# Tagged Questions

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### 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 ...
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### 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
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### 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 ...
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### 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 ...
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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 ...
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### 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 ...
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### 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 ...
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### 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 $M_h$, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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 ...