A dimension condition on the cohomology of a homogeneous space

The rational cohomology of a homogeneous space G/K admits a homomorphism from H∗(BK) induced from the classifying map G/K→BK of the principal K-bundle G→G/K. Assume the Lie group is K connected, so that π1(BK)=0; then the space G/K is formal in the sense of rational homotopy theory if and only if H∗(G/K) is a free … Read more

Gradient of spectral function on noncompact homogeneous space

Let (M,g) be a noncompact Riemannian manifold whose isometry group acts transitively on M, i.e. a (not necessarily normal) homogeneous space. Let eλ(x,y) be the integral kernel of f↦∫λ0dEν(f) where dEν is the spectral measure of the (non-negative) Laplacian associated to (M,g). Is there a relatively simple proof that ∫M|∇xeλ(x,y)|2gdy=∫Meλ(x,y)⋅Δxeλ(x,y)dy ? Note that the differentiation is … Read more

Tight bound on spectral gap of compact homogeneous manifold?

This paper by Peter Li proves a bound on the spectral gap of the Laplacian on a compact homogeneous manifold of diameter d: λ1≥c/d2, where c=π2/4. Can this bound be strengthened in a large number of dimensions, or is it essentially tight? In other words, if we restrict to manifolds of dimension N what is … Read more

Cartan geometry: jet space perspective on the tractor bundle

Let G a Lie group and H⊂G a Lie subgroup. For simplicity we assume that the adjoint action of H on g/h is faithful. Let M a differentiable manifold of the same dimension as G/H. A (H,G)-Cartan geometry on M is defined as a reduction of the structure group of the frame bundle of M … Read more

Decomposition of fiber product of GG-sets in GG-orbits

I have posted an identical question in MSE few days ago, but maybe this site is a better adress to discuss this problem: Let G be a finite group and K,H≤G two subgroups. Then the right quotients G/H and G/K become naturally left G-sets via ρ:G×G/H→G/H,(g,fH)↦gfH (we will work only with left G-sets and omit … Read more

Invariant measure on affine charts of complex Grassmannian

Consider the complex Grassmannian U(n)/U(k)×U(n−k) with it’s U(n)-invariant measure. The affine chart corresponding to i1,…,ik is given by n×k matrices for which the submatrix given by columns corresponding to i1,…,ik is the k×k identity matrix Ek. Consider for example the chart given by 1,…,k which consists of matrices (Ik|X), where X is arbitrary k×(n−k) complex … Read more

An analogy of product formula for homogeneous space?

$\DeclareMathOperator\Sel{Sel}$Let $E$ be an elliptic curve defined over a number field $K$ with full $2$-torsion. The classical complete $2$-descent method tells that the $2$-Selmer group $\Sel_2(E/K)$ can be identified with the set of locally solvable everywhere homogeneous spaces. More precisely, we consider a quadratic twist $$E:y^2=x(x-na)(x+nb)$$ defined over $\mathbb Q$, where $n,a,b,\frac{a+b}{2}$ are positive odd … Read more

Noncommutative Erlangen Program

Has Klein’s “Erlangen Program” been generalized/extended to the noncommutative setting (say, à la Connes)? Is there a classification of “noncommutative klein geometries” at least in very low dimension? Answer Some steps towards generalizing/extending “Erlangen Program” to the noncommutative setting have been taken by V. Kisil. See eg. here and his later papers. AttributionSource : Link … Read more

$K$-Theory Of Aloff–Wallach Spaces

Aloff–Wallach spaces were discussed in this question. They are quotients of SU(3) by U(1) indexed by a lattice of rank 2. Am I correct in guessing that the $K$-theory group $K_0$ of these spaces is the same for all elements in the lattice? (I am also assuming here, perhaps falsely, that the algebraic, topological, and … Read more

Conformally flat homogeneous spaces

Let’s say we have a homogeneous space H∖G. Is it possible to tell whether this homogeneous space admits a conformally flat metric just from its group structure? I am particularly interested in a situation when H∖G is maximally-noncompact, i.e. H is a maximally compact subgroup of G. I hope, my question does not sound too … Read more