### Abstract

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.

Original language | English (US) |
---|---|

Pages (from-to) | 313-353 |

Number of pages | 41 |

Journal | K-Theory |

Volume | 25 |

Issue number | 4 |

DOIs | |

State | Published - Dec 1 2002 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*K-Theory*,

*25*(4), 313-353. https://doi.org/10.1023/A:1016036724442

}

*K-Theory*, vol. 25, no. 4, pp. 313-353. https://doi.org/10.1023/A:1016036724442

**Equivariant-bivariant Chern character for profinite groups.** / Baum, P.; Schneider, P.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Equivariant-bivariant Chern character for profinite groups

AU - Baum, P.

AU - Schneider, P.

PY - 2002/12/1

Y1 - 2002/12/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=26844544037&partnerID=8YFLogxK

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U2 - 10.1023/A:1016036724442

DO - 10.1023/A:1016036724442

M3 - Article

AN - SCOPUS:26844544037

VL - 25

SP - 313

EP - 353

JO - K-Theory

JF - K-Theory

SN - 0920-2036

IS - 4

ER -