2
votes
1answer
171 views

fundamental group and torus action

Let $T$ be the complex torus acting on a complex connected algebraic variety $X$ and let $p \colon X\rightarrow Y$ be a good quotient for this action. For any $y\in Y$ we have a sequence $p^{-1}(y) ...
2
votes
2answers
357 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 $X$ ...
2
votes
2answers
218 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 ...
5
votes
1answer
228 views

Conjugation of homogeneous spaces

Let $X$ be a smooth irreducible algebraic variety over the field of complex numbers ${\mathbb{C}}$. Let $x\in X({\mathbb{C}})$. Let $\tau$ be an automorphism of ${\mathbb{C}}$ (not necessarily ...
2
votes
1answer
483 views

Union of disjoint Vitali Sets…

We say $X$ is a Vitali set if there exists a countably dense subgroup, $\Gamma$, of the additive group $\mathbb{R}$, such that $X$ is a selector of the partition of $\mathbb{R}$ canonically associated ...
4
votes
4answers
472 views

Countable Dense Sub-Groups of the Reals…

Can countable dense additive subgroups of the reals be well-ordered up to isomorphism by inclusion? If so, is $\mathbb{Q}$ the smallest (up to isomorphism) countable dense subgroup of the reals, and ...
7
votes
4answers
2k views

Fundamental group of Lie groups

Let $T$ be a torus $V/\Gamma$, $\gamma$ a loop on $T$ based at the origin. Then it is easy to see that $$2 \gamma = \gamma \ast \gamma \in \pi_1(T).$$ Here $2 \gamma$ is obtained by rescaling ...
0
votes
1answer
405 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
3answers
675 views

Cohomology rings of GL_n(C), SL_n(C)

Can anyone provide me with the reference for the following fact (idea of the proof will be appreciated too): Cohomology ring with $\mathbb Q$-coefficients of a group $G$ (I don't know precisely what ...
3
votes
1answer
309 views

Principal bundle for contractible group is weak homotopy equivalence for ind schemes

This is may be obvious, but I am not comfortable with ind-schemes. I have an ind-scheme $X$ over $\mathbb{C}$. Every point has a neighborhood which can be written as an ascending union of regular ...
8
votes
1answer
449 views

Cohomology map induced by the group actions on homogeneous vector bundles

Here is a topological question which seems quite elementary. The answer to this question may be useful e.g. in estimating the orders of the automorphism groups of some algebraic varieties and in ...