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An extension of Steinberg's Theorem to biquotient pairs of subgroups

Marcus Zibrowius

TL;DR

This paper extends Steinberg's theorem to biquotient pairs of subgroups $(H_1,H_2)$ of a compact Lie group $G$, proving vanishing of higher Tor groups $\mathrm{Tor}_i^{\mathrm{R}G}(\mathrm{R}H_1,\mathrm{R}H_2)$ under a strict (and a relaxed lax) biquotient condition. The author develops a strategy that reduces to maximal-rank torus cases, uses a diagonal reduction and Segal's theory of supports, and then propagates the result to arbitrary connected subgroups via torus-enlargement and localization techniques. These Tor-vanishing results yield concrete descriptions of the complex $K$-theory of biquotients through Hodgkin's spectral sequence, notably $\mathrm{K}^0(H_1\backslash G/H_2) \cong \mathrm{R}H_1 \otimes_{\mathrm{R}G} \mathrm{R}H_2$ and $\mathrm{K}^1(H_1\backslash G/H_2) \cong \mathrm{Tor}_1^{\mathrm{R}G}(\mathrm{R}H_1,\mathrm{R}H_2)$, with vanishing in the maximal-rank case. The work connects representation theory, algebraic topology, and geometric aspects of biquotients, and furnishes a framework for explicit computations in the $K$-theory of biquotients.

Abstract

We study the derived tensor product of the representation rings of subgroups of a given compact Lie group G. That is, given two such subgroups H_1 and H_2, we study the tensor product of the associated representation rings R(H_1) and R(H_2) over the representation ring RG, and prove a vanishing result for the associated higher Tor-groups. This result can be viewed as a natural generalization of the Theorem of Steinberg that asserts that the representation rings of maximal rank subgroups of G are free over RG. It my also be viewed as an analogue of a result of Singhof on the cohomology of classifying spaces. We include an immediate application to the complex K-theory of biquotient manifolds.

An extension of Steinberg's Theorem to biquotient pairs of subgroups

TL;DR

This paper extends Steinberg's theorem to biquotient pairs of subgroups of a compact Lie group , proving vanishing of higher Tor groups under a strict (and a relaxed lax) biquotient condition. The author develops a strategy that reduces to maximal-rank torus cases, uses a diagonal reduction and Segal's theory of supports, and then propagates the result to arbitrary connected subgroups via torus-enlargement and localization techniques. These Tor-vanishing results yield concrete descriptions of the complex -theory of biquotients through Hodgkin's spectral sequence, notably and , with vanishing in the maximal-rank case. The work connects representation theory, algebraic topology, and geometric aspects of biquotients, and furnishes a framework for explicit computations in the -theory of biquotients.

Abstract

We study the derived tensor product of the representation rings of subgroups of a given compact Lie group G. That is, given two such subgroups H_1 and H_2, we study the tensor product of the associated representation rings R(H_1) and R(H_2) over the representation ring RG, and prove a vanishing result for the associated higher Tor-groups. This result can be viewed as a natural generalization of the Theorem of Steinberg that asserts that the representation rings of maximal rank subgroups of G are free over RG. It my also be viewed as an analogue of a result of Singhof on the cohomology of classifying spaces. We include an immediate application to the complex K-theory of biquotient manifolds.
Paper Structure (14 sections, 31 theorems, 35 equations)

This paper contains 14 sections, 31 theorems, 35 equations.

Table of Contents

  1. Acknowledgements
  2. Representation rings
  3. Abelian groups
  4. Arbitrary compact groups
  5. Connected groups
  6. Good groups and Steinberg's Theorem
  7. Supports
  8. Biquotient pairs
  9. Reduction to diagonal
  10. The biquotient condition can be checked on finitely many intersections
  11. Enlarging tori while keeping intersections almost intact
  12. Reduction to toroidal subgroups
  13. The proof
  14. K-theory of biquotients

Key Result

Theorem 1

Suppose the fundamental group of $G$ is torsion-free. If $H_1$ intersects every conjugate of $H_2$ trivially, then $\mathop{\mathrm{Tor}}\nolimits_i^{\mathrm{R}G}(\mathrm{R}H_1, \mathrm{R}H_2)$ vanishes for all $i > \mathop{\mathrm{rank}}\nolimits G - (\mathop{\mathrm{rank}}\nolimits H_1 + \mathop{\

Theorems & Definitions (62)

  • Theorem 1
  • Theorem : Steinberg Steinberg:Pittie
  • Theorem : Singhof Singhof:DCM
  • Corollary 2
  • Theorem 3
  • Lemma 4
  • Definition 5
  • Lemma 6
  • proof
  • Lemma 7
  • ...and 52 more