### Abstract

Alperin's weight conjecture [1] admits a reformulation in terms of the cohomology of a functor on a category obtained from a subdivision construction applied to a centric linking system [7] of a fusion system of a block, which in turn can be interpreted as the equivariant Bredon cohomology of a certain functor on the G-poset of centric Brauer pairs. The underlying general constructions of categories and functors needed for this reformulation are described in 1 and 2, respectively, and 3 provides a tool for computing the cohomology of the functors arising in 2. Taking as starting point the alternating sum formulation of Alperin's weight conjecture by Knorr-Robinson [11], the material of the previous sections is applied in 4 to interpret the terms in this alternating sum as dimensions of cohomology spaces of appropriate functors, using further work of Robinson [15, 16, 17].

Original language | English |
---|---|

Pages (from-to) | 495-513 |

Number of pages | 18 |

Journal | Mathematische Zeitschrift |

Volume | 250 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jul 2005 |

### Keywords

- CANCELLATION THEOREMS
- BLOCKS
- DADE

### Cite this

*Mathematische Zeitschrift*,

*250*(3), 495-513. https://doi.org/10.1007/s00209-004-0753-x

**Alperin's weight conjecture in terms of equivariant Bredon cohomology.** / Linckelmann, Markus.

Research output: Contribution to journal › Article

*Mathematische Zeitschrift*, vol. 250, no. 3, pp. 495-513. https://doi.org/10.1007/s00209-004-0753-x

}

TY - JOUR

T1 - Alperin's weight conjecture in terms of equivariant Bredon cohomology

AU - Linckelmann, Markus

PY - 2005/7

Y1 - 2005/7

N2 - Alperin's weight conjecture [1] admits a reformulation in terms of the cohomology of a functor on a category obtained from a subdivision construction applied to a centric linking system [7] of a fusion system of a block, which in turn can be interpreted as the equivariant Bredon cohomology of a certain functor on the G-poset of centric Brauer pairs. The underlying general constructions of categories and functors needed for this reformulation are described in 1 and 2, respectively, and 3 provides a tool for computing the cohomology of the functors arising in 2. Taking as starting point the alternating sum formulation of Alperin's weight conjecture by Knorr-Robinson [11], the material of the previous sections is applied in 4 to interpret the terms in this alternating sum as dimensions of cohomology spaces of appropriate functors, using further work of Robinson [15, 16, 17].

AB - Alperin's weight conjecture [1] admits a reformulation in terms of the cohomology of a functor on a category obtained from a subdivision construction applied to a centric linking system [7] of a fusion system of a block, which in turn can be interpreted as the equivariant Bredon cohomology of a certain functor on the G-poset of centric Brauer pairs. The underlying general constructions of categories and functors needed for this reformulation are described in 1 and 2, respectively, and 3 provides a tool for computing the cohomology of the functors arising in 2. Taking as starting point the alternating sum formulation of Alperin's weight conjecture by Knorr-Robinson [11], the material of the previous sections is applied in 4 to interpret the terms in this alternating sum as dimensions of cohomology spaces of appropriate functors, using further work of Robinson [15, 16, 17].

KW - CANCELLATION THEOREMS

KW - BLOCKS

KW - DADE

U2 - 10.1007/s00209-004-0753-x

DO - 10.1007/s00209-004-0753-x

M3 - Article

VL - 250

SP - 495

EP - 513

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

SN - 0025-5874

IS - 3

ER -