1
vote
1answer
29 views

Naturality condition for connecting homomorphisms?

I've been reading about the Mayer-Vietoris sequence, but I don't follow a certain naturality condition. Suppose two spaces can be written as $X=X_1^\circ\cup X_2^\circ$ and $Y=Y_1^\circ\cup ...
0
votes
1answer
63 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
1
vote
1answer
29 views

How can I compute $Tor\left(Z_{p},Z_{q}\right)$?

I am self-studying Vick's Homology theory, and now it is on the topic of free resolution. Since I am not familiar with it, I have little ideas about how to compute $$Tor\left(Z_{p},Z_{q}\right)$$ ...
0
votes
1answer
25 views

$[X,F] \to [X,E] \to [X,B] $ is exact sequence of pointed sets

how to show: if $F \to E \to B $ is a fibration then for any space $X$ the sequence $[X,F] \to [X,E] \to [X,B] $ is exact sequence of pointed sets. any hints, thanx.
0
votes
0answers
21 views

Homology of the fixed points of the singular complex of a G-space

Suppose $X$ is a topological space and $G$ a finite group acting on it. We can form the singular complex $C_\bullet(X),$ and then taking homology gives singular homology: $H_*(X) = h_* C_\bullet(X).$ ...
2
votes
1answer
49 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
38 views

Relative singular chains basis

If $(X,A)$ is a pair, then $S_k(X,A):=S_k(X)/S_k(A)$ is free on the singular simplicies of $X$ with image not contained in $A$. Why is this so? I tried to give a proof by checking the mapping property ...
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 ...
0
votes
2answers
42 views

Topological dimension and derham cohomological dimension

If G is a compact complex manifold then does the topological dimension bound the deRham cohomological dimension below? By derham cohomological dimension, I mean the largest extended natrual number ...
0
votes
1answer
26 views

Is it possible to compute homology groups of a space given the Pontryagin ring?

Or similarly, given the cohomology ring of a space, is it possible to compute its cohomology groups? I'm mainly interested in integer and mod 2 homology and cohomology.
1
vote
0answers
36 views

existence of a cofiber sequence

can anyone help me with this problem. thanx. Show that there are cofiber sequence $S^{n+3} \to S^{n+2} \to \sum^{n}\mathbb{C}P^2$ for each $n \in \mathbb{Z}^+$. Conclude that a space of the form ...
1
vote
0answers
69 views

Can group cohomology be used to study fiber bundles?

Is (non-abelian) cohomology used to study vector and principal bundles? Can you give me a text or an article? For example: Consider a vector bundle $E$ with fiber $V$ and base manifold $M$. Consider ...
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 ...
3
votes
0answers
32 views

Reference request: where can I find illustrative, concrete examples of the use of the Eilenberg–Moore spectral sequence?

Pursuant to advice at When does cohomology take pullbacks to pushouts?, I tried to use the Eilenberg–Moore spectral sequence in the simplest conceivable example, for the Hopf bundle $S^3 \to ...
0
votes
0answers
32 views

Book of Pullbacks and Pushouts

what books can I consult for properties of pullback and pushouts in algebraic topology? I need to understand the theory of homotopy in algebraic topology and I started to study pullbacks and push ...
-1
votes
1answer
53 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
4
votes
0answers
78 views

A question about the universal coefficient theorem.

Or rather a couple of questions. Let $X$ be some topological space, $R$ be a (unital) PID and $G$ be an $R$-module. Am I correct in understanding that the singular cochain complexes ...
1
vote
1answer
47 views

Queston on the definition of singular homology

From the Hatcher's can someone told me why $\sigma$ has singularities ? Thank you
4
votes
1answer
87 views

Existence and homotopies of embeddings between simplicial complexes

Let $K$ and $L$ be simplicial complexes, $m=\dim K$, and $h:|K|\rightarrow |L|$ be a continuous map. Show that $h$ is homotopic to a map carrying $K$ into $L^{(m)}$, the $m$-skeleton of $L$. I'm ...
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 ...
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.
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 ...
5
votes
0answers
154 views

Uniqueness Theorems in Axiomatic Homology Theory

Milnor states in his paper 'On axiomatic homology theory' the following uniqueness theorem: If $H$ is a homology theory (in the sense of the Steenrod-Eilenberg axioms) on the category $\mathscr{W}$ ...
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.
1
vote
1answer
50 views

Homology groups

I have to compute the groups $H_q(S^{3},S^1)$ (Singular Homology) I am new in the subject, i have compute some basics groups, but i dont know how to start with this one, if someone could help me, ...
1
vote
1answer
42 views

An exact homology sequence associated with a principal SO(n) bundle

Suppose $P$ is a principal $SO(n)$ bundle, X is its base space. Why is there an exact sequence in homology groups $$ 0 \to H^1(X;\mathbb{Z}_2) \to H^1(P;\mathbb{Z}_2) \to H^1(SO(n);\mathbb{Z}_2)\to ...
3
votes
1answer
88 views

Induced isomorphism of homology implies isomorphism with coefficients in any group?

