# Tagged Questions

**33**

votes

**1**answer

799 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.
...

**8**

votes

**0**answers

334 views

### Homology of Lie groups

Let $G$ be a Lie group and $G^{\delta}$ the underlying group (with discrete topology). Obviously, we have a continuous map of groups $i:G^{\delta}\rightarrow G$ which induces a map between classifying ...

**4**

votes

**0**answers

132 views

### Cohomology of Bott-Samelson varieties?

How is the cohomology of Bott-Samelson varieties (desingularizations of Schubert Varieties ) usually calculated? Let's fix the Lie group to be $GL_n(\mathbb{C})$ or $SL_n(\mathbb{C})$ here.
Is there ...

**0**

votes

**1**answer

160 views

### Compact Lie groups with only 3 dimensional cohomology generators

Let $M$ be a compact connected semi-simple Lie group. Then by Hopf's Theorem $H^*(M;\mathbb Q)=\Lambda[\omega_1,...,\omega_s]$ where $\omega_i\in H^i(M;\mathbb Q)$ , $i\ge 3$ is odd.
For which $M$, ...

**2**

votes

**0**answers

153 views

### Generator of $\pi_3(SU(4))$ in Mimura-Toda

In this paper of Mimura and Toda, tables are given for low-dimensional homotopy groups of $SU(3)$, $SU(4)$ and $Sp(2)$. As far as I understand it, Theorem 6.1 gives the generator of $\pi_3(SU(4))$ as ...

**16**

votes

**2**answers

557 views

### What's the relationship between these two isomorphisms involving G and T?

Let $G$ be a compact connected Lie group with maximal torus $T$ and Weyl group $W$. Recall the following two isomorphisms.
Isomorphism 1: $R(G) \cong R(T)^W$, where $R(-)$ denotes the representation ...

**7**

votes

**0**answers

124 views

### $v_1$-periodic homotopy and principal bundle classification

This question came back to my mind while pondering this MO question. The classification of principal bundles is seriously difficult because of our lack of understanding of homotopy groups of compact ...

**0**

votes

**0**answers

281 views

### A Generalized De Rham cohomology

Edit According to the comment of Alex Degtyarev, I deleted the last part of the previous version.
Let $E$ be a real vector space. The complex valued $k$- tensors on $E$ is denoted by ...

**3**

votes

**0**answers

100 views

### Why “non-linear similarity” is the same as equivalence of representations for connected Lie groups?

Let $G$ be a compact Lie group and $V$ a finite-dimensional orthogonal $G$-representation. Write $S^V$ for the quotient $D(V)/S(V)$, where $D(V)$ and $S(V)$ are the unit disk and sphere in $V$, ...

**0**

votes

**1**answer

247 views

### Computing the fundamental group of a flag variety

Let $G$ be a compact and connected and simply connected Lie group and $\mathfrak{g}$ be its Lie algebra and $x\in\mathfrak{g}^*$. How can we compute the fundamental group of $G/G_x$ where $G_x$ is ...

**1**

vote

**1**answer

191 views

### Finding the 2nd homotopy group $\pi_2(G^\mathbb{C}/P)$

Let $G$ be a compact connected and simply connected Lie group and $G^\mathbb{C}$ be the complexification of Lie group (with is diffeomorphic with $G^\mathbb{C}\cong T^*G$) then I am looking for ...

**3**

votes

**0**answers

91 views

### Equivariant Poincare Series of Based Loop Group of SU(2)

Let $\Omega SU(2)$ denote the based loop group of $SU(2)$, and consider the action of $S^1$ on $\Omega SU(2)$ as a maximal torus of $SU(2)$. (This is not the "loop rotation" action.) Is there an ...

**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 ...

**7**

votes

**2**answers

254 views

### Integral versus real (universal) characteristic classes

I'm pretty confused about the precise relation of the integral and the real cohomology of the classifying space $BG$ of a compact Lie group $G$. The natural map $H^n(BG;\mathbb{Z})\to ...

**9**

votes

**2**answers

422 views

### Cohomology ring of a flag variety and representation theory

I'm interested in the cohomology ring $H^*(G/B)$ of a flag variety $G/B$, where $G$ is a complex semi-simple Lie group and $B$ the Borel subgroup. Borel (1953) showed that this ring is isomorphic to ...

**6**

votes

**1**answer

856 views

### Double coset formulas for Orthogonal groups [Solved]

According to Madsen-Brumfiel "Evaluation of the Transfer and the Universal Surgery Classes" Inventiones mathematicae 32 (1976): 133-170 Theorem 3.11, we can compute
the composition
...

**9**

votes

**1**answer

167 views

### Are compact simple groups homotopically non-abelian?

Take a compact connected simple centreless Lie group $G$.
Can the commutator map $G\times G\to G$ sending $(x,y)$ to $[x,y]$ be homotopic to a constant map?
I am interested mostly in the case, ...

