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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3
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0answers
20 views

Grading of Cech-de Rham cohomology

I am currently self-studying Bott and Tu. In chapter 2 the Cech-de Rham cohomology is introduced and I thought I had understood it well enough. However when I got to chapter 3 on spectral sequences I ...
1
vote
1answer
23 views

Definition of Coboundary

I was reading these lecture notes from Duke University and found a typo I think. If it's not a typo then I'm really confused. Anyway, on page $95$ shouldn't $$B^p=\operatorname{im}\delta^{p+1}:C^{...
1
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0answers
16 views

Generalization of de Rham cohomology, or cohomology for non-smooth case

Let $\Omega\subseteq \mathbb{R}^{3}$ be a simply connected domain and $f:\Omega\to \mathbb{R}^{3}$ be a vector field on $f$. If $f$ is smooth vector field and $\nabla\cdot f=0$, then $f=\nabla\times g$...
4
votes
0answers
44 views

What's a cohomology that's not defined from a cochain complex?

According to Wikipedia: In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups associated to a topological space, ...
-3
votes
0answers
59 views

What is cohomology? [on hold]

What is cohomology? Can someone describe the meaning and concept of cohomology in a visual non-technical tongue?
3
votes
1answer
86 views

Show that the homology induced homomorphism $f_*:H_3(RP^3)\rightarrow H_3(S^2\times S^1)$ is a zero map.

Let $f:\mathbb{RP}^3\rightarrow S^2\times S^1$ be a continuous map. Prove that induced map $f_*:H_3(\mathbb{RP}^3)\rightarrow H_3(S^2\times S^1)$ is a zero map. I found that the third homology of ...
6
votes
1answer
63 views

For $n\geq 2$, any continuous map $f:\mathbb{C}P^n\rightarrow S^2$ induces the zero map on $H_2(*)$

I am working through an old qualifying exam from another university. My course did not cover as much material as what is on this test (e.g. we did not cover cohomology). So I am just working through ...
2
votes
1answer
28 views

Cohomology ring of $G$ based on its Sylow.

I have a bunch of notes made from a professor about cohomology that states that If $S$ is a $p$-Sylow subgroup of $G$ ($\vert G \vert <\infty$), then $$H^{\ast}(G,\mathbb{F}_p)\leq H^{\ast}(...
0
votes
0answers
26 views

A doubt in Whitehead's proof about cohomology with local coefficients [on hold]

In the proof of Theorem 4.9 says that $p^*:H^n(X_n,X_{n-1};G|X_n) \to Hom(H_n(\widetilde{X}_n, \widetilde{X}_{n-1}),G_0)$ has image $Hom^{\pi}(H_n(\widetilde{X}_n, \widetilde{X}_{n-1}),G_0)$. ...
2
votes
1answer
28 views

Map inducing zero on first cohomology is nullhomotopic (plus assumptions on fundamental group and universal cover)

Currently I am studying for a topology exam next week and came across an exercise where I could need some hints (cf. here): Let $X$ be a path-connected space with $\pi := \pi_1(X,*)$ abelian and ...
2
votes
1answer
64 views

Counter-example for $\tilde{H} (X/A) \cong H (X, A)$?

Yo! I was not able to find a counter-example to $$\tilde{H} (X/A) \cong H (X, A)$$. It's a well known fact that for cofibrations $A \hookrightarrow X$ (or more generally whenever $A$ is a deformation ...
-3
votes
0answers
24 views

What is the definition for the subgroup $Z^p(K;G)$ in cohomology? [on hold]

I am self-studying cohomology. And I figured that the kernel of a cohomology group $H^p(K;Z)$ called $Z^p(K;Z)$ cocycle subgroup is a set when the p-coboundary $d$ operator acts on $C^p(K)$ a p-...
8
votes
2answers
85 views

Homology and cohomology of 7-manifold

I have the following problem: Let $M$ be a connected closed $7$-manifold such that $H_1(M,\mathbb{Z}) = 0$, $H_2(M,\mathbb{Z}) = \mathbb{Z}$, $H_3(M,\mathbb{Z}) = \mathbb{Z}/2$. Compute $H_i(M,\...
1
vote
1answer
44 views

Classifying space of $GL_{n}(\mathbb{F})$?

