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Equivariant bicycles on singular spaces. (English. Abridged French version) Zbl 0719.19003
The authors announce in this note some of their results concerning the geometric description of the equivariant Kasparov groups. These are generalizations of earlier results obtained by the first author and R. Douglas [Proc. Symp. Pure Math. 38, 117-173 (1982; Zbl 0532.55004)] and then extended by A. Connes and G. Skandalis [C. R. Acad. Sci., Paris, Sér. I 292, 871-876 (1981; Zbl 0529.58030); Publ. Res. Inst. Math. Sci. 20, 1139-1183 (1984; Zbl 0575.58030)]. The Connes- Skandalis construction of the groups $$KK(X,Y)$$ as sets of equivalence classes of bivariant cycles (bicycles) is extended into the wider context when both X and Y are Thom-Mather stratified spaces; the new construction relies on Goresky’s $$\pi$$-fiber condition and on the concept of a normally non-singular map between stratified spaces. Transversality is then used to give a direct construction of the intersection product $KK(X,Y)\otimes_{Z}KK(Y,W)\to KK(X,W).$ The map $$\mu$$ : KK(X,Y)$$\to KK(C_ 0(X),C_ 0(Y))$$ constructed by Connes and Skandalis is shown to be an isomorphism which transforms the intersection product into that of Kasparov.
The authors’ method applies to the equivariant case as well, yielding the groups $$KK_ G(X,Y)$$ (for G a compact Lie group). The definition of the corresponding intersection product $KK_ G(X,Y)\otimes_{Z}KK_ G(Y,W)\to KK_ G(X,W)$ uses mainly the geometric Bott periodicity. As a byproduct, geometric realizations of the cobordism groups $$\Omega$$ (X), of the bivariant cobordism groups $$\Omega\Omega(X,Y)$$, as well as of their equivariant version $$\Omega \Omega_ G(X,Y)$$, are derived.

MSC:
 19K35 Kasparov theory ($$KK$$-theory) 19L47 Equivariant $$K$$-theory 55N22 Bordism and cobordism theories and formal group laws in algebraic topology 57N80 Stratifications in topological manifolds