KK-theory of S7/Q8S^7/Q_8 and a counterexample to a result of P.M. Akhmet'ev

A simple counterexample is presented to a proposition which is used in the arguments given by P. M. Akhmet'ev in his work on the Hopf invariant and Kervaire invariant. The counterexample makes use of the $K$-theory of the quotient of the 7-sphere by the quaternion group of order 8.Comment: Version published in Morfismos Vol. 13, No. 2, 2009, pp. 55-6

K-theory and elliptic operators

This expository paper is an introductory text on topological K-theory and the Atiyah-Singer index theorem, suitable for graduate students or advanced undegraduates already possessing a background in algebraic topology. The bulk of the material presented here is distilled from Atiyah's classic "K-Theory" text, as well as his series of seminal papers "The Index of Elliptic Operators" with Singer. Additional topics include equivariant K-theory, the G-index theorem, and Bott's paper "The Index Theorem for Homogeneous Differential Operators". It also includes an appendix with a proof of Bott periodicity, as well as sketches of proofs for both the standard and equivariant versions of the K-theory Thom isomorphism theorem, in terms of the index for families of elliptic operators. A second appendix derives the Atiyah-Hirzebruch spectral sequence. This text originated as notes from a series of lectures given by the author as an undergraduate at Princeton. In its current form, the author has used it for graduate courses at the University of Oregon.Comment: 50 page

Harmonic spinors on homogeneous spaces

Let G be a compact, semi-simple Lie group and H a maximal rank reductive subgroup. The irreducible representations of G can be constructed as spaces of harmonic spinors with respect to a Dirac operator on the homogeneous space G/H twisted by bundles associated to the irreducible, possibly projective, representations of H. Here, we give a quick proof of this result, computing the index and kernel of this twisted Dirac operator using a homogeneous version of the Weyl character formula noted by Gross, Kostant, Ramond, and Sternberg, as well as recent work of Kostant regarding an algebraic version of this Dirac operator.Comment: 7 page

Twisted representation rings and Dirac induction

Extending ideas of twisted equivariant $K$-theory, we construct twisted versions of the representation rings for Lie superalgebras and Lie supergroups, built from projective $\Z_{2}$-graded representations with a given cocycle. We then investigate the pullback and pushforward maps on these representation rings (and their completions) associated to homomorphisms of Lie superalgebras and Lie supergroups. As an application, we consider the Lie supergroup $\Pi (T^{*}G)$, obtained by taking the cotangent bundle of a compact Lie group and reversing the parity of its fibers. An inclusion $H \hookrightarrow G$ induces a homomorphism from the twisted representation ring of $\Pi(T^{*}H)$ to the twisted representation ring of $\Pi(T^{*}G)$, which pulls back via an algebraic version of the Thom isomorphism to give an additive homomorphism from $K_{H}(\mathrm{pt})$ to $K_{G}(\mathrm{pt})$ (possibly with twistings). We then show that this homomorphism is in fact Dirac induction, which takes an $H$-module $U$ to the $G$-equivariant index of the Dirac operator \dirac \otimes U on the homogeneous space $G/H$ with values in the homogeneous bundle induced by $U$.Comment: 26 pages. Shortened the paper and cleaned up problems with cocycles vs. cohomology classes, Proposition 2, and other minor issue

Representation rings of Lie superalgebras

Given a Lie superalgebra \g, we introduce several variants of the representation ring, built as subrings and quotients of the ring R_{\Z_2}(\g) of virtual \g-supermodules (up to even isomorphisms). In particular, we consider the ideal R_{+}(\g) of virtual \g-supermodules isomorphic to their own parity reversals, as well as an equivariant K-theoretic super representation ring SR(\g) on which the parity reversal operator takes the class of a virtual \g-supermodule to its negative. We also construct representation groups built from ungraded \g-modules, as well as degree-shifted representation groups using Clifford modules. The full super representation ring SR^{*}(\g), including all degree shifts, is then a \Z_{2}-graded ring in the complex case and a \Z_{8}-graded ring in the real case. Our primary result is a six-term periodic exact sequence relating the rings R^{*}_{\Z_2}(\g), R^{*}_{+}(\g), and SR^{*}(\g). We first establish a version of it working over an arbitrary (not necessarily algebraically closed) field of characteristic 0. In the complex case, this six-term periodic long exact sequence splits into two three-term sequences, which gives us additional insight into the structure of the complex super representation ring SR^{*}(\g). In the real case, we obtain the expected 24-term version, as well as a surprising six-term version of this periodic exact sequence.Comment: 36 pages, 1 figure, uses Payl Taylor's diagrams package. Updated with minor correction

