# Questions tagged [homology-cohomology]

Use this tag if your question involves some type of (co)homology, including (but not limited to) simplicial, singular or group (co)homology. Consider the tag (homological-algebra) for more abstract aspects of (co)homology theory.

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### $h_*:S_*(X\times Y)\to S_*(X\times Y)$ a homomorphism of natural chain complexes. If $h_0=Id$, show that $h_*$ is homotopic to $Id$

Let $X$ and $Y$ be topological spaces and $h_*:S_*(X\times Y)\to S_*(X\times Y)$ a homomorphism of natural chain complexes. If $h_0=Id$, show that $h_*$ is homotopic to $Id$. To solve this problem I ...
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### Need a description of the map $i:\mathbb{C}P^2\vee\mathbb{C}P^2\rightarrow \mathbb{C}P^2\times \mathbb{C}P^2$ on $2$nd Cohomology level

Let $i:\mathbb{C}P^2\vee\mathbb{C}P^2\rightarrow \mathbb{C}P^2\times \mathbb{C}P^2$ be the inclusion map. I want to know what is $i^*$ on $2$nd Cohomology level. Seems it should be an isomorphism ...
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### free group acting on the 3D sphere

Let's say I have a 3 dimensional sphere $S^2$ and I define the following transformation $R_1$ rotate the sphere around the north-south axis of $\frac{\pi}{2}$ (north and south are fixed points till ...
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### Definition of equivariant chain complex.

What do we mean by equivariant chain complex? Is it a chain complex with some property ? I looked in many references and i did not find a definition of the expression "equivariant chain complex", I ...
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### A function $f:X \to Y$ which induces isomorphisms on all homology groups but $X$ and $Y$ have non-isomorphic cohomology rings.

I am trying to find a function $f:X \to Y$, where $X$ and $Y$ are path connected, such that $f$ induces isomorphisms on all homology groups, but $X$ and $Y$ have non-isomorphic cohomology rings. It ...
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### Prove that $f_*: H_2(T; \mathbb{Z}/2)\to H_2(K; \mathbb{Z}/2)$ is the trivial mapping.

Let us denote by $T$ the torus and $K$ the bottle of Klein. Let $f:T\to K$ be a continuous function. Prove that $f_*: H_2(T; \mathbb{Z}/2)\to H_2(K; \mathbb{Z}/2)$ is the trivial mapping. I know that ...
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### To Understand The Cohomology ring $H^*(\mathbb{R}P^n;\mathbb{Z})$

I am trying to understand the Cohomology ring structure of $\mathbb{R}P^n$ with $\mathbb{Z}$ coefficients using the Coholomogy ring of of $\mathbb{R}P^n$ with $\mathbb{Z}/2\mathbb{Z}$ coefficients. ...
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### Any embedding $\mathbb{R} P^n\hookrightarrow \mathbb{R} P^{\infty}$ induces a surjection of the singular cohomology rings.

On page 79 of "Vector Bundles and K-Theory" Hatcher makes use of the claim in the title where he considers cohomology with integer or $\mathbb{F}_2$ coefficients. Why is this claim true? I know ...
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### Twisted Cech Cohomology on the Circle

Exercise 10.7 in Bott and Tu involves calculating the Cech cohomology, valued in $\mathbb{Z}$, for a particular open cover of the circle. In particular, we have a good cover consisting of $U_0, U_1$, ...
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### Let $f : (X, A) → (Y, B)$ be a map such that both $f : X → Y$ and $f : A → B$ are homotopy equivalences.

Let $f : (X, A) → (Y, B)$ be a map such that both $f : X → Y$ and $f : A → B$ are homotopy equivalences. Show that $f_∗ : H_n(X, A) → H_n(Y, B)$ is an isomorphism for all $n$. I know that there is a ...
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### homology groups of the quotient of $\mathbb{S}^2$ obtained by identifying north and south poles to a point

Compute the homology groups of the quotient of $\mathbb{S}^2$ obtained by identifying north and south poles to a point. I have already calculated the homology groups in one way, but I would like to ...
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