# Tagged Questions

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### cohomology ring of grassmannian

Let $G_k(\mathbb{R}^n)$ be the grassmannian consisting of all $k$-subspaces in $\mathbb{R}^n$. How to compute the cohomology ring $$H^*(G_k(\mathbb{R}^n);\mathbb{Z})$$ and what is the result?
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### smash product of Eilenberg-Maclane spaces

Let $G$ be an abelian group and $K_n=K(G,n)$ be the Eilenberg-Maclane space. How to obtain $K_m\wedge K_n$ is $(m+n-1)$-connected? (Hatcher's book page 404)
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### Meyer-vietoris sequence to compute the compact cohomology for Möbius strip

How do you use Meyer-vietoris sequence to compute the compact cohomology for MÃ¶bius strip without the bounding edge? Please give detail math. In particular explain how inclusion map is used. On page ...
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### Showing the image of $H^j(X;\mathbb C^\times)$ lies in the torsion subgroup of $H^{j+1}(X;\mathbb Z)$

Let $X$ be a (compact, if necessary) topological space. Then from the short exact sequence of constant sheaves $$0 \to \mathbb Z \to \mathbb C \to \mathbb C^\times \to 0$$ we have a connecting ...
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### Some questions about cellular homology and cohomology

Consider the CW structure on $\mathbb{RP}^n$ given by one cell in every dimension. This gives rise to the cellular complex $C_\bullet(\mathbb{RP}^n)$ which is generated by a single element $c_i$ for ...
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### 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 ...
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### Simple example of $X$ with torsion in $H^1(X,\mathbb{Z})$?

Question: Is there a simple example of a space $X$ possessing torsion in its first integral cohomology group $H^1(X,\mathbb{Z})$? For reasonable spaces $X$, e.g. CW-complexes, one has ...
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### An isomorphism between homology and cohomology with $\mathbb{Z}_2$ coefficients

In the proof of a theorem we did in a class (namely: if $M$ is an odd-dimensional, closed manifold, then $\chi(M)=0$), there's the following step: $$H_k(M;\mathbb{Z}_2)\cong H^k(M;\mathbb{Z}_2)$$ ...
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### The cohomology ring of the nerve of a category associated to a vector space

Let $n\ge2$, and let $V$ be an $n$-dimensional vector space over a field $k$. Consider the category $\mathcal{C}$ whose objects are nonzero, proper subspaces of $V$, and whose morphisms are ...
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### 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 ...
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### Cohomology Calculation

A couple of days ago I asked this Question on calculating hypercohomology I tried a similar example for $(\mathbb{C}^*)^2$, and I have a couple of questions. Here is my calculation: We have a ...
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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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### 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 $a$, ...
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 ...