The integral cohomology groups of configuration spaces of pairs of points in real projective spaces

We compute the integral homology and cohomology groups of configuration spaces of two distinct points on a given real projective space. The explicit answer is related to the (known multiplicative structure in the) integral cohomology---with simple and twisted coefficients---of the dihedral group of order 8 (in the case of unordered configurations) and the elementary abelian 2-group of rank 2 (in the case of ordered configurations). As an application, we complete the computation of the symmetric topological complexity of real projective spaces of dimension 2^i + d for d=0,1,2.Comment: Published version: Morfismos Volumen 14, N\'umero 2, 201

CR structures on open manifolds

We show that the vanishing of the higher dimensional homology groups of a manifold ensures that every almost CR structure of codimension $k$ may be homotoped to a CR structure. This result is proved by adapting a method due to Haefliger used to study foliations (and previously applied to study the relation between almost complex and complex structures on manifolds) to the case of (almost) CR structures on open manifolds

The K-theory of abelian symplectic quotients

Let T be a compact torus and (M,\omega) a Hamiltonian T-space. In a previous paper, the authors showed that the T-equivariant K-theory of the manifold M surjects onto the ordinary integral K-theory of the symplectic quotient M \mod T of M by T, under certain technical conditions on the moment map. In this paper, we use equivariant Morse theory to give a method for computing the K-theory of the symplectic quotient by obtaining an explicit description of the kernel of the surjection \kappa: K^*_T(M) \onto K^*(M \mod T). Our results are K-theoretic analogues of the work of Tolman and Weitsman for Borel equivariant cohomology. Further, we prove that under suitable technical conditions on the T-orbit stratification of M, there is an explicit Goresky-Kottwitz-MacPherson (``GKM'') type combinatorial description of the K-theory of a Hamiltonian T-space in terms of fixed point data. Finally, we illustrate our methods by computing the ordinary K-theory of compact symplectic toric manifolds, which arise as symplectic quotients of an affine space \C^N by a linear torus action.Comment: 15 pages; typos correcte

The integral cohomology of configuration spaces of pairs of points in real projective spaces

We compute the integral cohomology ring of configuration spaces of two points on a given real projective space. Apart from an integral class, the resulting ring is a quotient of the known integral cohomology of the dihedral group of order 8 (in the case of unordered configurations, thus has only 2- and 4-torsion) or of the elementary abelian 2-group of rank 2 (in the case of ordered configurations, thus has only 2-torsion). As an application, we complete the computation of the symmetric topological complexity of real projective spaces of dimensions of the form 2^i+j for non-negative i and j with j<3.Comment: The results in this paper include those in arXiv:1004.0746, but the methods are different; here we depend on Bockstein spectral sequence calculations. While arXiv:1004.0746 deals only with additive structures, we now obtain full descriptions of the relevant cohomology rings. Further, this paper is more condensed than arXiv:1004.0746, from 41 pages in the latter, to the current 28 page

Surjectivity for Hamiltonian G-spaces in K-theory

Let G be a compact connected Lie group, and (M,\omega) a Hamiltonian G-space with proper moment map \mu. We give a surjectivity result which expresses the K-theory of the symplectic quotient M//G in terms of the equivariant K-theory of the original manifold M, under certain technical conditions on \mu. This result is a natural K-theoretic analogue of the Kirwan surjectivity theorem in symplectic geometry. The main technical tool is the K-theoretic Atiyah-Bott lemma, which plays a fundamental role in the symplectic geometry of Hamiltonian G-spaces. We discuss this lemma in detail and highlight the differences between the K-theory and rational cohomology versions of this lemma. We also introduce a K-theoretic version of equivariant formality and prove that when the fundamental group of G is torsion-free, every compact Hamiltonian G-space is equivariantly formal. Under these conditions, the forgetful map K_{G}^{*}(M) \to K^{*}(M) is surjective, and thus every complex vector bundle admits a stable equivariant structure. Furthermore, by considering complex line bundles, we show that every integral cohomology class in H^{2}(M;\Z) admits an equivariant extension in H_{G}^{2}(M;\Z).Comment: 27 pages. Revised with more general versions of the Atiyah-Bott lemma and the Kirwan surjectivity theorem, as well as a proof of equivariant formality (formerly only a conjecture