# Tagged Questions

This tag is for questions relating to cohomology groups and cochain complexes.

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### lift of antiholomorphic involution of Riemann surface to its Jacobian's cohomology

Start from a connected closed Riemann surface $\Sigma_g,$ obtained as the (symmetric) covering of an open and/or unoriented surface $\Sigma,$ namely $\Sigma=\Sigma_g/\Omega,$ where $\Omega$ is an ...
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### How to prove $H^2(\mathfrak{g}, J(\mathfrak{g}))\neq0$, where $J(\mathfrak{g})$ is the augmentation ideal of $\mathfrak{g}$?

$\mathfrak{g}$ is a finite-dimensional semisimple Lie algebra over a field $k$ with $\mathrm{char}k=0$. $J(\mathfrak{g})$ is the augmentation ideal of $\mathfrak{g}$. That is, the kernel of ...
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### how to compute the de Rham cohomology of the punctured plane just by Calculus?

I have a classmate learning algebra.He ask me how to compute the de Rham cohomology of the punctured plane $\mathbb{R}^2\setminus\{0\}$ by an elementary way,without homotopy type,without ...
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### Top de Rham cohomology

I just realized that I never really understood why $H_{dR}^n(M, \mathbb{R}) = \mathbb{R}$ if $M$ is compact and $H_{dR}^n(M, \mathbb{R}) = \{0\}$ if $M$ is not compact (provided that's true?). I'm ...
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### de Rham cohomology of $\mathbb{R}P^n$ via action by $SO(m+1)$

In lecture, my teacher proved the theorem that given a smooth $G$-action by a compact, connected Lie group on a manifold $M$, the de Rham cohomology of the $G$-invariant differential forms $H^p_G(M)$ ...
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### Problem with cohomology (II)

Let $G$ be a group and $K$ be subgroup of $G$, Let $A$ be $G$ module. Let $X=\{X_n\}$ be free resolution of $\mathbb{Z}$. Let $\delta$ represents collectively the homomorphism induced in Hom sequence ...
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### simple question about cohomology group

Let's consider compact 4-manifold $M^{4}$ and point $P \in M$. Then (use Mayer-Vietoris) inclusion $i\colon M\setminus P \to M$ induce isomorphism $i^{*}\colon H^2(M) \to H^2(M\setminus P)$. Let's ...
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### Differential forms as functionals on curves

Please give me a reference to a book or lecture notes where the following stuff is studied. Let $M$ be a Riemann surface with boundary $\partial M$ (but not necessarily, any smooth $n$-dimensional ...
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### Definition of the relative de-Rham cohomology and its generalization

Let $M$ be a smooth manifold and $N$ be its smooth submanifold. We say that two closed forms $\omega_1$ and $\omega_2 \in \Lambda^k(M)$ are equivalent if their difference is an exact form from ...
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### Cohomological ($p$-)dimension of a pro-$p$ group

I have a question concerning the cohomological dimension and $p$-dimension of a pro-$p$-group. Let's first recall the definitions of that The cohomological dimension $cd \ G$ of a pro-finite group ...
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### Do the cyclic or Hochschild homologies satisfy the addition axiom of Eilenberg Steenrod?

Do the cyclic or Hochschild homologies satisfy the addition axiom of ES? If so please provide a reference or proof (reference is preferable).
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### Relative de Rahm cohomology computation for two disjoint circles embedded in R^2

Consider a submanifold $Y$ of $\mathbb{R}^2$ formed by two disjoint embedded copies of $S^1.$ Compute $H^{\bullet}_{dR}(\mathbb{R}^2,Y).$ In this case the long exact sequence splits, and we can ...
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### Finite generation of Tate cohomology groups

Let $G$ be a finite group, and let $F$ be a complete resolution for $G$. In other words, $F$ is an acyclic chain complex of projective $\mathbb{Z}G$-modules together with a map ...
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### Non abelian $H^1(G,A)$ problem.

Let $G,A$ be groups. We do not assume that $A$ is abelian. For $f,g\in Z^1(G,A)$, we write $f\backsim g$ if there is an $a\in A$ such that $g(x)=a^{-1}f(x)\ ^xa$ (we use the pre-exponential notation ...
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### confused, Universal Coefficient Theorem (cohomology)

This is bad, but I was applying the UCT to a small complex and didn't seem to work. Namely the chain complex $0 \rightarrow \mathbb{Z} \rightarrow \mathbb{Z} \rightarrow 0$ where the nonzero map is, ...
I have the following setup. The CW-complexes $\Gamma_n$ are equipped with maps $\gamma_n\colon\Gamma_{n+1}\rightarrow\Gamma_{n}$ and it is known that the rank of their first cohomology groups is ...
Related to my previous question here. Let $X$ be a topological space and let $H_c^{\bullet}(X)$ denote its singular cohomology with compact supports (rational coefficients). Let $U$ be an open subset ...