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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6
votes
2answers
144 views

Is there an analogue of Eilenberg-Maclane spaces for homology?

Let $G$ be a group and $n$ a positive integer. A connected topological space $Y$ is called an Eilenberg–MacLane space of type $K(G, n)$, if $\pi_n(Y) \cong G$ and all other homotopy groups of $Y$ are ...
5
votes
1answer
59 views

Try to generalize a problem in Hatcher: finite vs. infinite CW-complexes

While solving a problem in Hatcher I got this doubt in my mind, In the 2nd chapter (Homology) Hatcher asked us to prove the following question... If $X$ is a finite dimensional CW-complex then, ...
1
vote
0answers
59 views

Representation of sum of homology classes

Let $X$ be a path-connected topological space, let $x, x' \in H_{k}(X)$ for $k>0$ be represented by two connected manifold i.e. there exist two compact oriented connected manifolds $M$, $N$ and two ...
5
votes
1answer
58 views

Not null homotopic map from $S^3$ to $S^2 \vee S^2$

I'm asked to present a continuous function $\alpha \colon S^3 \rightarrow S^2 \vee S^2$ s.t. it is not null homotopic but taken both projections $pr \colon S^2 \vee S^2 \rightarrow S^2$ the ...
2
votes
1answer
33 views

Does every n-chain have a homology class?

I was under the impression that not every (singular) $n$-chain has a homology class, since $H_n(X) = Z_n(X)/B_n(X)$, and not every $n$-chain is an $n$-cycle. But I came across the following in ...
8
votes
1answer
60 views

Can (singular) homology classes always be represented by images of closed manifolds?

My intuition tells me that if $A \in H_2(M;\mathbf Z)$, then $A$ can be represented by a map $\Sigma \to M$, where $\Sigma$ is a closed (= compact boundaryless) surface, i.e., the connected sum of ...
1
vote
1answer
35 views

Mayer-Vietoris sequence in reduced homology.

By using the Mayer-Vietoris sequence in reduced homology : $...\overset{\Delta_{n+1}}{\longrightarrow} \tilde{H_n}(A)\overset{E_{n}}{\longrightarrow} \tilde{H_n}(X_1)\times ...
3
votes
1answer
65 views

Homology of mapping cone

Let $f:X\to Y$ be a map, and $\text{cone}(f) = CX \sqcup_f Y$ its mapping cone. Let $(H_n)_{n\in \Bbb{Z}}, (\partial_n)_{n\in \Bbb{Z}}$ be a homology theory with values in the category of $R$-modules. ...
7
votes
0answers
116 views

Fundamental class of the connected sum of two closed orientable manifolds

I need to find a representation of $ [M \mathbin\sharp N] \in H_n(M \mathbin\sharp N) $ in terms of the fundamental classes $[M]$ and $[N]$. My idea is that $$ [M \mathbin\sharp N] = [M]+[N]$$ ...
3
votes
1answer
45 views

cup product in cohomology ring of a suspension

Let $X$ be a CW-complex. Let $\Sigma$ be suspension. Let $R$ be a commutative ring. Is the cup product of $$ H^*(\Sigma X;R)$$ trivial? How to prove? Where can I find the result?
1
vote
1answer
70 views

Homology functors defined on $\mathsf{Top} \times \mathsf{Top}$ in Eilenberg-Steenrod axioms?

The Eilenberg-Steenrod axioms state that a homology theory is a sequence of functors $H_n : \mathsf{Top} \times \mathsf{Top} \to \mathsf{Ab}$ satisfying some additional properties. What I don't ...
6
votes
4answers
280 views

An introduction to algebraic topology from the categorical point of view

I'm looking for a modern algebraic topology textbook from a categorical point of view. Basically, I'd like a textbook that uses the language of functors, natural transformations, adjunctions, etc. ...
2
votes
1answer
41 views

Homology of homotopy fiber of degree map between spheres

From Hatcher's Spectral Sequences: Compute the homology of the homotopy fiber of a map $S^k → S^k$ of degree $n$, for $k,n > 1$. Here's where I am: For $k > 1$, the sphere $S^k$ is ...
3
votes
0answers
19 views

What is $HC_0(\operatorname{Spec} k[x,y]/(xy))$?

