Pure and Applied Mathematics Quarterly

Volume 12 (2016)

Number 2

$K$-homology and Fredholm operators II: elliptic operators

Pages: 225 – 241

DOI: https://dx.doi.org/10.4310/PAMQ.2016.v12.n2.a2


Paul F. Baum (Department of Mathematics, Pennsylvania State University, University Park, Penn., U.S.A.)

Erik van Erp (Dartmouth College, Hanover, New Hampshire, U.S.A.)


We give a simple and direct proof of the Atiyah–Singer–Kasparov theorem in $K$-homology, which reduces the full theorem for elliptic operators to the special case of Dirac operators. This is done by proving commutativity of a triangle of abelian groups.


elliptic operator, Dirac operator, $K$-homology, Thom isomorphism, Atiyah–Singer index theorem

The first author was partially supported by NSF grant DMS-0701184.

The second author was partially supported by NSF grant DMS-1100570.

Received 20 November 2016

Published 9 February 2018