0
votes
1answer
57 views

History of five lemma

I am interested in the history of five lemma. Who was first to prove it and What was the purpose of proving it ? http://en.wikipedia.org/wiki/Five_lemma
2
votes
1answer
43 views

The space $\Delta^n$ with all faces of the same dimension.

If the space $A$ is obtained from $\Delta^n$ by identifying all faces of the same dimension; What is a $\Delta$-complex structure on the space $A$? And how can you compute the Simplicial Homology ...
0
votes
1answer
35 views

Tor of submodule

Let $R$ be a $CRing$. If $i:A \rightarrow B$ is the inclusion of a $R$-subalgebra A into an $R$-algebra $B$, then what is ther relationship between: $Tor_{A^e}$ and $Tor_{B^e}$?
1
vote
0answers
26 views

Coproducts and Hochschild

I $\{X_i\}$ is a small family of associative $\mathbb{C}$-algebras and $X$ is their free product. Then I have two questions: 1) Why is $X$ their coproduct? 2) Is the Hochschild homology of X ...
2
votes
1answer
77 views

Question about the definition of homology

i have this paragraphe: Can someone explaine me what it means ? if i understand $H_n$ measure the numbers of holes with dimension $n$ but what about $H_0$ what is the relation between the holes of ...
1
vote
1answer
35 views

Question about relative singular homology groups

I know that the sphere $S^{\infty}$ is contractible, but why if $H$ is a Hilbert space then we have $$H_q(H,S^{\infty})=0, q\in \mathbb{N}?$$ Please help me Thank you
0
votes
1answer
31 views

Homotopy between two homomorphisms and homology

If I have two chain complexes $C$ and $D$ and I suppose that there is a homotopy between $\phi, \psi:C \rightarrow D$ (i.e there is a sequence of homomorphisms $(K_n: C_n\rightarrow D_{n+1})$ such ...
5
votes
1answer
98 views

Isomorphism in cohomology is an isomorphism in homology

Let $f:X \to Y$ be a continuous map between topological spaces and $R$ some coefficients. From the universal coefficient theorem for homology we immediatly get, that if $H_*(f,\mathbb{Z})$ is an ...
2
votes
1answer
28 views

What does the $\Omega$ represent in $\Omega S^{n}$?

To put my question in context, I'm reading Hatcher's book on Spectral sequences is which is say " The suspension homomorphism $E$ is the map on $pi_{i}$ induced by the natural inclusion map ...
-1
votes
1answer
48 views

Relative homology of ball and sphere

What is the result of $H_k(B^n,S^{n-1}; \mathbb{A })$ and in any book can i found the proof ? And what about $H_n(S^{n};\mathbb{A})$ (sigular homology of the sphere )?? Please help me. Thank you
1
vote
1answer
45 views

Queston on the definition of singular homology

From the Hatcher's can someone told me why $\sigma$ has singularities ? Thank you
8
votes
1answer
110 views

Universal Coefficient Theorem - what kind of morphisms?

Let $G$ be an $R$-module, where $R$ is a P.I.D., and let $X$ be a topological space. We have the exact sequence $$0 \rightarrow H_n(X) \otimes G \rightarrow H_n(X; G) \rightarrow ...
1
vote
1answer
28 views

Small question on relative holology

if $Y\subset X$ , what is $\ker \delta$ such that $\delta: H_k(X,Y)\rightarrow H_k(Y)$ ? is it $\ker \delta = H_k(X,Y)$ ? $\delta$ is the usual connecting homomorphism from the long exact sequence ...
0
votes
1answer
65 views

Homology and Exact Sequence

I have this exact sequence: $$0\stackrel{f}{\rightarrow} H_k(X,C)\stackrel{g}{\rightarrow} H_k(X,A)\stackrel{h}{\rightarrow} 0$$ Can I say that $H_k(X,A)=H_k(X,C)$ and why? Please; Thank you.
2
votes
0answers
68 views

Definition for a bar resolution for a module over a dg category

Let $ \mathcal{A}$ be a dg category and define a right $ \mathcal{A}$ module to be a dg functor $ M: \mathcal{A}^{op} \rightarrow dif\ k$ where $dif\ k$ is the category of differential $k$ modules ...
0
votes
2answers
57 views

The singular homology and cohomology of a topological space with coefficients in a zero characteristic field.

