# Tagged Questions

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

43 views

204 views

### Universal coefficient theorem with ring coefficients

The universal coefficient theorem for cohomology reads: $$0 \to Ext(H_{n-1}(C), R) \to H^n(C;R) \to Hom(H_n(C), R) \to 0,$$ where $C$ is a chain complex of free abelian groups and $R$ is a ring. It ...
56 views

### Is there any simple example that $lim^1$ terms appear?

limit of cohomology does not behave well in the sense that there will be $lim^1$ term. Is there any simple example that $lim^1$ terms appear? Thanks!
318 views

### Why are de Rham cohomology and Cech cohomology of the constant sheaf the same

I am comfortable with de Rham cohomology, sheaves, sheaf cohomology and Cech cohomology. I am looking to prove the following theorem: If $M$ is a smooth manifold of dimension $m$, then we have ...
242 views

50 views

### Topological dimension and derham cohomological dimension

If G is a compact complex manifold then does the topological dimension bound the deRham cohomological dimension below? By derham cohomological dimension, I mean the largest extended natrual number ...
53 views

44 views

### Existence of a suitable cover for $S^{2}$ and a given sheaf

I am trying to find a Leray covering for the 2-sphere with respect to the sheaf $\mathcal{F}=\mathbb{Z}$. I am also assuming that a contractible open covering satisfies $H^{i}(U,\mathbb{Z})$ for all ...
79 views

### Question about Relative Cohomology

I need help with the following question please: Suppose that a space $X \subseteq Y$ retracts onto some subspace $A \subseteq X$. When do I have $H^\ast ( Y,X) \cong H^\ast (Y,A)$? Thanks.
35 views

### Cohomology of Circle from unreduced Eilenberg-Steenrod Axioms

I would like to compute the cohomology groups of $S^1$ straight from the unreduced Eilenberg-Steenrod axioms. My motivation is to be able to calculate the cohomology group of spheres in any dimension ...
108 views

### Cohomological Whitehead theorem

Let $X$ and $Y$ be CW complexes (resp. Kan complexes) and let $f : X \to Y$ be a continuous map (resp. morphism of simplicial sets). The following seems to be a folklore result: Theorem. The ...
123 views

### Solving an exercise in Milnor-stasheff's “characteristic classes”

I am trying to solve the following exercise (which is an exercise in Milnor-Stasheff's book). It basically says the following: Let $M =S^n$ be the $n$-sphere and let $TM$ be its tangent ...
88 views

### De Rham cohomology of $T^*\mathbb{CP}^n$

I am a bit rusty on my de Rham cohomology, and I'm hoping that someone here could help me. I want to find the cohomology of $T^*\mathbb{CP}^n$ (seen as a real manifold). Now, this should be equal to ...
57 views

### $S^{1}$-bundles over $\mathbb{RP}^2$

How many $S^1$-bundles over $\mathbb{RP}^2$ do exist? Is it true that there exist only two bundles - trivial and not?
48 views

### Can we compare cohomology rings with different coefficients?

I have an example sheet that asks me to compute the cohomology rings for two spaces, say X and Y, with coefficients in $\mathbb{Z}$ and $\mathbb{Z}_d$ respectively. It then asks whether X and Y are ...
68 views

### Relating Ext groups of abelian groups and group cohomology

One can define $\mathrm{Ext}$-groups in the category of abelian groups (not $\mathbb{Z}[G]$-modules) and group cohomology in very similar ways. The second, group cohomology, can be computed in the ...
27 views

### Show that $H^{\prime} \cap A$ is a homomorphic image of $M(G)$

Let $H$ be a group and $A$ be a central subgroup of $H$ of finite index. Let $G =H/A$. Show that $H^{\prime} \cap A$ is a homomorphic image of $M(G)$. Here $H^{\prime}$ denotes the commutator ...
118 views

### How do you compute group cohomology in practice?

If you have a finite group $G$ and a finite $G$-module $K$, and you need to know $H^1(G,K)$ or $H^2(G,K)$, how do you do it? Do you use a computer algebra system? (If so, which one?) Do you use a ...
67 views

### Dolbeault cohomology on torus

Let $T=\mathbb{C}/\Gamma$ where $\Gamma$ is a lattice of $\mathbb C$. Given that $H_{dR}^1(T)=\mathbb{C}^2$. Prove that $H^{1,0}_\bar{\partial}(T)=\mathbb{C}$. I have no idea what to do. Can someone ...
100 views

### If M is a non-orientable closed connected 3 manifold prove H1(M) is an infinite group.

This is an example from a question sheet (non-assessed) of a university class. If M is a non-orientable, closed, connected 3 manifold, prove $H_1(M;\mathbb{Z})$ is an infinite group. I know that since ...
21 views

### second hochschild cohomology and extension

i started learning the theorem that says there is a one to one correspondence between Ext(A,M) and H^2(A,M). however, the proof is not clear. I managed to show that there is a well-define map U from ...
### Why do we have $H_2(X,\mathbb Z)\cong\mathbb Z$ for the quintic threefold $X\subset\mathbb P^4$?
Let us work over $\mathbb C$. In this article by S. Katz, it is stated that for a quintic threefold $X\subset \mathbb P^4$ one has $$H_2(X,\mathbb Z)\cong\mathbb Z.$$ Can anyone help me to see why ...
I am just starting to learn the basics of cohomology and am confused about the construction of the cohomology groups. So given a group $G$, the idea is you take a projective resolution of \$P_0 = ...