# Tagged Questions

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### Cohomology Ring of Klein Bottle over $\mathbb{Z}_2$

I am trying to show that the cohomology ring of the Klein bottle with $\mathbb{Z}_2$ coefficients is $H^*(K,\mathbb{Z}_2) \cong \mathbb{Z}_2[x,y]/(x^3,y^2, x^2y)$. What I know: ...
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### $C_g \simeq SX$ and $C_h \simeq SY$ [on hold]

Hi need some help with this problem: Let $f : X \to Y$ . Then we can form the cofiber sequence $X \to Y \to C_f \to C_g \to C_h$ where $g: Y \to C_f$, $h: C_f \to C_g$, and $i: C_g \to C_h$. Show ...
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### Compute the induced map on $\mathbb{CP}^n$

Let $d>0$ and $f:\mathbb{C}^{n+1}\rightarrow \mathbb{C}^{n+1}$ be given by $f(z_0,...,z_n)=(z_0^d,...,z_n^d)$. Let $F:\mathbb{CP}^n \rightarrow \mathbb{CP}^n$ be the induced map by $f$. Compute ...
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### Relative cohomology of a vector space module non-zero vectors

I am trying to explicitly compute the relative cohomology groups $H^m(\mathbb R^n,\mathbb R^n_0;\mathbb Z)$, where $\mathbb R^n_0$ is all the non-zero vectors in $\mathbb R^n$. I think that the answer ...
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### Is $H^0(S^0;G)\simeq G\oplus G$ or $G$?

In the article on topospaces for the (co)homology of spheres, it says $H^0(S^n,G)\simeq H^n(S^n,G)\simeq G$. Is this true when $n=0$? I think not, for if we view $S^0$ as the union of two ...
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### Degree of an induced map on $\mathbb{CP}^n$

Let $r :\mathbb{C}^{n+1} \rightarrow \mathbb{C}^{n+1}$ be the map $r(z_0, z_1,\ldots, z_n)=(-z_0, z_1,\ldots, z_n)$. $r$ induces a map $\bar r : \mathbb{CP}^n \rightarrow \mathbb{CP}^n$. What is the ...
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### Why is $H^2(\mathbb{R}P^2,\mathbb{Z})\simeq\mathbb{Z}_2$?

Why is the second cohomology group of $X=\mathbb{R}P^2$ with $\mathbb{Z}$-coefficients $\mathbb{Z}_2$? We can put the usual $\Delta$-structure on $X$ with two vertices, three $1$-simplices, say $a$, ...
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### Wedge Sum of Two Spheres Homotopy Equivalent to a Compact Manifold?

Let $X=S^2$v $S^2$ (wedge sum). The homology groups are $H_0(X,\mathbb{Z})= \mathbb{Z}$, $H_1(X,\mathbb{Z})= 0$, and $H_2(X,\mathbb{Z})= \mathbb{Z} \oplus\mathbb{Z}$. I can see that $X$ is not ...
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### Explititly evaluating the Poincaré duality

Let $M$ be a closed 2-dimensional manifold (a surface). Assume that I have a more or less explicit expression for a Čech 2-cocyle $h_{ijk} \in H^2(M, G)$. I want to know the expression of $h_{ijk}$ as ...
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### 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 ...
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### Cohomology with coefficients in a commutative ring, how are the chain groups defined?

I have been studying a course in algebraic topology that follows Hatcher's textbook on the subject. I have some queries as to how certain things are defined. The first part of the text defines the ...
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 ...
I want to show that the spaces $S^1 \vee S^1 \vee S^2$ and $S^1 \times S^1$ do not have the same homotopy type. I calculated their homologies and cohomologies and they turn out to be equal. So I ...