Does anybody know how to compute $HC_0(\operatorname{Spec} k[x,y]/(xy))$? Here $HC_0(-)$ is the zeroth cyclic homology group. I'm curious since $\operatorname{Spec} k[x,y]/(xy)$ can be viewed as the ...
2
votes
1answer
60 views

Eilenberg–Maclane space: nonvanishing cohomology

I would like to prove (self study) that $H^{np}(K(\mathbb{Z},n),\mathbb{Z})\neq 0$, where $n$ is even , $p\geq1$ and $K(\mathbb{Z},n)$ is the Eilenberg–Maclane space. I used the adjunction between $[ ...
6
votes
1answer
61 views

When does covering preserve rational cohomology?

Let $X$ and $Y$ be compact manifolds. $p:X \rightarrow Y$ is a covering. Generally it is not true that $$ H^* (X , \mathbb{Q} ) = H^* (Y , \mathbb{Q} )$$ For instance, if $Y$ is a sphere with $y$ ...
2
votes
1answer
41 views

$H_n(S^n,A)$ is not trivial

Let $(H_n)_{n\in \Bbb{Z}}, (\partial_n)_{n\in \Bbb{Z}}$ be an ordinary homology theory with values in the category of $R$-modules. Let $A\subset S^n$ be a proper subset. Then $H_n(S^n, A)$ is not ...
1
vote
2answers
67 views

Splitting of Singular Homologies

In Singular homology, let $C_n(X)$ be the free abelian group generated by all the $n$-siimplices of the topological space $X$. Let $U$ be a subspace of $X$, then we have a spliting sequence ...
0
votes
1answer
68 views

homology groups of a torus

How can I find the homology group of a torus without using cellular homology and the CW complex ? in other words , how can i calculate the homology groups of a torus using only relative homology ? I ...
4
votes
1answer
62 views

If $M$ is a 4-dimensional, compact, simply connected manifold with boundary, what is the topology of $\partial M$?

I know if we have a compact simply connected 3-manifold with boundary, then the boundary will be $S^2$. The argument is as following 1) $H_1(M)\simeq 0$; 2) Poincare duality $H_2(M,\partial ...
1
vote
1answer
36 views

Understanding the generator of $H_1 (S^1 \times I)$

I'm trying to work with the space $X=S^1\times I$. It is obvious that $X\simeq S^1$ and therefore $H_1(X)=H_1(S^1)=\mathbb Z$, but I want the properties of $X$ itself. I would assume that a ...
0
votes
1answer
36 views

$\tilde H_0 \oplus \mathbb Z =\mathbb Z\oplus \mathbb Z$

Generaly $\tilde H_0 \oplus \mathbb Z =H_0$. (reduced homology and homology) I'm interested in the specific case $H_0 =\mathbb Z \oplus \mathbb Z$ or a little more generally $H_0 =\bigoplus_{1\leq ...
2
votes
0answers
43 views

References for equivariant cohomology

I am studying the paper An introduction to equivariant cohomology and homology, follwing Goresky, Kottwitz, and Macpherson - Julianna S. Tymoczko but there are too many gaps. I can't link most of ...
1
vote
1answer
48 views

An short exact sequence of $\mathfrak{g}$ of which head and tail are in category $\mathcal{O}$.

Let $\mathfrak{g}$ be a finite-dimensional, semisimple Lie algebra over $\mathbb{C}$. Let $$ 0\rightarrow X \rightarrow Y \rightarrow Z \rightarrow 0 $$ be a short exact sequence of ...
0
votes
2answers
53 views

Does the singular homology functor preserve injectivity and surjectivity?