**8**

votes

**1**answer

213 views

### The Image of the Mod 2 Homology of BSp in the Homology of BSO

I'm essentially trying to figure out exactly what the title asks for. I've been scouring old Seminaires Henri Cartan and books by Stong to try to see exactly how to do this, but the combination of ...

**12**

votes

**3**answers

734 views

### Why do we need a $G$-universe?

Let $G$ be a compact Lie group. Before defining $G$-prespectra, we have to define a $G$-universe $\mathcal U$.
Question: Why do we need a $G$-universe?
A $G$-universe is defined to be a countably ...

**2**

votes

**0**answers

94 views

### Can class in $H^4(BT)$ be realized as the second Chern class of a principal SU(2) bundle?

The question in the title, to which I add some clarification. Can every class in $H^4(B\mathbb{T})$ be realized as the second Chern class of a principal $SU(2)$ bundle?
$B\mathbb{T}$ is the ...

**3**

votes

**2**answers

382 views

### Equivariant Cohomology of a Complex Projective Variety

Suppose that I have a complex projective variety $X$ endowed with an algebraic action of a complex torus $T$. Suppose also that the set $X^T$ of fixed points is finite. I would like to relate the ...

**2**

votes

**1**answer

280 views

### Thom-Gysin Sequences and Stratifications

Let $X$ be an affine algebraic variety over $\mathbb{C}$, and let $G$ be a semisimple complex linear algebraic group acting by variety automorphisms with finitely many orbits. The decomposition of $X$ ...

**4**

votes

**4**answers

197 views

### Stratifications and Cohomology Computations

I am interested in references and suggestions concerning the use of stratifications in topology to inductively compute topological invariants. I would appreciate a fairly introductory reference on the ...

**8**

votes

**3**answers

743 views

### HIgher Homotopy Groups and Representation Theory

Let $G$ be a compact Lie group, and $g$ its associated Lie algebra.
In what ways do the higher homotopy groups $\pi_{n}(G)$ with $n>1$ appear in the representation theory of $G$?
As an example, ...

**7**

votes

**3**answers

460 views

### Homotopy classes of maps to Lie groups

In Physics one often encounters maps from a certain manifold $M$ to a Lie group $G$. For example, in gauge theories, this gives a gauge transformation, wich is a symmetry of a theory. It is then ...

**2**

votes

**2**answers

235 views

### Stabilizers for Nilpotent Adjoint Orbits of Semisimple Groups

Let $G$ be a connected, simply-connected, complex, semisimple Lie group with Lie algebra $\frak{g}$. Suppose that $X\in\frak{g}$ is a nilpotent element (ie. that $ad_X:\frak{g}\rightarrow\frak{g}$ is ...

**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

**2**answers

537 views

### Group cohomology of orthogonal groups with integer coefficient

I would like to know the group cohomology of orthogonal groups $SO(n)$, which is the topological cohomology of the classifying space of the group:
$H^*(BSO(n);\mathbb{Z}) = $ ? (for example for ...

**4**

votes

**1**answer

150 views

### Representations of Finite Subgroups on Homology

Suppose that $G$ is a connected, simply-connected, complex, semisimple Lie group, and that $H$ is finite subgroup. Consider the left-multiplicative action of $H$ on $G$, and the resulting ...

**5**

votes

**2**answers

288 views

### Lie groups bundle

Given compact Lie groups $H \subset K \subset G$, there is a fiber bundle
$ \frac{K}{H} \rightarrow \frac{G}{H} \rightarrow \frac{G}{K}$.
Do you have a simple proof of this?

**3**

votes

**1**answer

140 views

### Compact homogeneous spaces that admit a self map of degree >1

It is well known that compact manifolds of negative sectional curvature don't admit self-maps of degree $>1$. At the same time positively curved manifolds such as $S^n$ and $\mathbb CP^n$ clearly ...

**6**

votes

**3**answers

892 views

### $\pi_1$ Sequence of Topological Groups

Consider a connected topological group $G$ (not necessarily Lie). You have some maps $G\times G\to G$, such as projection to either summand, or multiplication $(g,h)\mapsto gh$. Now let's look at a ...

**6**

votes

**1**answer

518 views

### G-equivariant Whitehead's Theorem

Suppose $X$ is a CW complex and $Y$ is a subcomplex. Let $G$ be a compact Lie group that acts on $X$ and $Y$. Suppose further that the CW structures on $X$ and $Y$ are $G$-stable. Moreover assume ...

**6**

votes

**3**answers

1k views

### Homology versus cohomology of Lie groups

A central advantage of cohomology theory over homology -- at least in terms of richness of structure and strength as an invariant -- is the additional ring structure from the cup product. Recall that ...

