considerations to smooth $3$-manifolds. So the manifold $M$ we consider here is endowed with smooth coordinate charts $(x,y,z)$. What I have ...
0
votes
2answers
36 views

$H^1_{dR}(S^1 \times S^1) = H^1_{dR}(S^1) \oplus H^1_{dR}(S^1)$ without Künneth?

As in the title: I am trying to derive $H^1_{dR}(S^1 \times S^1) = H^1_{dR}(S^1) \oplus H^1_{dR}(S^1)$. First let me share my thoughts: I am trying to derive that $$ H^1_{dR}(S^1 \times S^1) = ...
0
votes
1answer
41 views

Requirements for Mayer-Vietoris

This question might be a duplicate -- but as I don't find an entry (maybe because of the lack of a good keyword) I open this question. Besides, this questions arises when trying to prove Proposition ...
0
votes
0answers
19 views

Finding closed non-exact differential forms

I have been thinking about this for quite some time now. I worked on the example of the torus and determined the generators of $H^n_{dR}(T)$ by trial and error. It worked but it wasn't satisfying. I ...
1
vote
0answers
27 views

Homology CW complex

I have strong intuition that the following fact is true: If $X = \bigcup_{n\in \mathbb N} X_n$ is a CW-complex (and $X_n$ its $n$-skeleton) then $$ \tilde H_n (X) = \tilde H_n(X_{n+1}). $$ ($\tilde ...
5
votes
0answers
50 views

Intuitive Aproach of Dolbeault Cohomology

I would like to understand an intuitive approach to the definitions of Dolbeault Cohomology (using $\partial$ and $\bar{\partial}$) similar to the one given here. All suggestions are welcome.
3
votes
1answer
74 views

Intuitive Approach to de Rham Cohomology

The intuition behind homology may be summarized in a sentence: to find objects without boundary which are not the boundary of an object. This has geometric meaning and explains the algebraic boundary ...
2
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0answers
45 views

Reduced homology and colimits

I would like to prove that colimits commute with reduced homology, i.e that if $ X = \operatorname{colim}\limits_{n \in \Bbb{N}} X_n$, then $$ \displaystyle\tilde H_k(X) = ...
2
votes
2answers
68 views

The singular homology and cohomology of manifolds vanishes in high dimensions

Let $M$ be an $n$-manifold. It seems that there are two results that the $p$-th singular homology and cohomology of $M$ are zero if $p>n$. But I can not find them in my books of algebraic ...
1
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0answers
55 views

Uncertainties on the details of the Connor-Floyd isomorphism and Formal Group Laws

Let $\Omega^{\bullet}(-)$ be the complex cobordism cohomology. $\Omega^n(X) = \{ (M, f) | f: M \to X \}$ where cobordant maps are identified, $M$ and $X$ are smooth manifolds, and dim($M $) = $n$. ...
8
votes
2answers
203 views

What's the difference between cohomology theories of varieties and topological spaces

There is defined several cohomology theories for algebraic varieties, but in the situation is very different for topological spaces (up to homotopy) for which there is only one cohomology theory for ...
1
vote
1answer
43 views

cohomology ring of a quotient space

what is the cohomology ring $$ H^*((S^3\times S^3\setminus \{e\})/(a,b)\sim (ab,b^{-1});\mathbb{Z}_2)? $$ Here the unit $e$ and the product $ab$ is of the Lie group $S^3=Sp(1)$.
1
vote
1answer
29 views

cohomology ring of some spaces [closed]

What is the cohomology ring $$ H^*(\mathbb{R}^2\times S^1\setminus (0,1);\mathbb{Z})? $$ $$ H^*(\mathbb{R}^2\times S^3\setminus (0,e);\mathbb{Z})? $$ Here $0=(0,0)\in \mathbb{R}^2$, $1=\exp(i0)\in ...
2
votes
1answer
39 views

Direct Sum of Homology Groups and Connected Sums - Something's gone wrong.

I know that for any surfaces, $K, L$ forming $K$#$L$ (where # denotes the connected sum) is done by deleting a disk from each and gluing together the boundary. I also know a few facts about the ...
1
vote
1answer
37 views

Degree of an antipodal map

Let $f:S^n\to S^n$ a continuous map, $n>0$; we consider the induced homomorphism $f_* : H_n(S^n)\to H_n(S^n)$, and, recalling $H_n(S^n)\simeq\mathbb Z$, define $deg(f)\doteq f_*(1)$. I'm asked to ...
1
vote
0answers
28 views

cohomology of labelled configuration space & relation with braid space [closed]

Let: $M$ be a manifold (if we want, we can let $M=S^2$ , $S^1\times \mathbb{R}$, etc.); $(X,*)$ be a pointed topological space. $F(M,k)$ be the ordered configuration space of $k$-tuples on $M$; ...
2
votes
0answers
23 views

Short exact sequence for topological join: split needed

I am desperately trying to solve the following problem: Let $X$ and $Y$ be topological spaces and $X * Y$ their join. Prove that there is a short exact sequence $$ 0 \to \tilde{H}_k(X * Y) \to ...
2
votes
0answers
26 views

cohomology of orbit space by a free group action

Let $G$ be a group. Let a principal $G$-bundle $G\to E\to B$. Then we have a fiber sequence $G\to E\to B\to BG$. Let $k$ be a field. Suppose $H^*(BG;k)$ and $H^*(E,k)$ are known. How to get ...
1
vote
0answers
32 views

cohomology of orbit space

Let $p$ be an odd prime. Let $T^p=S^1\times\cdots \times S^1$ be the $p$-dimensional torus. Then $$H^*(T^p;\mathbb{Z}_p)=\otimes_pH^*(S^1;\mathbb{Z}_p)=\otimes_p\Lambda_{\mathbb{Z}_p}[a].$$ Here ...
2
votes
2answers
91 views

