8
votes
1answer
109 views
+50

Homology Whitehead theorem for non simply connected spaces

(One version of) the Whitehead theorem states that a homology equivalence between simply connected CW complexes is a homotopy equivalence. Does the following generalisation hold true? Suppose ...
0
votes
1answer
58 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
24 views

Homology group of S1

In Algebraic Topology by Hatchers, the first example of simplicial homology group is created using a segment $a$ which two endpoints are identified, generating the circle $S^1$. The definition of the ...
0
votes
0answers
31 views

Question about singular homology

in order to prove that $H_0(X)\simeq \mathbb{F}$, $\mathbb{F}$ is the unitary commutative ring we have to prove that $C_0(X)/B_0(X)\simeq \mathbb{F}$ since we have that $C_0(X)$ is generated by the ...
3
votes
1answer
35 views

Computing the Euler Characteristic of the $n$-sphere

Let $n\ge 2$. Compute the Euler characteristic of the $n$-sphere $S^n$ using the standard triangulation of the $n+1$-simplex. I know the union of the proper faces of the $(n+1)$-simplex is ...
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 ...
7
votes
1answer
44 views

Equality of rank for homology and cohomology groups via the universal coefficient theorem

I'm having trouble understanding a passage from the proof of Corollary 3.37 in Hatcher's Algebraic Topology, namely the fact that the universal coefficient theorem implies $$ ...
2
votes
1answer
46 views

$S^{1}$-bundles over $\mathbb{RP}^2$

How many $S^1$-bundles over $\mathbb{RP}^2$ do exist? Is it true that there exist only two bundles - trivial and not?
0
votes
0answers
19 views

worked out examples in borel-moore homology

I'm trying to learn about BM homology. I've found a few references and they are all quite abstract and high-powered. I am actually familiar with derived categories and sheaves so I don't mind thinking ...
2
votes
0answers
40 views

Homology of A Compact Manifold [duplicate]

I am working on a compact manifold $M$ of dimension $m$. Moreover, I suppose that $M$ is oriented of fundamental class $ [M] \in H_m (M)$. I want to show that I have an isomorphism $H_\ast (M, ...
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 ...
0
votes
1answer
25 views

If the reduced homology group is free of rank what does this say about the actual reduced homology of the give space?

If the reduced homology group $\tilde{H}_{i}((\Delta^{n})^{k})$ said to be free of rank $\binom{n}{k+1}$ when $i=k$ what does this mean in practice; what will the actual reduced homology be?
1
vote
1answer
39 views

What does it mean for a relative homology group to be free of rank?

What does it mean for the relative homology group $\tilde{H}_{i}((\Delta^{n})^{k})$ to be free of rank $\begin{pmatrix}n\\k+1\end{pmatrix}$? And what does this mean the actual relative homology is?
0
votes
1answer
48 views

Homology isomorphism of $H_n(S^d\times X)$ and $H_{n-1}(S^{d-1}\times X)$

$X$ is an arbitrary space, $d\geq 1$. The existence of such isomorphism in the title supposedly follows from the Mayer-Vietoris sequence of $(S^d\times X,S^d_{+}\times X,S^d_{-}\times X)$: ...
-1
votes
1answer
49 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
0
votes
0answers
32 views

Show that they are not the boundaries of any disjointly embedded disks.

This is an exercise in Hatcher's topology book. It's in Page 176, problem 4(b). In the unit sphere $S^{p+q-1}$,let $S^{p-1}$ and $S^{q-1}$ be the subspheres consisting of points whose last $q$ and ...
1
vote
1answer
36 views

Definition of the relative boundary map

According to Hatcher (page 115), since the boundary map $\partial: C_n(X)\rightarrow C_{n-1}(X)$ takes $ C_n(A)$ to $ C_{n-1}(A)$, it induces a quotient boundary map. I am trying to reformulate this ...
2
votes
0answers
23 views

uniqueness of Hopf invariant

(Hopf invariant, page 427 of A. Hatcher's Algebrac Topology): Let $f: S^m\longrightarrow S^n$ with $m\geq n$. We can form a CW-complex $C_f$ by attaching a cell $e^{m+1}$ to $S^n$ via $f$. The ...
4
votes
1answer
51 views

Degree modulo 2 and usual degree

If $f:S^n\rightarrow S^n$ is a map, why do we get its $\mathbf{Z}_2$-degree by taking its $\mathbf{Z}$-degree modulo 2? I am taking the definition of degree via homology theory
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
72 views

Covering of orientable surface (Hatcher)

The following is an exercise from Hatcher, Algebraic Topology, that I'm struggling with (exercise 2.2.23): Show that if the closed orientable surface $M_g$ of genus $g$ is a covering space of $M_h$, ...
1
vote
2answers
78 views

Homology of orientable surface of genus $g$

I came across the problem of computing the homology groups of the closed orientable surface of genus $g$. Here Homology of surface of genus $g$ I found a solution via cellular homology. This seems ...
3
votes
2answers
78 views

Non-zero degree on circle $\Rightarrow$ surjective on disk

Recently I came across the following problem that I cant's solve: Let $f: (D^n, S^{n-1}) \rightarrow (D^n, S^{n-1})$ be a continuous map such that $f|_{S^{n-1}}$ has non-zero degree. Show that $f$ is ...
2
votes
2answers
64 views

Non-trivial element of $H_n(S^n)$ covers all of $S^n$

I have a question about singular homology of the $n$-sphere that I'm getting nowhere with: Prove the following: Any cycle $c$ that represents a non-trivial class in the $n$-th singular homology ...
3
votes
1answer
93 views

Isomorphism in homology of $\mathbb{R} P^2$

I have a question about the homology of the real projective space $\mathbb{R} P^2$ with which I'm having some trouble: Let $f: \mathbb{R}P^2 \rightarrow \mathbb{R}P^2$ be a map which induces an ...
5
votes
2answers
243 views