**14**

votes

**1**answer

341 views

### Proof for which primes H*G has torsion

In 1960 Borel proved a beautiful result:
Theorem. Let G be a simple, simply connected Lie group. Suppose that p is a prime that does not divide any of the coefficients of the highest root (expressed ...

**2**

votes

**4**answers

815 views

### Commutativity of the fundamental group of any Lie Group [closed]

How do we formally prove that the fundamental group of any Lie group is always commutative?

**7**

votes

**6**answers

2k views

### connected compact semisimple lie group finite fundamental group

I was told that the fundamental group of a connected, compact, semisimple Lie group is finite, with the outline of a possible way to prove this fact. Is there any source however that fleshes this out ...

**11**

votes

**5**answers

792 views

### When are all centralizers in a Lie group connected?

Let $G$ be a compact connected Lie group acting on itself by conjugation,
$$ G\times G\to G,\qquad (\sigma,h)\mapsto \sigma h \sigma^{-1}.$$
The fixed point set of a closed subgroup $H\le G$ equals ...

**10**

votes

**1**answer

523 views

### 3rd homotopy group of a compact Simple Lie Group

Suppose $G$ is a compact simple Lie group with Lie algebra $\mathfrak g$. Then we know that $\pi_3(G)=Z$. Now suppose that $H_\alpha$ is a co-root vector in correspondence with a root $\alpha$. So it ...

**13**

votes

**1**answer

478 views

### free loop space and invariant forms

Cartan proved that for a connected compact Lie group $G$ the left invariant differential forms yield the correct cohomology of $G$. The same argument works for a connected compact $G$-manifold: the ...

**6**

votes

**0**answers

212 views

### Cohomology of T^{n}/W for compact Lie groups

Let $G$ be a compact, connected and simply connected.
Let $T\subset G$ be a maximal torus and let $W$ be the corresponding
Weyl group.
Then we have the diagonal action of $W$ on $T^{n}$ for $n\ge ...

**6**

votes

**5**answers

1k views

### Weight lattice and the first fundamental group

Let $G$ be a compact connected Lie group and let $T$ be a maximal torus of $G$, with Lie algebras $\frak{g}$ and $\frak{t}$ respectively. Then, $\frak{t}$ can be considered as a Cartan subalgebra of ...

**0**

votes

**0**answers

238 views

### Is $GL_{S^1}(H)$ (for $H$ Hilbert space ) path-connected?

Due to the negative answer to my last question I want to know if at least the following is true:
Let H be an infinite dimensional separable complex Hilbert space with $S^1$-action. Let ...

**2**

votes

**1**answer

325 views

### Theorem of Kuiper for Hilbert spaces with group action

Let $H$ be an infinite dimensional separable complex Hilbert space with Lie group action (I am mostly interested in the case $S^1$). Let $\text{Gl}_{G}(H)$ be the space of invertible, bounded and ...

**3**

votes

**3**answers

1k views

### Group cohomology of compact Lie group with integer coeffient

It is known that group cohomology class $H^d[U(1),Z]$ is Z for even d and 0 for odd d.
Do we know $H^d[G,Z]$ for $G=SO(3)$, $SU(2)$ and other compact Lie group?
Also is the Borel-group-cohomology ...

**5**

votes

**2**answers

745 views

### Calculate the group cohomology classes $H^d[U(1)\rtimes Z_2, Z]$ and $H^d[U(1)\rtimes Z_2, Z_T]$

I would like to know what are the group cohomology classes $H^d[U(1)\rtimes Z_2, Z]$ and $H^d[U(1)\rtimes Z_2, Z_T]$, and/or how to calculate them.
It can be shown that $H^d[U(1), Z]$ is $Z$ for ...

**2**

votes

**1**answer

271 views

### Sg: How to Show this Sequence is Exact?

Hi,All:
I am seeing a result in which the following sequence, in the context of the genus-g surface Sg, is described as being exact:
1-->Tg-->$M^{(2)}g$-->$Sp^{(2)}(2g,\mathbb Z)$-->1
Where :
i)Tg ...

**8**

votes

**1**answer

217 views

### Branch cuts of $GL_n^+(\mathbb{R})$

Branch cuts
Let $GL_n^+(\mathbb{R})$ denote the group of $n\times n$ real matrices with positive determinant. Topologically, $GL_n^+(\mathbb{R})$ is connected, and
$$ \pi_1(GL_2^+(\mathbb{R})) = ...

**6**

votes

**2**answers

387 views

### Cohomology of finite quotients of Lie groups

Let $G$ denote the $Spin(n)$ group with $n>4$ and let $\Gamma$ be a cyclic subgroup $G$ of a prime order $p >2$. When does the projection $G \to G/\Gamma$ induce a surjection
between ...

**11**

votes

**2**answers

729 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 ...