0
votes
1answer
218 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
1answer
182 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
0answers
85 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
0answers
78 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
2answers
243 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
2answers
352 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
1answer
618 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
1answer
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
1answer
207 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
3answers
711 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
0answers
91 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 ...
2
votes
2answers
361 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 ...
1
vote
1answer
239 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
4answers
189 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
3answers
689 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
3answers
407 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
2answers
219 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
1answer
197 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 ...
2
votes
2answers
510 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 ...
3
votes
1answer
145 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
2answers
279 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
1answer
139 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
3answers
878 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
1answer
485 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
3answers
964 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
1answer
337 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
4answers
737 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?
6
votes
6answers
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
5answers
733 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 ...
9
votes
1answer
491 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
1answer
461 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 ...
5
votes
0answers
206 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
6answers
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
0answers
233 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
1answer
320 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
3answers
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 ...
4
votes
2answers
674 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
1answer
270 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
1answer
215 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
2answers
382 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 ...
10
votes
1answer
628 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 ...
6
votes
4answers
999 views

$H_2$ of a simply connected Lie group vanishes

How do I show that the $H_2$ of a simply connected Lie group vanishes? (I don't want to use that $\pi_2(Lie group) = 0$, since this is what I want to prove. And I don't want to use the classification ...
7
votes
3answers
1k views

Lie subgroups of SU(3)

Apart from images of representations of subgroups of SU(2), what are the Lie subgroups of SU(3)? Where should I look for a reference?
6
votes
6answers
1k views

cohomology of BG, G compact Lie group

It has been stated in several papers that $H^{odd}(BG,\mathbb{R})=0$ for compact Lie group $G$. However, I've still not found a proof of this. I believe that the proof is as follows: --> $G$ compact ...
2
votes
1answer
379 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 $H^{1,2}(G,\mathbb{Z})\cong 0$, ...
0
votes
2answers
572 views

Looking for general approaches to show connectedness of topological groups

Let $G$ be a topological group. One general approach to show that $G$ is connected is the following: For every subgroup $H\leq G$ (not necessarily closed) we have a projection map: $$ \pi: ...
15
votes
1answer
681 views

Word maps on compact Lie groups

Let $w=w(a,b)$ be a non-trivial word in the free group $F_2 = \langle a,b \rangle$ and $w_G \colon G \times G \to G$ be the induced word map for some compact Lie group $G$. Murray Gerstenhaber and ...
11
votes
1answer
503 views

Lie's third theorem via differential graded algebras?

Dennis Sullivan, "Infinitesimal computations in topology", Publ. IHES: At the end of section 8, he writes, among other things, roughly the following. Let $\mathfrak{g}$ be a (finite-dimensional, ...
9
votes
5answers
2k views

Does a finite-dimensional Lie algebra always exponentiate into a universal covering group

Hi, I am a theoretical physicist with no formal "pure math" education, so please calibrate my questions accordingly. Consider a finite-dimensional Lie algebra, A, spanned by its d generators, ...
0
votes
1answer
266 views

What does the weights of Lie group mean?

Let $\Delta=\{\alpha_1,\alpha_2\}$ be the simple root system of the exceptional Lie group $G_2$ with $\alpha_1$ is short and $\alpha_2$ is long, so ...