Homology of connected sum of real projective spaces

Let $A_k=RP^2\sharp RP^2\sharp \cdots \sharp RP^2$ be a connected sum of $k$ copies of real projective space. With coefficients in $\mathbb{Z}$, it is clear $H_n(A_k)=0$ when $n\geq2$ and ...
0
votes
0answers
52 views

Image of the map on homology induced by a covering

Let $X$ and $Y$ are two compact connected oriented 2dim smooth manifolds, and $\pi\colon X\to Y$ is an unramified covering of degree $d$. Consider the induced map $\pi_* \colon H_1 (X,\mathbb Z) \to ...
2
votes
0answers
44 views

Homology of $S^1$ with local coefficients

I'm trying to compute the cellular chain complex $C_{*}(\tilde{S^1})$ where $\tilde{S^1}$ is the universal cover of $S^1$, as a $\mathbb{Z}[t, t^{-1}]$ module. This can then be used to compute the ...
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 ...
1
vote
1answer
127 views

Torus minus disk does not retract

I'd like to show the following (intuitively clear) fact: Given a torus $T^2$ and an embedded disk $D\subset T^2$ (put a disk in the middle of the square whose edges we identify to get the torus), ...
2
votes
1answer
89 views

Real projective plane: $f_*$ isomorphism $\implies f$ surjective

Suppose we have a continuous map $f: \mathbb{R}P^2\rightarrow \mathbb{R}P^2$ that induces an isomorphism in homology $f_*: H_p(\mathbb{R}P^2) \rightarrow H_p(\mathbb{R}P^2)\ \ \ \forall p$. How do I ...
1
vote
1answer
38 views

Why are invariants of Homology 3-Spheres interesting?

I see how invariants (of any kind of mathematical objects) are interesting in general, since classification is interesting, but mostly in case there is a vast variety of such objects that we do not ...
3
votes
2answers
80 views

Rigorous application of Mayer-Vietoris to a quotient of $S^{2}\times I$

As part of exercise 3.3.24 in Hatcher's Algebraic Topology, I computed the homology groups of the space resulting from $S^{2}\times I$ by identifying $S^{2}\times\{0\}$ and $S^{2}\times\{1\}$ via a ...
9
votes
0answers
240 views

Mapping cylinder is a CW complex

If you read the question entirely, it is not a duplicate. The first time I asked the question, I already gave the link to the similar question and explained why the answer is not satisfying. If you ...
3
votes
0answers
65 views

Concrete non trivial computation of Morse homology

I am studying Morse homology and have found only examples on spheres and tori so far. Of course the homology of these manifolds is better understood by other more standard methods, so I am having ...
3
votes
3answers
172 views

Why homology with coefficients?

I am currently studying a bit of homology theory (on topological spaces). Let $H_n(X)$ denote the singular homology groups of the topological space $X$, then as you know we can define the singular ...
3
votes
0answers
42 views

Homological Interpretation of the intersection number

Let $M^r$ and $N^s$ be smooth submanifolds of the smooth manifold $V^{r+s}$ and $M,N,V$ compact, connected and orientable. The Thom-Isomorphism together with the Tubular Neighbourhood Theorem gives me ...
1
vote
1answer
75 views

Algebra prerequisites for Homology Theory

I am a first year graduate student in Mathematics. I am planning to take a graduate course on Homology Theory. My background is Point Set Topology (material covered in Part 1 of Munkres) and the ...
3
votes
1answer
25 views

Nonexistence of map taking boundary to torus to wedge of circles homeomorphically

Specifically, the question says to consider the torus $T$ as a square with the usual identifications, with two opposite boundary edges labelled $a$ and the other two edges labelled $b$, and consider ...
0
votes
0answers
22 views

Affine operations on affine simplices

First of all some definitions: Let $\sigma=[v_0,v_1, ... , v_p]:\Delta_p \to \Delta_q$ be an affine simplex. Let $v \in \Delta_p$ The cone on $\sigma$ from v is defined as ...
2
votes
0answers
43 views

Chain homotopy inverse to inclusion

I am currently trying to solve the following problem: Given a simplicial complex $K$ (being the union of simplices such that any face of a simplex in $K$ is also in $K$ and any two simplices intersect ...
3
votes
0answers
68 views

Products of homology groups

In the topology course I attend we work with the following, rather unusual definition of homology groups: For topological spaces $X,Y$ define the presheaf $\mathcal{F}_Y$ on $X$ as follows: Let ...
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.
2
votes
0answers
60 views

Mayer-Vietoris of pair (X,C)

I would like to know if i can use Mayer-Vietoris with this form: Let X be a topological space and A, B be two subspaces whose interiors cover X and $C\subset A\cap B$. We get the exact sequence ...
4
votes
1answer
100 views

Homology of the product

I have to prove that $$H_q(X\times\partial I^n,X\times\{p_0\})=H_{q-n}(X)$$ for $X$ a topological space. I tried using induction, but I didn't go too far, and think that using some exact sequence ...
1
vote
1answer
74 views

simple question about cohomology group

Let's consider compact 4-manifold $M^{4}$ and point $P \in M$. Then (use Mayer-Vietoris) inclusion $i\colon M\setminus P \to M$ induce isomorphism $i^{*}\colon H^2(M) \to H^2(M\setminus P)$. Let's ...
5
votes
0answers
118 views

Characterising singular homology among a more general class of cosimplicial spaces

Is there a way to characterise (up to isomorphism) the simplicial spaces $F: \Delta \to \underline{\text{Top}}$ with $F( \underline{n}) \subset \mathbb{R}^{n+1}$ compact and $F(\underline{0})$ a ...