To carry through the proof of the APS index theorem as outlined in the introduction a reasonably good understanding of the heat kernel,

is needed. In particular the cases P = 3“ 3+, P = 3+8~ are important. The analysis of these kernels will start with the case d X = 0, which is very standard. However the use of blow-up techniques to define a space (the heat space) on which the heat kernel is quite simple is not so usual, although philosophically it is just a slight extension of Hadamard’s method. This blow-up approach is very much in the same style as the treatment of the 6-calculus and therefore has the advantage that it generalizes very readily to the case of the heat kernel for a 6-metric, which is the important case for the APS theorem. For other generalizations of this approach to the heat kernel see [57] and [30]. 7.1. Heat space.