# Tagged Questions

**37**

votes

**2**answers

1k views

### $H^4(BG,\mathbb Z)$ torsion free for $G$ a connected Lie group

Recently, prompted by considerations in conformal field theory, I was let to guess that for every compact connected Lie group $G$, the fourth cohomology group of it classifying space is torsion free.
...

**2**

votes

**0**answers

97 views

### The transfer map $H_*(BSO(3))\rightarrow H_*(BO(2))$: reference request

All cohomology and homology will be $Z/2$ coefficient. The restriction map
$H^*(BSO(3))\rightarrow H^*(BO(2))$ is well-known to be the inclusion of
the Dickson invariant $Z/2[w_2,w_3]$ into the ...

**3**

votes

**1**answer

216 views

### Cohomology of Projective Classical Lie Groups

Let $G$ be a compact, connected, simply-connected Lie group with centre $Z(G)$, and consider the Lie group $G/Z(G)$. I believe that for $G$ a classical group, the Lie group $G/Z(G)$ is sometimes ...

**4**

votes

**1**answer

338 views

### Cohomology Ring of the Flag Manifolds, Cartan Subalgebras, and Weyl Groups

I've recently read the following line in an interesting paper:
It is well-known that the cohomology ring of a flag variety $G/B$ is isomorphic to the quotient ring of the ring of polynomial ...

**5**

votes

**3**answers

593 views

### Continuous cohomology of semi-simple Lie group.

Let $G$ be a real connected semi-simple Lie group. Let $M$ be a finite dimensional representation of it. Are there general criteria when the continuous cohomology groups $H_{cont}^q(G,M)$ vanish?
A ...

**11**

votes

**2**answers

738 views

### Torsion for Lie algebras and Lie groups

This question is about the relationship (rather, whether there is or ought to be a relationship) between torsion for the cohomology of certain Lie algebras over the integers, and torsion for ...

**2**

votes

**1**answer

405 views

### Generator of Lie Group cohomology in degree 3

This is my first question.
Take a simple, connected, compact, simply connected Lie group $ G$ (dim $G\geq 3$).
The cohomology of $G$ with integer coefficients is
$H^{1,2}(G,\mathbb{Z})\cong 0$, ...

**4**

votes

**1**answer

529 views

### smooth cohomology of Lie groups

Let $G$ be a Lie group and $A$ a smooth $G$-module. Define $C^n(G,A)=\{ f: G^n \to A|~f~\text{is smooth}\}$ and $\partial^n: C^n \to C^{n+1}$ by the standard formula as used in the cohomology of ...

**0**

votes

**1**answer

413 views

### Natural embedding GL_n(C) -> C^{n^2} \ {0} induces zero on cohomology

The question looks like an exercise in elementary algebraic topology, but I didn't manage to solve it. I am considering this question because it is a toy example in a problem I'm thinking about.
...

**3**

votes

**1**answer

398 views

### Relation of Lie Groups and Cohmology Theories via Formal Group Laws

There is a standard process (for example explained here) to obtain a formal group law form a complex oriented cohomology theory.
For a Lie group G one can choose coordinates at the unit and expand ...

**11**

votes

**0**answers

389 views

### To what extent does (co)homology of groups made discrete depend on set theory?

There's a well-known paper by Milnor, "On the homology of Lie groups made discrete," that discusses the relation between the homology of a Lie group $G$ and the underlying discrete group $G^\delta$. ...

**3**

votes

**2**answers

347 views

### What is the homology of the real coordinate ring of SO(n,R)? Other compact matrix groups?

As someone whose knowledge of cohomology is patchy and picked up on a need-to-know basis, and whose algebraic geometry is even worse, I wondered if someone could help with this question. (I ran into ...

**6**

votes

**2**answers

380 views

### Can one calculate the (co)homology of the loopspace of a Lie group from its Lie algebra?

Compact connected simply-connected Lie groups have so much structure that you can calculate their cohomology from their Lie algebras using Lie algebra cohomology (certain Ext-groups) and similarly ...

**11**

votes

**4**answers

972 views

### How to compute the (co)homology of orbit spaces (when the action is not free)?

Suppose a compact Lie group G acts on a compact manifold Q in a not necessarily free manner. Is there any general method to gain information about the quotient Q/G (a stratified space)? For example, I ...