Paul Baum

Riemann-Roch for Singular Varieties

The basic tool for a general Riemann-Roch theorem is MacPherson’s graph construction, applied to a complex E. of vector bundles on a scheme Y, exact off a closed subset X. This produces a localized Chern character1 ch x y (E.) which lives in the bivariant group \( A{\left( {X \to Y} \right)_\mathbb{Q}} \) For each class α∈A * Y, this gives a class

THE COHOMOLOGY OF QUOTIENTS OF CLASSICAL GROUPS

ch_X^Y\left( {E.} \right) \cap \alpha \in {A_ * }{X_\mathbb{Q}}

On the Zeroes of Meromorphic Vector-Fields

whose image in \( {A_ * }{Y_\mathbb{Q}} \) is \( {\sum {\left( { - 1} \right)} ^i}ch\left( {{E_i}} \right) \cap \alpha \) The properties needed for Riemann-Roch, in particular the invariance under rational deformation, follow from the bivariant nature of ch x y E.

A proof of the Baum-Connes conjecture for p-adic GL(n)

Publisher Summary This chapter discusses the Chern character for discrete groups. H j (X;Q) is the j-th Cech cohomology group of X with coefficients the rational numbers Q. The key property of this classical Chern character is that it is a rational isomorphism. Cyclic cohomology can be used to define the delocalized equivariant cohomology of X. The traditional homotopy quotient Chern character gives a map, which is always surjective, for compact X. The chapter also discusses twisted homology and K homology. K homology is the homology theory associated to the Z × BU spectrum. A concrete realization of this theory is obtained by using the K-cycle definition.

On the Baum–Connes conjecture in the real case

THE HOMOLOGY and cohomology rings of the classical compact Lie groups so(n), SU(n), Sp(n) are well known (for example see Bore1 [3]). Most of these groups have non-trivial centers, and in [4], Bore1 investigated the quotients of these groups by central subgroups, in particular, calculating cohomology rings with Zp coefficients, p prime. In this paper we pursue the investigation of these quotients further, extending Borel’s results. We obtain extra information on the integral cohomology and we completely determine the diagonal maps in cohomology with 2, coefficients p prime, and the action of the Steenrod algebra in any quotient of one of these groups by a central subgroup. This information leads to some applications such as the fact that homotopy equivalent compact connected simple groups are isomorphic, and a technique to prove facts about vector fields on real projective spaces, (see

Toeplitz Operators and Poincaré Duality

9). These results on vector fields may be applied to prove non-immersion theorems for real projective spaces. In particular, it is shown that if n = 2’ + 3, r 2 3, then P*-l and P” do not immerse in R2”-7. Mahowald [9] and Sanderson [12] have shown that P” does immerse in RZn-‘, so that this is the best possible immersion for P” and P”-‘. The diagonal maps in the cohomology of these quotient groups with Z, coefficients are usually non-commutative, where p divides the order of the central subgroup, the only exceptions being low dimensional and PSp(n) for odd n (mod 2), (where PG denotes the quotient of G by its center). In particular ifp is an odd prime and p divides the order of the central subgroup, then the diagonal map in cohomology modp is non-commutative. Araki [2] has shown that the exceptional groups which have 3-torsion in homology have non-commutative diagonal maps in cohomology mod 3. One may conjecture: If a Lie group G hasp-torsion in its homology (p an odd prime) then the diagonal map in H*(G; Z,) is not commutative.

The Hecke algebra of a reductive p-adic group: a geometric conjecture

We give a proof that the geometric K-homology theory for finite CWcomplexes defined by Baum and Douglas is isomorphic to Kasparov’s Khomology. The proof is a simplification of more elaborate arguments which deal with the geometric formulation of equivariantK-homology theory.

Expanders, exact crossed products, and the Baum–Connes conjecture

Let M be a compact complex analytic manifold and let x be a holomorphic vector-field on M. In an earlier paper by one of us (see [2]) it was shown that the behavior of x near its zeroes determined all the Chern numbers of M and the nature of this determination was explicitly given where x had only nondegenerate zeroes. The primary purpose of this note is to extend this result to meromorphic fields, or equivalently to sections s of T⊗L where T is the holomorphic tangent bundle to M and L is a holomorphic line bundle. We will also drop the non-degeneracy assumption of the zeroes of s, but we treat only the case where s vanishes at isolated points {p}.