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.

learn more… | top users | synonyms (2)

3
votes
2answers
60 views

can we derive integral cohomology from rational cohomology and mod p cohomology?

Let $X$ be a topological space. If we know that for $\mathbb{F}=\mathbb{Q}$ and $\mathbb{Z}/p$, for any prime $p$, $$ H^*(X;\mathbb{F})=0$$ for any $*\geq n+1$, can we conclude that $$ ...
3
votes
1answer
46 views

A simpler definition of the snake map?

I would like to ask whether the following definition of the connecting morphism in the long exact sequence in homology of a pair $(X,A)$ is correct. First, define relative cycles and boundaries via ...
1
vote
1answer
47 views

Hatcher, Thm 2.13

Theorem 2.13. If $X$ is a space and $A$ is a nonempty closed subspace that is a deformation retract of some neighborhood in $X$, then there is an exact sequence $$\cdots \longrightarrow \tilde H_n ...
3
votes
1answer
30 views

Higher homology groups relative a lower dimensional subspace

One often works with reduced homology, which (in the case of say, smplicial homology) is defined as the homology relative a point. Now at every grade except zero, the reduced homology objects are ...
1
vote
0answers
19 views

Reduced homology isomorphic to homology relative to a point

I'm going over reduced homology following tom Dieck's Algebraic Topology. I don't understand where the short exact sequence in the excerpt below comes from. $j:(X,\emptyset)\hookrightarrow (X,P)$ is ...
11
votes
0answers
65 views

Why would I define Alexander–Spanier cohomology?

I think I can motivate the definitions of simplicial, singular, de Rham, Čech, and sheaf (co)homology, more or less. I might want to understand bordism, and start by trying to understand ...
0
votes
0answers
14 views

Let $K\leq G$ a closed subgroup of a compact lie group G, where do I find examples in that $H^{1}(K)=0$?

Let $K\leq G$ a closed subgroup of a compact lie group G, where do I find examples in that $H^{1}(K)=0$? $H^{1}(K)$ is the first de Rham cohomology group of $K$.
7
votes
2answers
188 views

Intuition of non-free homology groups?

According to Soft Question - Intuition of the meaning of homology groups and Wikipedia, homology groups can be thought of as counting the number of holes in a given dimension. For instance, a wedge of ...
2
votes
0answers
30 views

Morphism induced in cohomology of a covering space

It is a basic question but I'm stuck. If $p:M\rightarrow N$ is a $m$-fold unramified covering between surfaces, why the morphism induced by $p$ in cohomology at level 2 with coefficients in ...
3
votes
1answer
43 views

Isomorphism on top homology for closed, orientable manifolds

If $M$ and $N$ are orientable, closed $n$-manifolds, then $H_n(M)$ and $H_n(N)$ are generated by the fundamental classes $[M]$ and $[N]$. We can think of them as the sum of the faces of triangulations ...
1
vote
1answer
32 views

Discussion of local homology groups on Hatcher's Algebraic Topology

In p. 126 of Hatcher's Algebraic Topology, there is a discussion after theorem 2.26 about local homology groups. In particular, he says that the local homology groups of $X$ at a point $x \in X$ are ...
1
vote
1answer
60 views

Hochschild (co)homology and derived functors

Suppose that $A$ is (complex) unital algebra. We will consider $A-A$ bimodules $M$: such a bimodule is the same as (say) left $A \otimes A^{op}$ module. Let us define $C_n(A,M)$ as $M \otimes ...
1
vote
0answers
73 views

Relative (co)homology of closed submanifolds

Let $M$ be a closed orientable manifold of dimension $n$ and $A$ a closed submanifold. I obtained these equivalences on relative homology and cohomology groups (with coefficients in arbitrary ring): ...
7
votes
0answers
72 views

Vector space identity from Chow's “You Could Have Invented Spectral Sequences”

In Chow's You Could Have Invented Spectral Sequences (3rd page, left column) appears the following isomorphism of vector spaces: $$\frac{Z_d}{B_d}\cong \frac{Z_d+C_{d,1}}{B_d+C_{d,1}}\oplus ...
1
vote
0answers
29 views

What is the poinacre dual of the projectivization of a line bundle inside the projectivization of the sum of two line bundles?

Let $S:= \mathbb{CP}^1 \times \mathbb{CP}^1$. Let $TS \approx L_1 \oplus L_2$, where $L_1$ and $L_2$ are the pullback of $T\mathbb{CP}^1$ wrt to the respective projection maps. Note that ...
2
votes
0answers
47 views

Intuitive Approach to Sheaf and Cech Cohomology

Sheaf and Cech cohomology $H^*(X,\mathcal{F})$ (which give the same result when applied to good enough topological spaces) are a useful generalisation of the concepts of de Rham and Dolbeault ...
2
votes
2answers
73 views

First Cohomology Group

Is it true that the first cohomology group of a differentiable manifold with finite fundamental group is trivial? If so, could you explain why? Thanks very much
0
votes
1answer
25 views

Prove that the set of all vector fields $V(S^1)$ is a free $C^{\infty}(S^1)$-module

I need to prove that the set of all vector fields, $X:S^1\to TS^1$ name it: $V(S^1)$, is a free $C^{\infty}(S^1)$-module. So i need a basis $\frac {d}{dx_1},...,\frac {d}{dx_n}$ for $V(S^1)$.It's easy ...
4
votes
0answers
58 views

Homology of $\mathbb{R}\setminus A_+$. [duplicate]

Let $A$ be the unit circle in the $xy$ plane in $3$-dimensional real space and let $A_+$ be a semicircle. I have to compute the homology of $\mathbb{R}^3\setminus A_+$. I was thinking that ...
0
votes
0answers
20 views

Why is a curve bounding a punctured torus nullhomologous?