Computing $\pi_4(S^3)$ using Serre spectral sequence

I'm following Davis & Kirk's computation that $\pi_4(S^3)=\mathbb Z/2$ using the Serre spectral sequence but I'm having problems at the very end. We consider a homotopy fibration $X\to S^3 \to ...
5
votes
1answer
131 views

Question on proof of Lefschetz Fixed Point Theorem (from Hatcher Theorem 2C.3)

In Hatcher's statement of the Lefschetz Fixed Point Theorem (2C.3), he has a hypothesis that the space $X$ in question must be a retract of a finite simplicial complex. The first part of the proof ...
1
vote
0answers
53 views

Hopf invariant and homotopy groups of spheres

I am trying to understand how to use Hopf invariant, to calculate $\pi_{4n-1}(S^{2n})$. I've started with defining a new space $X$ adjoining $D^{4n}$ to $S^{2n}$ via a map $\phi\in\pi_{4n-1}(S^{2n}) ...
0
votes
0answers
46 views

The induced map on the de Rahm cohomology of a surjective submersion.

Let $M,N$ be two smooth manifolds and $f: M \rightarrow N$ a surjective submersion (so $f$ and $f_*$ both surjective everwhere). It is straightforward to show that then the pullback of $k$-forms: ...
0
votes
1answer
58 views

Homology class and Betti number for a compact manifold with boundaries

If I take $Q=\mathbb{Z}_N \equiv \mathbb{Z}/(N \mathbb{Z})$, for a genus-g 2-dimensional Riemann surface $\Sigma$, I should have $$H_1(\Sigma; \mathbb{Z}_N)=\prod^{2g}_1 \mathbb{Z}_N,$$ So, ...
3
votes
1answer
48 views

Borel-Moore Homology and Kunneth Formula

Given two algebraic varieties $X$ and $Y$. It is true that $H^{BM}_n(X\times Y) \cong \bigoplus_{i+j=n} H^{BM}_i (X)\otimes_\mathbb{Q} H^{BM}_i(Y)$. I think that the proof is similar to the one in ...
1
vote
0answers
26 views

Is there a complete Link Invariant for links with N crossing.

Are there known examples of pairs $\left(f, N\right)$, where $f$ is a link invariant that is known to be complete when restricted to link diagrams that have at most $N$ crossings? (Ideally, f should ...
1
vote
0answers
23 views

group cohomology of permutation groups

Let $\Sigma_k$ be the permutation group of order $k$. Let $F$ be a field. What is the cohomology $$ H^*(\Sigma_k;F)=H^*(K(\Sigma_k,1);F)=H^*(B\Sigma_k;F)? $$ For $F=\mathbb{Z}/p\mathbb{Z}$ for prime ...
3
votes
0answers
35 views

Upgrading Leray–Hirsch to Künneth for principal bundles

The Leray–Hirsch theorem says that given a fiber bundle $F \to E \to B$ such that $H^*(F)$ is free (as a module over whatever coefficient ring $k$) and, for each $n \geq 0$ there is a set of classes ...
1
vote
1answer
82 views

Betti number and the homology class - what determines the coefficient $Q$?

From Wikipedia: For a non-negative integer $k$, the $k$th Betti number $b_k(X)$ of the space $X$ is defined as the rank (number of generators) of the abelian group $H_k(X)$, the $k$th homology group ...
6
votes
0answers
66 views

Leray's theorem up to some degree

I am interested in the proof of Leray's theorem that relates Čech cohomology and sheaf cohomology. The theorem states that if we have a space $X$, a sheaf $\mathcal{F}$ and a covering of $X$ such ...
3
votes
2answers
75 views

cohomology of classifying space of cyclic group

(1). Let $p$ be a prime number. Let $B\mathbb{Z}_p$ be the classifying space of the discrete group $\mathbb{Z}_p$. How to obtain $$ H^*(B\mathbb{Z}_p;\mathbb{Z}_p)=\mathbb{Z}_p[t]\otimes \Lambda[e]? ...
4
votes
1answer
54 views

Calculating the second cohomology group for trivial group action

Let $G$ be a finite group acting trivially on $\mathbb{R}^*$. How can I compute $H^2(G,\mathbb{R}^*)$? It seems that direct calculations are somewhat hopeless, but the answer should be simple anyway.
1
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0answers
44 views

Cohomology calculation or computation?

I have a terminology question: Does one compute the cohomology of a group, or does one calculate it? Is it more common to speak of cohomology calculation or cohomology computation? Thanks for your ...
2
votes
2answers
70 views

K-theory formulation of the index theorem

The Atiyah-Singer index theorem is one of the most important results in twentieth century's mathematics. It states that for an elliptic differential operator $D$ on a smooth, oriented compact manifold ...
1
vote
1answer
41 views

Relative homology groups

I have to compute the homology groups $H_{n}(X,A)$ when $X$ is $S^{2}$ or $S^{1}\times S^{1}$ and $A$ is a finite set of points in $X$. So, I write the exact long sequence : $...\rightarrow ...
0
votes
0answers
23 views

Basic question about abelianization of Homotopy Groups and Homology [duplicate]

When precisely, is the homology group: $$H_n(T)$$ of a topological space, $T$, isomorphic to the abelianiation of the corresponding homotopy group $\pi_n$? Does this only occur when $n=1$, or is ...