I have a field with zero characteristic, like $K=\mathbb{C},\mathbb{R}$ and I want to show that the homology groups and cohomology groups with coefficients in these fields satisfy: $$H_n(X,K) \approx ...
0
votes
0answers
59 views

Exact sequences and spectral sequences

We have the well-known theorem for cohomological spectral sequences as follows: Theorem: Let $(E_r , d_r )$ be a third quadrant spectral sequence and let $E^{p,q}_2‎\Rightarrow‎ H^n(Tot(M)$. a) If ...
6
votes
0answers
102 views

Morita-invariance of Hochschild (co)homology.

Ok, I'm reading this paper by Christian Kassel on associative algebras and Hochschild (co)homology and on page 19 he says that Hochschild homology is Morita-invariant, by which he means that if $R$ ...
1
vote
0answers
36 views

Rank of homology group

$x_0$ is the unique a global minimum and let $c=f(x_0)$ in a Hilbert space, let $\theta$ be an other critical point of $f$ non minimum. the Morse type number of $x_0$ is ...
3
votes
1answer
53 views

Relative homology

Let $E$ be a real banach space if $E=Y\oplus Z$ and if $S^{m-1}$ is the sphere on $Y$ ($\dim Y =m $) why $H_{m-1}(E \setminus Z)\simeq H_{m-1}(S^{m-1})$ ? please thank you
0
votes
2answers
66 views

Question on relative homology

i have this: where $|\tau|$ is the support of the chain $\tau$, i don't understand the first part why $[\sigma]=0$ in $H_{m-1}(\phi^{c+\varepsilon},\emptyset)$ ??? Please, thank you.
4
votes
1answer
129 views

Finding example of quasi isomorphism that has no quasi inverse

Between differential graded algebra $V,W$, a chain map $f\colon V\to W$ induces homomorphism between its homology. If this becomes an isomorphism between the homology of $V,W$, call this quasi ...
0
votes
1answer
92 views

Example on relative homology

I am trying to prove that $$H_p(B_{n+1},S_n;\mathbb{A}) \cong \left\{\begin{array}{ll} H_{p-1}(S_n,\mathbb{A}) & \text{if } p\geq2\\\ 0&\text{if } p=1, n\geq 1\\ \mathbb{A} &\text{if } ...
1
vote
1answer
37 views

Question on relative homology [duplicate]

i have that $H_p(X,Y)$ is isomorphic to $Z_p(X,Y)/(B_p(X)+C_p(Y))$, where $Z_p(X,Y)=\lbrace \sigma\in C_p(X), \partial\sigma\in C_{p-1}(Y)\rbrace$ and i want to deduce that $H_0(X,Y)$ is the free ...
0
votes
1answer
110 views

Relative homology and path connected space

I want to prove that if $X$ is a path connected space and if $Y$ is nonempty then $$H_0(X,Y)\simeq 0$$ it is sayed that we have this chain: $H_0(Y)\rightarrow H_0(X)\rightarrow H_0(X,Y)\rightarrow 0$ ...
0
votes
1answer
47 views

Small question on relative homology

Let $(X,Y)$ be a paire of topological space how to prove that the boundary map $\partial: C_p(X)\rightarrow C_{p-1}(X)$ send $C_p(Y)$ on $C_{p-1}(Y)$ if i take a singular p-simplexes of $Y$ than why ...
0
votes
1answer
66 views

Question on “Homotopy invariance”

i have this from Hatcher's book "Algebric topology" And i don't understand why $\displaystyle \partial P(\sigma)=\sum_{j\leq i}(-1)^i(-1)^j F\circ (\sigma\times ...
0
votes
1answer
92 views

Question on singular homology

please where i can found the prove of this: If $X$ is a topological space and $(X_{\alpha})_{\alpha\in I}$ is the family of it's path connected components. Prove that for each $n\in \mathbb{N}$, ...
1
vote
0answers
55 views