I read (simplified) the following in a paper by Fintushel and Stern: The curve $\alpha$ bounds a punctured (*) torus and thus is null-homologous. Is this just the consequence of applying the ...
1
vote
0answers
30 views

Relative singular homology $H(M,\partial M)$ for a manifold $M$?

Let $M$ be an orientable manifold. What can be said about the relative homology $H(M,\partial M)$? Perhaps one can calculate the homology using excision?
0
votes
0answers
19 views

Suspension of a cup product

Suppose we have a multiplicative cohomology theory $E$. Hence we have suspension isomorphisms $E^n(X, o) \to E^{n+1}(\Sigma X, o)$. Take two elements $x, y \in E^*(X, o)$ of degree's $i,j$ with ...
0
votes
0answers
22 views

On 2-cocycles of a finite group

Suppose that $G$ is an abelian finite group and $A$ is a finite $G$-module. Let us consider the notation $a^g$ for the action of $g \in G$ on $a \in A$. My question is that if $f \in B^2(G,A)$ is a ...
4
votes
1answer
29 views

question about a line in Hatcher about long exact sequence in cohomology

On p.210 Example 3.11 of Hatcher's Algebraic Topology book, he makes the assertion in the following screenshot - where $Y$ is any space, and $I$ is the unit interval. My question is why this is the ...
1
vote
0answers
34 views

cohomology ring of Lie algebras: multiplication

If $\mathfrak{g}$ is a Lie $R$-algebra, then the Chevalley-Eilenberg complex defines the cohomology modules $H^k(\mathfrak{g})$. If $H^\ast(\mathfrak{g})=\bigoplus_kH^k(\mathfrak{g})$, then the ...
1
vote
0answers
17 views

Building a homology/cohomology from generalized tensor invariants

It is known from Tensor Algebra that for a given Tensor $A$ the quantities $I_1 = tr(A)$, $I_{dim(A)} = det(A)$ are the invariants of the Tensor. More precisely, if $\lambda_i$ are the eigenvalues of ...
6
votes
2answers
173 views

A map which is trivial on homology but not on cohomology?

Is there a map $f:X\to Y$ of connected CW-complexes which induces the trivial map $f_*=0:H_i(X,\Bbb Z) \to H_i(Y,\Bbb Z)$ for all $i\ge 1$, but with the property that the induced map on cohomology ...
1
vote
0answers
18 views

Calculate the hochschild homology

Could you help me to calculate the hochschild homology of the following chain complex: $0 \longleftarrow M \longleftarrow M\otimes Z[i] \longleftarrow M\otimes Z[i]\otimes Z[i]\longleftarrow $ where ...
0
votes
0answers
26 views

Is there a proof of domain invariance by applying cohomology theory?

Domain invariance theorem. Given a injective continuous map $f : U \to \mathbb{R}^n$, where $U$ is a nonempty open subset of $\mathbb{R}^n$, then $f$ is open. Almost in all textbooks of algebraic ...
0
votes
0answers
12 views

1-skeleton cellular homology, viewed as a graph

I recently learned the definition of n-skeleton from Hatcher's Algebraic Topology. As I was reading about cellular homology, it seemed like 1-skeletons, viewed as graphs, must be connected by ...
3
votes
1answer
39 views

First examples for topology of non-Hausdorff spaces

I have absolutely no intuition about non-Hausdorff spaces. I would like to understand the topology of non-Hausdorff spaces (in particular spaces obtained by "bad" group actions). As a first example, ...
3
votes
0answers
26 views

Crossed product in relative cohomology

First let me fix some notations: $\Delta^p$ will be a standard $p$-simplex, $\Sigma_p(X)$ the set of all continuous maps $\sigma:\Delta^p \to X$ (where $X$ is some topological space) Let $S_p(X,R)$ ...
3
votes
2answers
193 views

Can the cohomology ring of the two-fold torus be calculated abstractly?

In our lectures, we are given an unusual definition of cohomology and cup products which makes explicit calculations a bit tedious (that is, even more tedious than usual). For the $n$- and ...
2
votes
1answer
72 views

homology over fields

Is it true that the homology of a manifold with field coefficients determines the homology over the integers? I know that by the universal coefficient theorem that $H_k(X; \mathbb{F}) \cong H_k(X; ...
9
votes
1answer
90 views

In what sense is $p$-adic Hodge theory related to ordinary (complex) Hodge theory?

I started reading about $p$-adic Hodge theory in the notes of Brinon and Conrad. I quote (page 7): The goal of p-adic Hodge theory is to identify and study various “good” classes of $p$-adic ...
0
votes
0answers
38 views

Equivalences between categories $\mathcal{K}^b(\text{Injectives})$ and $\mathcal{D}^b(\mathcal A)$ if $\mathcal{A}$ has enough injectives

I have the following question: Let $\mathcal{A}$ be a abelian category and $\mathcal{I}$ be the full subcategory of injective objexts of $\mathcal{A}$. Assume that $\mathcal{A}$ has enough ...
1
vote
0answers
52 views

integral cohomology ring of real projective space

What is the cohomology ring $$ H^*(\mathbb{R}P^\infty;\mathbb{Z})?$$ $$ H^*(\mathbb{R}P^n;\mathbb{Z})?$$ for mod 2 coefficient, the answer is on Hatcher's book and Proving that the cohomology ring ...
2
votes
1answer
55 views

Which maps on the zeroth homology 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
15 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
64 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
97 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) → ...
3
votes
1answer
58 views

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
24 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
52 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
31 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
49 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
43 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
36 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 ...