Fadell-Husseini index theory: ring coefficients vs field coefficients
by
Pavle V. M. Blagojevic
Mathematical Institute SANU, Belgrade, Serbia
Coauthors: This is a part of the join project with Günter Ziegler.

The Fadell-Husseini index of a G-space X with coefficients in the field / ring R, Index_{G, R}(X), is the kernel ideal of the map in equivariant cohomology

H*(G, R) ≅ HG*(pt, R)→ HG*(X, R)

induced by the projection X→ pt. The contraposition of the basic property: if there is a G-map X→ Y then Index_{G, R}(X) ⊇ Index_{G, R}(Y), gives a criterion for the non-existence of G-maps:

IndexG, R(X)\nsupseteq IndexG, R(Y) ⇒  there is no G-map X→ Y.

Thus, the ideal valued index theory of Fadell and Husseini is a method for discussing the existence of G-maps between G-spaces.

Classically Fadell-Husseini index theory is considered with field coefficients. Motivated by the geometric combinatorial problems, for the first time, the indexes with the ring Z coefficients are used, explicitly computed and confronted by the related indexes with F₂ coefficients. In this talk we emphasize:

• the differences between indexes with ring and field coefficients,

• difficulties reflected by the problem of computing cohomology of group with Z coefficients, and

• the Z coefficient results which can not be obtained with the use of F₂ coefficients.

Date received: May 18, 2008