Question on Singular homology

i have this example : The homology of the space $X=\lbrace x \rbrace$ . for all $p\geq 0$, there is a unique singular p-simplex $\sigma_p:\Delta_p\rightarrow X$, and for $p>0$ we have ...
1
vote
0answers
43 views

Isomorphism of relative homology

I have $2$ pair of chain complexes $(X_1,Y_1), (X_2,Y_2)$. And $$H_i(X_1,\mathbb Z) \cong H_i(X_2,\mathbb Z), H_i(Y_1,\mathbb Z) \cong H_i(Y_2,\mathbb Z).$$ Is it true that $$H_i(X_1, Y_1,\mathbb Z) ...
0
votes
1answer
62 views

Why call them cycles and boundaries?

I have a small question. Why we have this designation: $n$-cycles for $Z_n$ and $n$-boundaries for $B_n$ ? Why they are called cycles and boundaries ? ...
0
votes
0answers
61 views

acyclic implies identity null-homotopic?

I have proved the following for a chain complex $\mathcal{C}_{*}$ where the $\mathcal{C}_i$ are free $\mathbb{Z}$ modules, $\mathcal{C}_i = 0$ for $i>0$. The identity map on $\mathcal{C}_{*}$ is ...
1
vote
0answers
54 views

Question on Snake lemma

we have Short exact sequence of chain complexe $0\rightarrow C\xrightarrow[]{f}D\xrightarrow[]{g}E\rightarrow 0$ i want to prove that there existe a longue exact sequence of modules $$...\rightarrow ...
2
votes
2answers
70 views

Prove that $H_n(A \sqcup B) \cong H_n(A) \oplus H_n(B)$ for all $n \in \mathbb{Z}$.

Let $A, B$ be topologyical spaces. Then, I want to prove that $H_n(A \sqcup B) \cong H_n(A) \oplus H_n(B)$ for all $n \in \mathbb{Z}$. I know how to prove this from the Mayer-Vietoris theorem, but I'm ...
3
votes
0answers
98 views

Simplicial homology for n-simplex

I've just started to study homology theory. And I'm trying to calculate all $H_n(\Delta_N)$ for some $N$. I know that the number of $m$-simplex in $N$-simlex is $b_{N,m}={N+1 \choose ...
0
votes
0answers
41 views

A general form of $H_i(\Gamma,\mathbb{Z})$

Given a group $\Gamma (\subset PGL_d(K))$, is there some general form to express $H_i(\Gamma,\mathbb{Z})$ in terms of it?
3
votes
1answer
55 views

Proof that derived functors don't depend on choice of resolution.

Can somebody help me out with this? Let $X$ be an object in an abelian category $A$ with enough injectives, let $0 \rightarrow X \rightarrow M^{\bullet}$ be an injective resolution , let $0 ...
5
votes
1answer
126 views

What is the integral homology of $\mathrm{GL}_2(\mathbb{Z}[i])$?

I am currently trying to compute homology groups of general linear groups over the ring of integers of an imaginary quadratic number field. As I would like to check my results I would like to know if ...
1
vote
5answers
201 views

Split-Lemma for chain complexes

Suppose $k$ is a field and $A$, $B$ and $C$ are chain complexes of $k$-vector spaces, i.e., objects in $\mathbf{Ch}(k\text{-}\mathbf{Vect})$. Is there are chain complex version of the split lemma, ...
3
votes
1answer
492 views

Does taking the direct limit of chain complexes commute with taking homology?

Suppose I have a directed system $C_i$, $i\in\mathbb{N}$ of chain complexes over free abelian groups (bounded below degree $0$) $$C_i=0\rightarrow C^{0}_{(i)}\rightarrow ...
3
votes
2answers
85 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 ...
6
votes
5answers
179 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
68 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
126 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
128 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
72 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
70 views

Surjectivity in little diagram

Given the following commutative diagram of exact sequences $$ \begin{array} & & 0 & 0 & 0 &\\ & \downarrow & \downarrow & \downarrow &\\ 0 \rightarrow & A ...
4
votes
0answers
102 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
139 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
222 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} ...