If $\alpha\colon C \to C'$ is a map of chain complexes (of free abelian groups) that induces an isomorphism on homology $a_{*} \colon H_n(C) \simeq H_n(C')$, then I know that $\alpha$ induces an ...
0
votes
1answer
94 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
38 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 ...
8
votes
0answers
84 views

Is there anything to be learned from the spectrum of a cohomology ring?

Given some topological space, $X$, is there any benefit to studying $Spec(H^*(X))$, or is everything we care about already available "in the algebra"? As $H^*$ is a graded ring, does this question ...
0
votes
1answer
111 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: $H_0(Y)\rightarrow H_0(X)\rightarrow H_0(X,Y)\rightarrow 0$ ...
1
vote
1answer
78 views

Homology and cohomology are basically the same

Is my following understanding correct: A chain complex $(C,\partial)$ is a family $\{C_i\}_{i\geq 0}$ of $R$-modules ($R$ is a given ring) together with a family of $R$-module homomorphisms ...
0
votes
1answer
48 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
1answer
51 views

Mayer-Vietoris for a cover without triple intersections

Let $M = \bigcup_i U_i$ be a cover with open sets $U_i$ such that for for distinct $i,j,k$ we always have $U_i \cap U_j \cap U_k = \emptyset$. I would like to show the existence of the following ...
0
votes
1answer
33 views

Topological interpretation of a zero map.

I have a pair of complexes of modules $A_{\bullet}$ and $B_{\bullet}$, and I want to create a new complex pretty trivially by shifting one complex by 1, and then taking the direct sum of the ...
1
vote
1answer
64 views

Help with Cohomologies and Homologies

My algebraic skills are very weak, so in answering please assume I know close to nothing about algebra, geometry, forms, or the like. I am trying to compute homologies and cohomologies. For ...
0
votes
1answer
254 views

Homology of wedge sum is the direct sum of homologies

I want to prove that $H_n(\bigvee_\alpha X_\alpha)\approx\bigoplus_\alpha H_n(X_\alpha)$ for good pairs (Hatcher defines a good pair as a pair $(X,A)$ such that $A\subset X$ and there is a ...
0
votes
1answer
64 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
55 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
71 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
101 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 ...
2
votes
2answers
204 views

Homology groups of the Klein bottle

I've seen this but didn't really understand the answer. So here is what I tried: According to this picture we have one 0-simplex - $[v]$, two 1-simplices - $[v,v]_a,[v,v]_b$ and two 2-simplices - ...
7
votes
1answer
101 views

$H_0(X)\simeq\Bbb{Z}^k$, where $k$ is the number of path components

I want to prove that $H_0(X)\simeq\Bbb{Z}^k$, where $k$ is the number of path components of $X$. What I tried Since $∂_0=0$, ...
2
votes
0answers
76 views

When do equivariant quasi-isomorphisms of chain complexes induce a quasi isomorphism on the tensor product

Suppose I have chain complexes $A,B,C,D$ where $A$ and $C$ have right $R$-module structures and $B$ and $D$ have left $R$-module structures, and that I have maps $f:A\to C$ and $g:B\to D$ which ...
3
votes
1answer
58 views

singular (co)homology over various fields of same characteristic

Is the following true: if $K$ and $F$ are fields with the same characteristic and $X$ is a topological space, then for any $n$ there holds $$\dim_K H_n(X;K) = \dim_F H_n(X;F)\text{ and }\dim_K ...
5
votes
0answers
63 views

Leray spectral sequence for complexes

Let $f:X\rightarrow S$ be a morphism of schemes. Let $0\rightarrow C_1 \rightarrow C_2 \rightarrow C_3 \rightarrow 0$ be an exact sequence of Abelian sheaves on $X$. Is there a general procedure to ...