I was wondering if the singular homology functor preserve injectivity and surjectivity? I've been trying to figure out a proof or counterexample for ages now and I just can't. This came up when I was ...
1
vote
0answers
28 views

$\beta_{q}=\dim H_{q}(X,\mathbb{Q})$

Let's define $\beta_{q}$ to be $q^{th}$-Betti number of X, i.e. the rank of of $H_{q}(X,\mathbb{Z})$, the $q^{th}$-homology of $X$. How can I see that $\beta_{q}=\dim H_{q}(X,\mathbb{Q})$, where ...
0
votes
0answers
52 views

cohomology ring of a subspace of real projective spaces

I learned $H^*(\mathbb{R}P^n;\mathbb{Z}_2)=\mathbb{Z}_2[a]/(a^{n+1})$, $|a|=1$, in topology class, when studying cell complex and cohomology. Now I want to find the cohomology ring ...
5
votes
1answer
41 views

What is the relation between cohomology of $K(\mathbf{Z}_2, n)$ and $K(\mathbf{Z}_2, n+1)$?

I'm reading a proof to a theorem, "Suppose a, b: $H^*(-, \mathbf{Z}_2)\to H^{*+k}(-, \mathbf{Z}_2)$ are two stable (commuting with suspension isomorphism) cohomology operation of degree k. If ...
7
votes
2answers
84 views

Top degree de Rham cohomology determines an orientation

Let $M^n$ be a smooth, compact, orientable, connected manifold. We know then that $H^n_{dR}(M^n)\simeq \mathbb{R}$ by the map $[\omega]\mapsto \int_{M^n} \omega$. I was wondering if, given an ...
3
votes
1answer
33 views

Relation between cohomology of Eilenberg- MacLane space and product of projective space

In an article, it says that "Consider the map $\mathbf{RP}^\infty\times\cdots\times\mathbf{RP}^\infty$(n copies) $\to$ $K(\mathbf{Z}_2,n)$", I think this map is the map related to killing homotopy to ...
1
vote
1answer
30 views

Codifferential and corresponding homology theory

This is the kind of a natural question which can come to mind after completing the standard course in differential geometry and homology theory: lety us start with a smooth manifold $M$. One can ...
3
votes
0answers
24 views

Multiplicative structure in the cohomological Leray-Serre spectral sequence — please elucidate a proof

Let $\pi \colon X \to B$ be a fibration with $B$ a path-connected CW complex. Write $B^p$ for the $p$-th skeleton of $B$ and set: $X_p = \pi^{-1}(B^p)$, $F_p^m = \ker [H^m(X) \to H^m(X_{p-1})]$, ...
1
vote
1answer
38 views

why is $H_0(A)\overset {i_*}{\to} H_0(X)$ injective?

Let $X$ be a a topological space, $A\subset X$. I've been told that it is "trivial" that if each path component of $X$ contains at most one path component of $A$ then $H_0(A)\overset {i_*}{\to} ...
2
votes
1answer
51 views

$R^nf_*\mathbb{Z}$ trivial for a morphism with hypersurface fibers.

I have some questions on local systems. If $f:X\to Y$ is a morphism of projective complex algebraic varieties, $Y$ being a curve, I want to prove that if the fibers of $f$ are smooth hypersurfaces in ...
2
votes
1answer
38 views

Prove $H_0 (X,A)=\bigoplus H_0 (X_i,X_i\cap A)$ for $X_i$ the path components

Let $X$ be a topological spcae, $X_i$ its path components, and $A\subset X$ a subspace. I'm interested in proving $H_0(X,A)=\bigoplus H_0(X_i,X_i\cap A)$. my work by now (I have completely proved ...
0
votes
0answers
20 views

Induced homology morphism of invertible linear transformation

I'm doing some excercises from Hatcher. I'm dealing with excercise 7 in section 2.2 (page 164 in PDF file): For an invertible linear transformation $f : \mathbb{R}^{n} \to \mathbb{R}^{n}$ show that ...
1
vote
1answer
44 views

is the homomorphism induced by the inclusion map is the inclusion map?

let $X$ be a topological space, $A\subset X$ subspace. Consider $i:\:A\to X$ the inclusion, and $i_* :\:H_n(A)\to H_n(X)$ the induced homomorphism. is $i_*$ the natural homomorphism $[a]\mapsto ...
0
votes
1answer
57 views

Why is the inclusion an isomorphism?

