0
votes
0answers
15 views

Question about Propety of Homology

I have this theorem, with the proof, but i dont understand, why they prove that $i_{1_*}, i_{2_*}$ are injective, we have that $i_{j_*},j=1,2$ are induced by an inclusion it is injective, so they are ...
5
votes
1answer
55 views

Definition of multiplication in Grothendieck ring

Let $X$ be a smooth variety over an algebraically closed field $k$ of dimension $n$. Consider the Grothendieck Group $K(X)$ of coherent sheaves on $X$, i.e. the free abelian group generated by ...
1
vote
1answer
40 views

Is the constant sheaf $\mathbb{Q}$ injective?

Let $X$ be a topological space, and let $\mathbb{Q}$ be the constant sheaf of abelian groups on $X$ associated to the group of rational numbers under addition. Is $\mathbb{Q}$ an injective object in ...
2
votes
1answer
140 views

Homotopy invariance in homology

i have this from Hatcher's book "Algebric topology" And i don't understand why we have $i-1$ in $(-1)^{i-1}$ and strict inequality in $P\partial(\sigma)$ ? Please. Thank you.
2
votes
0answers
23 views

An exercise in homology computation / What is the geometric fixed points of an Eilenberg Maclane Spectrum?

The question I want to ask has a reasonably elementary formulation and I think there is a good chance it can be answered in this form (by someone more computationally skilled than me, or perhaps by ...
1
vote
1answer
42 views

Property of Homology: group isomorphism

I have this proposition, but I don't understand how they use the axiom 5, since in the axiom 5; $f,g: (X,A)\rightarrow (Y,B)$ and in the theorem we have $f:(X,A)\rightarrow (Y,B)$, $g:(Y,B)\rightarrow ...
0
votes
1answer
40 views

A short exact sequence

I have this proposition, and I don't understand how to do to obtain the short exact sequence: where axiom 4 is:
4
votes
2answers
105 views

Property of homology

I have this proposition, and I have two questions: 1) Why $H_k=\text{Im} i_*\oplus \ker r_*$ ? 2) Why $j_*: \ker r_*\rightarrow H_k(X,A)$ ? Edit: For the second, I try the 1th theorem of ...
1
vote
2answers
61 views

Property of excision of Homology

Please what is the difference between these two excision property: Let $X$ a topological space, $A$ a sub-space of $X$ and $U\subset A$ such that $\overline{U}\subset \stackrel{\circ}{A}$ . The ...
2
votes
1answer
123 views

Homology and topological propeties

i have this theorem with it's proof but i don't understand the last part They use this proposition: My question is Why $\varphi^c\cap U_i$ is closed and pairwise disjoint ? where ...
1
vote
0answers
20 views

Extension to rational and real chains

In the paper on stable commutator length, D. Calegari says that generalized $\operatorname{scl}$ function can be extended by linearity to rational group $1$-chains and by continuity to real chains ...
0
votes
0answers
84 views

Question about Homology from the Chang's book: Methods in Nonlinear analysis

In the K.C.Chang's book in page $336$ of the book this corollary without prove there is a theorem before it but I don't know if it is a corollary of this theorem, how I can prove this ...
7
votes
1answer
137 views

Does the rank of homology and cohomology groups always coincide?

Let $(C_i)_{i \in \mathbb{Z}}$ be a chain complex of free abelian groups. Does the rank of the homology and cohomology groups of $(C_i)_{i \in \mathbb{Z}}$ always coincide, i.e. is ...
3
votes
1answer
52 views

Question about the Betti numbers

can someone explain me this definition from :http://en.wikipedia.org/wiki/Betti_number The $n^{th}$ Betti number represents the rank of the $n^{th}$ homology group, denoted $H_n$ "which tells us the ...
4
votes
1answer
46 views

Simplicial homology of the skeleton of a simplex

Let $n$ and $k$ two natural numbers. We consider the (abstract) simplicial complex $K$ on $n$ vertices $v_1,\dots,v_n$ and such that a subset of $\{v_1,\dots,v_n\}$ is a face of $K$ if and only if it ...
3
votes
1answer
41 views

Question about the proof of the universal coefficient theorem

When deriving the universal coefficient theorem, in class we proceeded as follows: We have the SES: $$0\to ...
0
votes
1answer
39 views

Empty set in a simplicial complex

Should the empty set be considered a simplex in a simplicial complex? Which justifications exist for the answer? I guess it is somewhat comparable to $1$ not being a prime number.
6
votes
1answer
125 views

Singular $\simeq$ Cellular homology?

