For the action of a locally compact and totally disconnected group G on a pair of locally compact spaces X and Y we construct, by sheaf theoretic means, a new equivariant and bivariant cohomology theory. If we take for the first space Y an universal proper G- action then we obtain for the second space its delocalized equivariant homology. This is in exact formal analogy to the definition of equivariant K-homology by Baum, Connes, Higson starting from the bivariant equivariant Kasparov K K -theory. Under certain basic finiteness conditions on the first space Y we conjecture the existence of a Chern character from the equivariant Kasparov K K -theory of Y and X into our cohomology theory made two-periodic which becomes an isomorphism upon tensoring the K K-theory with the complex numbers. This conjecture is proved for profinite groups G. An essential role in our construction is played by a bivariant version of Segal localization which we establish for K K -theory.
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