I was looking for the classifying space of the general linear group $GL_{n}(\mathbb{F})$ over a field (of characteristic either zero or positive, finite or infinite), but unfortunately I didn't manage ...
1
vote
1answer
39 views

Bott and Tu compact cohomology of the circle “differential forms in Algebraic Topology”

On page 27 of that book, it is claimed that the inclusion map $\delta$ which maps a form from the non-empty intersection of two open covers of the circle to the disjoint union of those covers has a ...
1
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0answers
21 views

Maps inducing identity in Hochschild and cyclic theories

Let $A$ be a unital algebra over $\mathbb{C}$, $M$ be an $A$ bimodule, $C^n(A,M)$ be a space off all $n$-linear maps $f:A^{n} \to M$ (to be called $n$ cochains) and define $b:C^n(A,M) \to C^{n+1}(A,M)$...
2
votes
1answer
39 views

On Steenrod's realization of cycles problem.

There is old problem of realization homology classes of (closed) manifold $M^n$ by fundamental classes of its submanifolds. Partially it was solved by René Thom in his "Quelques propriétés globales ...
2
votes
1answer
72 views

Homologically trivial immersion

Are there any examples when some manifold $N$ maps in other manifold $M$ as codimension 1 submanifold, its fundamental class is zero in the homology of M, but still this map $i\colon N\to M$ induces a ...
1
vote
0answers
46 views

Obstruction to lifting a map from the base space to the total space.

Suppose $\pi :E \to B$ is a fibration with fibre $F$ above a chosen base point. Then I am trying to solve when a map $f$ from a manifold $M$ to $B$ lift to a map $g:M \to E$. The answer given is they ...
2
votes
1answer
25 views

Functoriality of internal hom of chain complexes

The internal hom of chain complexes $[-,-]$ is supposed to form a bifunctor $$\operatorname{Ch}_\bullet(\mathsf{Mod}_R)^\mathrm{op} \times \operatorname{Ch}_\bullet(\mathsf{Mod}_R) \to \operatorname{...
2
votes
1answer
75 views

Show that if $\phi$ is a cocycle then $\phi(f\cdot g)=\phi(f)+\phi(g)$ for

This is an exercise from Hatcher: Let $X$ be a topological space, $G$ an abelian group. Regarding a cochain $\phi\in C^1(X;G)$ as a function from the paths in $X$ to $G$, show that if $\phi$ is a ...
3
votes
1answer
55 views

Stronger version of Acyclic Models Theorem

Let $\mathscr{C}$ be an abelian category. If $P_\bullet \in \operatorname{Ch}_{\geq 0}(\mathscr{C})$ is a bounded below complex of projectives, and $C_\bullet \in \operatorname{Ch}_{\geq 0}(\mathscr{C}...
4
votes
0answers
96 views

Cup/cap product: sheaf cohomology vs singular cohomology

Is anyone aware of a good resource which deals with how the cup/cap products of sheaf cohomology classes are a generalization of those in singular cohomology? I would say that I already understand the ...
0
votes
1answer
33 views

Homology group versus group homology

If we have a simplical complex $K$, then we are able to define $C_i(K)$ as the free abelian group over $\mathbb Z_2$ with the basis of all $i$-dimensional simplices. By using the boundary map we are ...
1
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1answer
41 views

Computing the homology of the torus with coefficients in $\Bbb F_p$, using two methods

I have some trouble to compute the homology of the torus with coefficients in $\Bbb F_p$ for $p$ a prime number. In particular I have a problem for $H_1$ : 1) The first way to compute it is to use ...
0
votes
0answers
21 views

