Tagged Questions

2
votes
2answers
68 views

Show that $[l_1 \cdot l_2 \cdot l_3 ] = [l_1 + l_2 + l_3] \in H_1(X)$ The first Homology group of X

Let $l_1$ , $l_2$ and $l_3$ be three paths in X with $l_1 (0) = l_3 (1)$, $l_1 (1) = l_2 (0)$ and $l_2 (1) = l_3 (0)$. Define the loop $l = l_1 \cdot l_2 \cdot l_3 $ (based at $l_1 (0)$). Show that ...
7
votes
5answers
103 views

(Elementary) applications of group (co-)homology

I am looking for an elementary example of a problem, for which one does not need many things to understand the question, but which can be solved with group homology or cohomology. My background is, ...
3
votes
1answer
50 views

Universal coefficient theorem for homology

When Hatcher discusses the universal coefficient theorem for homology (section 3.A, pg. 261), he first takes the exact sequence of chain complexes $$0 \rightarrow Z_n \xrightarrow{i_n} C_n ...
2
votes
2answers
70 views

Equivalence of categories and derived functors.

Don't know if this kind of a dumb question but let $A$ and $B$ be abelian categories and suppose they're equivalent: there are two functors $P: A \rightarrow B$ and $Q: B \rightarrow A$ satisfying the ...
2
votes
1answer
61 views

Resolutions of bimodules as $R^e$-modules.

Let $k$ be a commutative ring, let $R$ be a $k$-algebra, a $R$-Bimodule $M$ over $R$ is a $k$-module with two actions of $R$ on $M$, on the left and on the right, the classical example of this being ...
2
votes
0answers
56 views

Picturing resolutions of complexes

I got a question about resolutions of complexes, I just wanted to make sure I'm looking at them the right way. Let $\cdots \rightarrow P^{-1} \rightarrow P^{0} \rightarrow X \rightarrow 0 ...
1
vote
1answer
61 views

Surjectivity in little diagram

Given the following commutative diagram of exact sequences $$ \begin{array} & & 0 & 0 & 0 &\\ & \downarrow & \downarrow & \downarrow &\\ 0 \rightarrow & A ...
0
votes
0answers
39 views

Coinflation map for homology groups

I wanted to compute explicitly what the coninflation map for homology groups does. Heres the set up: $G$ is an abelian group, $H$ is a subgroup of finite index and $A$ is a $G$-module that has ...
4
votes
0answers
71 views

Cartan-Eilenberg resolutions, adapted classes and acyclic resolutions

I may get grilled for this but here I go: Let $\mathcal{A}$ be an abelian category with enough injectives. What I want to know is VERY VERY specific. Let's say I have a complex in $\mathcal{A}$ $0 ...
1
vote
1answer
106 views

Is there a quasi-isomorphism between a complex of sheaves and its Godement resolutions?

I have a doubt, I read somewhere that the Godement resolution of a sheaf $\mathcal{F}$ is a quasi-isomorphism $\mathcal{F} \rightarrow C^\bullet(\mathcal{F})$. Just right off the bat when I read ...
3
votes
1answer
145 views

Computing the homology of a torus by relative homology of a cylinder

I was trying to compute the homology of a torus with the long exact sequence for relative homology formed quotient, inclusion, and boundary $ \dots\to\tilde{H}_n(A)\overset{i_*}{\to} ...
1
vote
1answer
95 views

Relation between group zero-cohomology and the dual of group zero-homology

Let $\Gamma$ be a group and $A$ be an abelian group and let's take group zero-homology and zero-cohomology, $H_0(\Gamma,A)$, $H^0(\Gamma,A)$. Is there any relation between $H^0(\Gamma,A)$ and ...
2
votes
0answers
87 views

Adapted classes of objects and left (right) exact functors

I had a question about adapted classes of objects, I was confused by the definition and how it relates to left exact functors. Let $\mathcal{A}$ be an abelian category with enough injectives, let $F: ...
1
vote
0answers
53 views

Universal coefficient formula

Let $X$ be a compact manifold (so that all (co)homologyies have finite rank). The universal coefficient formula ($\mathbb{Z}$-coefficient for simplicity) says that we have the following short exact ...
34
votes
2answers
810 views

Algebraic Topology Challenge: Homology of an Infinite Wedge of Spheres

So the following comes to me from an old algebraic topology final that got the best of me. I wasn't able to prove it due to a lack of technical confidence, and my topology has only deteriorated since ...
5
votes
1answer
268 views

cohomology vs homology

I have learned the basic things about cohomology and homology. It seems that homology and cohomology both deal with the same objects, the complexes, but with a different choice of the indexes (for ...
1
vote
1answer
102 views

Universal coefficient theorem of relative homology

In Hatcher, Corollary 3A.4 stated a universal coefficient theorem for relative homology, i.e. the following short exact sequence splits: $0 \rightarrow H_n(X,A) \otimes_\mathbb{Z} G \rightarrow ...
1
vote
1answer
138 views

Homology calculation

When playing around with a homological calculation I came across a short exact sequence of the form $$ 0 \to \mathbb Z^{2g} \to H \to \mathbb Z / 2 \to 0 $$ My background in algebra is not very ...
5
votes
2answers
389 views

Relative homology groups of the torus

I have the following question to problem 2.1.17 in Allen Hatcher's "Algebraic Topology". So far I came up with the following exact sequences (for A and B): $$ \begin{aligned} 0&\rightarrow ...