# 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.

3,210 questions
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### Tensor product of a (co)chain and a cochain complexes

This is a question about the grading convention. Suppose we have a chain complex $$C_\bullet = 0\to C_n\stackrel{d_n}\to C_{n-1}\stackrel{d_{n-1}}\to...\to C_1\stackrel{d_1}\to C_0\to 0$$ and two ...
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### Definition of Prism Operator and homotopy of chain complexes

I am reading Algebraic Topology by Allen Hatcher. For reference: https://pi.math.cornell.edu/~hatcher/AT/AT.pdf On page 112, in the first sentence of the last paragraph, we find the following: The ...
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### Universal covering that induces zero on homologies

Let $p:\tilde{X}\rightarrow X$ be the universal covering space such that $p_*$ is zero on all homologies of dimension greater than zero. Does this imply that $X$ is $K(\pi_1(X),1)$? Working with the ...
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### Computing the homology of a simple chain complex

Let $R$ be a ring and $x\in R$ be a central element. Consider the complex $$0 \rightarrow R \xrightarrow{x} R \rightarrow 0$$ concentrated in degrees 1 and 0. Compute the homology of this complex. ...
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### Role of d^2 = 0 in chain complex

What is the motivation for requiring that the square of a differential be 0 for a complex, aside from enabling us to speak of the homology of a complex? Other homological notions like chain maps, ...
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### isomorphism for real vector bundles and complex vector bundles with inner product.

Let $B$ be paracompact. Call $\mathrm{Vect}_{\mathbb R}(B)$ the category of real vector bundles over $B$ and $\mathrm{Vect}_{\mathbb C,f}(B)$ be the category of complex vector bundles with an inner ...
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### Homology of the maps between complex projective spaces

Suppose $m > n$, and let $f : \mathbb CP^m \to \mathbb CP^n$ be continuous, the claim is that the induced map between the homology(over $\mathbb Z$) is zero. I have no clue why this should be ...
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### Why does this exact sequence exist?

I'm reading a proof and don't understand a certain part. Let $A^\bullet$ be a (cochain) complex of abelian groups. Let $I^\bullet$ be an injective resolution of an abelian group $B$. Then there is a ...
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### Is this an example of a local system

In Bott, Tu, one is asked to calculate the cohomology of the following sheaf: Exercise 10.7 (Cohomology with twisted coefficients). Let $\mathscr{F}$ be the presheaf on $S^1$ which associates to ...
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### Intuition behind the injectivity part of Hurewicz Theorem

The surjectivity part of Hurewicz Theorem is easy to understand: under the inductive hypothesis that all homotopy groups (of a CW-complex) up to dimension $n$ are trivial, it is clear (I believe) how ...
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### Homology of solvable Lie algebras

Let $\mathfrak{g}$ be a solvable lie algebra and $\lambda\in (\mathfrak{g}/[\mathfrak{g},\mathfrak{g}])^*$ be the character of $\mathfrak{g}$. How to compute homology for $\mathbb{C}_\lambda$, the ...
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### Cup product without cellular decomposition?

As we know, the cup product does not rely on the particular cellular decomposion of the space. So, is it possible to define the concept of cup product not rely on the cellular decomposition?
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### About Lie algebra cohomology and Ext group

Let $\mathfrak{g}$ be a Lie algebra over a field $K$. Then the Lie algebra cohomology of $\mathfrak{g}$ with coefficients in $V$ is the right derived functor of $V\mapsto V^\mathfrak{g}$ and can be ...
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### trivial modules of group rings

Let $R=\mathbb{F}_p[D]$ where $D$ is a finite group of order prime to $p$. Let $M$ be any simple $R$-module. If one knows that $H^0(D,M)=0$, is $M=0$? If not, under what further conditions can one ...
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### concerning about homology of a graph

I'm very lost in the following Bredon problem 9-pag 240 Let $G \subset S^n$ be a finite connected graph. Find $H_{\bullet}(S^n \setminus G)$. Any hint would be appreciated. Thanks.
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### Cohomology of quadric in $\mathbb{C}^4$ at infinity

How to prove that the singular cohomology of $X=V(xy-zw)\setminus 0 \subset \mathbb{C}^4$ is $$H^*(X,\mathbb{Z})=\{\mathbb{Z},0,\mathbb{Z},\mathbb{Z},0,\mathbb{Z}\}?$$ Preferably, I would like to know ...
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### Evaluating a symplectic form on $\pi_2$ or its image through the Hurewicz map

Let $(M,\omega)$ be a symplectic manifold. There are a priori two ways of evaluating $\omega$ on an element $A \in \pi_2(M)$: we can integrate $\omega$ on any representative $u : S^2 \to M$ of the ...
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### Group homology of action on left cosets

If $G$ is a group and $H$ a subgroup which is not normal. What is the homologies of the action of $G$ on the left coset space $G/H$? The action is multiplication from left. More precisely, $G/H$ is ...
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### The cohomology ring of two tori glued along the first circle

Let $X$ be the space consisting of two tori glued along the first circle: I interpret the space as $X=S^1_{(1)}\times S^1_{(2)}\sqcup S^1_{(3)}\times S^1_{(4)} / (S^1_{(1)}\sim S^1_{(3)})$. I want ...
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### Relationship between Betti number and Genus

I recently found out that Mathworld gave the same definition for both Betti number and Genus as: "the largest number of nonintersecting simple closed curves that can be drawn on the surface without ...
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### How is the generator of the first homology of the torus non-trivial?

Consider the above representation of the torus $X$. I need to show that if $\phi\in C^1(X;\mathbb Z)$ is the cochain that takes the value $1$ on the red lines with the orientation given by the ...
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### Serre fibration and Mittag Leffler condition

The proof I am concerned is 2.2.5 pg 37 Kochman Stable Homotopy. Let $R$ be a commutative ring. Let $S^n \rightarrow E \xrightarrow{p} B$ be Serre fibration. $B$ a CW complex, $B$ simpliy ...
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### $\mathbf{R}P^{n}/ \mathbf{R}P^{n-2} \simeq S^n \vee S^{n-1}$ iff $n$ is odd

Prove that $\mathbf{R}P^{n}/ \mathbf{R}P^{n-2} \simeq S^n \vee S^{n-1}$ iff $n$ is odd. One side is solved just calculating the homology groups of both sides. I can't see if $n$ is odd how to show ...
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### Reference for Universal Coefficient Theorem

I am looking for a proof of the following fact: let $C$ be a chain complex of real vector spaces, $C^*$ the dual cochain complex. Then $H^n(C^*) \cong H_n(C)^*$. That is, taking homology commutes with ...
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### Computing algebraic de Rham cohomology

Let $R=\mathbb C[x,y]/(y(x-a)(x-b)-1)$ where $a,b$ are distinct complex numbers. Show that the cohomology of the de Rham complex $$0\to R\to \Omega_{R/\mathbb C}\to 0$$ is $\mathbb C$ in degree zero ...
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