$0$-cohomology of a presheaf and the associated sheaf

Let $F$ be a presheaf of abelian groups on $X$ (topological space for example, nice enough) and $\tilde{F}$ the associated sheaf. Then if $f\in H^0(X,\tilde{F})$, we can find a cover $\mathfrak{U}=(...
3
votes
1answer
69 views

First cohomology group on a Riemann surface with all wedge products equal to zero

Sorry for the strong edit, but I realized my question had a easier formulation: Can there be a Riemann surface $X$ with the property $\sigma\wedge \tau=0$ for every $\sigma,\tau\in H^1(X,\mathbb{C})$?...
2
votes
0answers
45 views

singular homology of a simplicial complex [duplicate]

On Page 108 of the book Algebraic Topology, Allen Hatcher, the singular homology of a topological space $X$ is defined to be the homology of the chain complex by setting the $n$-chains $C_n(X)$ as the ...
4
votes
1answer
61 views

Natural isomorphism $\tilde H_i(X) \xrightarrow{\cong} \tilde H_{i+1}(\Sigma X)$ where $\Sigma X$ is the suspension of $X$.

Define $\Sigma X$ to be the quotient space of $[-1,1]\times X$ obtained by identifying ${0}\times X$ and ${1}\times X$ to two points respectively. For any homology theory (satisfying Eilenberg-...
2
votes
0answers
39 views

What are some examples of cohomology theories without a corresponding classifying space?

The general nonsense of cohomology theories is that each one "should" be presented by a classifying space, so that maps into this space give the cohomology (before passing to connected components). ...
1
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0answers
33 views

Why is topological K-theory equivalent to nonabelian cohomology with respect to the stable unitary group?

I was reading on the $n$Lab page for topological K-theory that taking cohomology of a smooth space with respect to the smooth $\infty$-stack $\mathbf{Vect}$ is equivalent to taking its cohomology with ...
2
votes
0answers
33 views

Duality between Thom space and a manifold embedded into a sphere

In a document https://www.math.purdue.edu/~gottlieb/Bibliography/53.pdf (s. 19) it is mentioned that there is a map $S^n \to M^+ \wedge \mathrm{Th}\left(\nu \left(M, S^n\right)\right)$, which gives a ...
1
vote
1answer
41 views

tom Diecks's proof of $H_1(X)\cong \pi_1(X,x_0)^{ab}$

My question is about tom Dieck's proof of Theorem 9.2.1 on page 227, which states that if $X$ is path connected, then the induced map $$h:\pi_1(X,x_0)^{ab}\to H_1(X)$$ is an isomorphism. Specifically,...
1
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0answers
82 views

Chain Homotopy in abelian category

When dealing with complexes of modules or groups, the following lemma is pretty easy: If $f,g :E\rightarrow E'$ are homotopic, i.e. $f-g=d'h+hd$ for some h, then $f,g$ induce the same homomorphism ...
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votes
0answers
12 views

Trying to understand an induced homomorphism

Let's say C is a simplicial complex, and let K be a subcomplex with the same underlying 0-skeleton. Let B(C) be a subcomplex of C, and B(K) be a subcomplex of K such that B(K) is a subcomplex of B(C) ...
0
votes
0answers
53 views

Is the cohomology ring of a CW complex computable?

There is a well-developed technology for computing the cohomology groups of a CW complex, cellular cohomology. It reduces the problem of computing cohomology to the two simpler problems of (1) ...
2
votes
0answers
46 views

Exciting applications of the Riemann-Roch-theorem for Riemann-surfaces

This semester I took a lecture on Riemann surfaces. The professor proved the Riemann-Roch-theorem (stated below). As an application of it, he proved elementary results, we did earlier in the course ...
3
votes
2answers
43 views

The Incluson Map $S_1\to S_1\times S_1$ Induces an Injection in First Homology.