Given an arbitrary CW-complex, are the singular chain complex $S_\ast(X)$ and cellular chain complex $C_\ast(X)$ homotopy equivalent or just quasi-isomorphic (some chain map induces isomorphisms on ...
2
votes
1answer
111 views

Homology of mapping telescope

It is stated here http://math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf that if $X$ is an increasing union of the type $X=\bigcup_{i \in I}X_i$ (where $X_i \subset X_{i+1}$), then we have an ...
1
vote
1answer
71 views

Period of a particular finite group

Let $G$ be a group fitting in the following exact sequence: $0 \to \mathbb{Z}/p \to G \to \mathbb{Z}/q^r \to 0.$ Here $q$ and $p$ are primes (not necessarily distinct). It is easy to check (by the ...
1
vote
0answers
34 views

chain homotopy equivalence between mapping cone complexes

Given continuous maps $f_i : X_i \to Y_i$ ($i=1, 2$) we may consider the singular chain cocomplexes $$ C^n(Y_i) \oplus C^{n-1}(X_i) $$ with boundary operator: $$ (u^n, v^{n-1}) \mapsto (-\delta u^n, ...
1
vote
0answers
48 views

A Question About Notation (Homology with Local Coefficients)

I am currently reading A J Berrick’s An Approach to Algebraic K-Theory, and I am stuck at one of the propositions there because he does not define homology with local coefficients. Proposition: ...
0
votes
0answers
42 views

Intuition of higher push-forward constant sheaves.

Let us consider the higher phsh-forward sheaves $R^if_*\mathbb{R}$ of a map $f:X\rightarrow Y$ between two compact manifolds. We assume that the fibers has a constant dimension, say $n$. I think ...
1
vote
0answers
12 views

The product of $E_2$-degenerate spectral sequences also $E_2$-degenerates?

Assume the Leray spectral sequence of a map $f_i:X_i\rightarrow B_i$ $E_2$-degenerates for $i=1,2$. Is it true that the Leray spectral sequence of the map $f_1\times f_2:X_1 \times X_2 \rightarrow B_1 ...
1
vote
0answers
54 views

In the Universal Coefficient Theorem, how does the cohomology generator relate to the homology generators?

Consider homology and cohomology of some space $X$ where the homology groups are finitely generated. Consider $tor(H^i(X))$, the torsion part of $H^i(X)$. How do the generators of $tor(H^i(X))$ ...
0
votes
0answers
18 views

How does one make cochain complex of sphere into an associative DGA?

Given the singular chain complex of the sphere $S^n$, $S^*(S^n)$, a reference says that one can use the Alexander Whitney product to make $S^*(S^n)$ into an associative differential graded algebra. ...
1
vote
1answer
30 views

Why is $S_{\ast}\left(X,A\right)$ free? [duplicate]

Why is $S_{\ast}\left(X,A\right)$ free? it is the quotient of two free groups $S_{\ast}\left(X\right)$ & $S_{\ast}\left(A\right)$
2
votes
0answers
39 views

Cohomology-Homology bilinear form of Seifert surfaces

Let $C_\ast$ be any chain complex of $R$-modules. Then for any $k\in\mathbb{Z}$ we obtain a $R$-bilinear map $$\langle-,-\rangle:H^k\!C_\ast\times H_kC_\ast\longrightarrow R, ...
5
votes
1answer
105 views

Space with prescribed local homology

Lets be $G_n$ sequence of abelian groups and $G_0 = \mathbb{Z}$. Is there topological space $X$ that local homology groups at every point are those $G_n$ ? ie. $$ \forall x\in X \; \forall ...
4
votes
0answers
87 views

Eilenberg-Moore Spectral Sequence for Homology with Coefficients in the Integers

I am trying to learn about the Eilenberg-Moore spectral sequence to compute homology and cohomology. I have been using Hatcher's book on spectral sequences and also McCleary's "A User's Guide to ...
3
votes
1answer
125 views

Universal coefficient theorem with ring coefficients

The universal coefficient theorem for cohomology reads: $$0 \to Ext(H_{n-1}(C), R) \to H^n(C;R) \to Hom(H_n(C), R) \to 0,$$ where $C$ is a chain complex of free abelian groups and $R$ is a ring. It ...
1
vote
1answer
61 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
75 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
0
votes
1answer
28 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.
1
vote
0answers
27 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
69 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
62 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
91 views

Question about the definition of homology

$\quad$ The functor $H_n$ measures the number of “$n$-dimensional holes” in the space (or simplicial complex), in the sense that the $n$-sphere $S^n$ has exactly one $n$-dimensional hole and no ...
0
votes
2answers
47 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
27 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
42 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
76 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
114 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
39 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
39 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
72 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
88 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
59 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
91 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
115 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 ...