Consider $X$ a path-connected space, $A\subset X$ a non-empty subset. My textbook makes the following claim without any explanation, and I wondered if you could help: it says that the inclusion $H_0 ...
1
vote
0answers
37 views

De Rham cohomology of $T^n$ using Künneth formula and Chevalley-Eilenberg theorem.

I want to calculate $H^*(T^n)$ with ring structure using both of these methods. Künneth formula gives $$ H^p(T^n)=H^p(S^1\times T^{n-1})=\bigoplus_{i+j=p}H^i(S^1)\otimes H^j(T^{n-1}) $$ for each ...
2
votes
0answers
20 views

Can the same surface have minimal genus in both a 3-manifold and a 4-manifold?

By a surface of minimal genus I mean in it's homology class: A surface $S_0$ embedded in a smooth manifold $M$ such that any other surface $S$ with $[S]=[S_0]\in H_2(M)$, we have $g(S)\geq g(S_0)$. ...
0
votes
0answers
30 views

Cohomology of conic bundle 3-folds

It is known that for a smooth cubic 3fold $X\subset \mathbb{P}^4$ we have $H^3(X,\mathcal{O}_X)$ (or if you prefer $H^{0,3}(X)=0$). Moreover, if I project off a line $l\subset X$ I can resolve the map ...
0
votes
1answer
29 views

Conditions on subgroups $H, K$ of an abelian group $G$ such that $G/K \cong H/(H \cap K)$

I am trying to prove the equivalence of two formulations of simplicial homology on a manifold $X$, both of which are defined as the quotient of a certain set of simplicial chains on $X$ by a certain ...
1
vote
2answers
104 views

Skew symmetry of indices in cocycles of Cech cohomology

In Cech cohomology, the coboundary operator $$\delta:C^p(\underline U, \mathcal F)\to C^{p+1}(\underline U, \mathcal F) $$ is defined by the formula $$ (\delta \sigma)_{i_0,\dots, i_{p+1}} = ...
2
votes
0answers
34 views

Question about computing cohomology of trivial action on $\mathbb{Z}_{4}$

I'm currently considering the trivial action of the group $G = \mathbb{Z}_{2}$ on the group $A = \mathbb{Z}/4\mathbb{Z}$. It is easy to show that $|C^{2}(G,A)|$ = $2^{8}$ and that $|B^{2}(G,A)| \leq ...
1
vote
0answers
24 views

Hochschild homology of a free commutative algebra

Let $V$ be a graded vector space over $k$. Let $Com(V)$ be the free commutative algebra over V. Let $HH_*(-,k)$ be the Hochschild homology with coefficients in $k$ functor. My questions are : $$ ...
1
vote
1answer
25 views

Any characterization of $H^2(\mathbb{Z}_n,\mathbb{Z}_m,\theta)$?

I've been reading chapter 7 of An Introduction to the theory of groups by Rotman related to Extensions and Cohomology, and there is something that is not completely clear to me. Given the exact ...
1
vote
1answer
49 views

Computing the homology groups of a quotient space of the sphere

I want to solve following question: Let $A$ denote the union of equatorial circle and the north pole on $S^2$. Let $X=S^2 / A$. Compute the homology groups of X. I calculated that $H_2(X) = \Bbb ...
7
votes
1answer
58 views

Boundary of boundary of singular cube is zero (Spivak)

At the bottom of page 99 of M. Spivak's Calculus on Manifolds he arrives at the formula $$\partial (\partial c)=\sum_{i=1}^n \sum_{\alpha=0,1} \sum_{j=1}^{n-1} \sum_{\beta=0,1} ...
2
votes
0answers
62 views

Reduced homology groups of a space which is the union of finitely many open subsets

This is exercise 33 (p.158) from section 2.2 in Hatcher's Algebraic Topology: Suppose the space $X$ is the union of open sets $A_1, \ldots, A_n$ such that each intersection $A_{i_1} \cap \cdots ...
2
votes
1answer
55 views

Poincaré lemma on a space with trivial homology group

Today I read about Poincaré's lemma from do Carmo's book Differential Forms and Applications. It says that A closed differential $k$-form on a contractible space is exact. I wonder if the ...