Let $T=S^1\times S^1$ be the torus and $A=S^1\times \{x_0\}$ be the "vertical" circle in the usual depiction of the torus as a tyre tube sitting "horizontally". Let $i:A\to T$ be the inclusion map....
0
votes
0answers
19 views

Let f and g are homotopic chain maps of (C,d)

Let $f$ and $g$ are homotopic chain maps of $(C,d)$, then $f_{*n}=g_{*n}: H_{n}(X) \to H_{n}(X')$. Can you give me the example showing that the converse is not true?
1
vote
1answer
17 views

Computation of the first group cohomology with coefficeint in $\mathbb Z_2$

How can I compute the first cohomology group $H^1(G,G\mathbb Z_2)$, where $G$ is the integer numbers i.e. $\mathbb Z$?
5
votes
1answer
74 views

The Euler Characteristic of $\mathbf RP^2$ is a Fraction.

Problem 22 in Section 2.2 in Hatcher's Algebraic Topology reads For $X$ a finite CW complex and $p:\tilde X\to X$ an $n$-sheeted covering space, show that $\chi(\tilde X)=n\chi(X)$. Here $\chi$ ...
1
vote
1answer
40 views

Proof of More Flexible Mayer-Vietoris for Calculating Homology Groups

On pg. 150 of Hatcher's Algebraic Topology, the author writes: Let $X$ be a topological space and $A$ and $B$ be subspaces of $X$ such that $X=A\cup B$. Suppose there are open subspaces $U$ and $V$ ...
1
vote
1answer
19 views

Relative Homology of the Mapping Cylinder w.r.t a Subspace

Given a continuous map $f:X\to Y$, the mapping cylinder of $f$ is defined as the space obtained from $(X\times I)\sqcup Y$ by identifying $(x, 1)$ with $f(x)$ for all $x$. Let $f:S^n\to S^n$ be a ...
0
votes
1answer
36 views

Proof of the cellular boundary formula

I'm trying to understand the proof in Hatcher (p. 141) of the cellular boundary formula. Now there's one thing that Hatcher does several times in his book and that I don't understand very well: he ...
1
vote
0answers
60 views

Proving a function isn't homotopic to a map to the boundary

Let $X$ be a compact Hausdorff space and $S$ a finite dimensional, convex, Hausdorff space. Moreover, let $f:X \to S$ be a closed, surjective map with $\tilde{H}^q(f^{-1}(s))=0$ for all $q\ge 0$ and $...
7
votes
2answers
113 views

Is Floer homology always isomorphic to the singular homology of some space?

After I studied Morse homology, I'm now studying the following Floer homology theories : 1) Symplectic Floer homology ; 2) Floer homology of lagrangians ; 3) Heegard-Floer homology ; ...
6
votes
3answers
99 views

Intuitive reason why the Euler characteristic is an alternating sum?

The Euler characteristic of a topological space is the alternating sum of the ranks of the space's homology groups. Since homeomorphic spaces have isomorphic homology groups, however, even the non-...
0
votes
1answer
39 views

Integral homology groups of the complexe projective n-plane

I am reading "Morse theory" by Milnor and on page 27 we have proved that the homotopy type of CP^n is of a CW-complex of the form : a 0-cell attached to a 2-cell attached to a 4-cell ... attached to a ...
1
vote
0answers
26 views

Mayer-Vietoris sequence for the union of an arbitrary number of topological sets. [closed]

The Mayer-Vietoris sequence for cohomological groups is of the form $$\dots\to H^i(A\cup B, G)\to H^i(A,G)\oplus H^i(B,G)\oplus H(A\cap B,G)\to H^{i_1}(A\cup B,G)\to\dots$$ How does it change when we'...
1
vote
1answer
41 views

Top cohomology of a non-orientable smooth surface with boundary.

I would like to know what the singular relative cohomology $H^2(M,\partial M;\mathbb{Z})$ of a smooth connected surface with boundary $M$ is. In the orientable case I did the following: The zero-th ...