### Abstract

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

Pages (from-to) | 1–10 |

Number of pages | 10 |

Journal | Homology, Homotopy and Applications |

Volume | 12 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2010 |

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### Keywords

- gluing problem

### Cite this

**The gluing problem does not follow from homological properties of Δ p (G).** / Libman, Assaf.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - The gluing problem does not follow from homological properties of Δ p (G)

AU - Libman, Assaf

PY - 2010

Y1 - 2010

N2 - Given a block b in kG where k is an algebraically closed field of characteristic p, Given a block b in kG where k is an algebraically closed field of characteristic p , there are classes α Q ∈H 2 (Aut F (Q);k × ) , constructed by Külshammer and Puig, where F is the fusion system associated to b and Q is an F -centric subgroup. The gluing problem in F has a solution if these classes are the restriction of a class α∈H 2 (F c ;k × ) . Linckelmann showed that a solution to the gluing problem gives rise to a reformulation of Alperin’s weight conjecture. He then showed that the gluing problem has a solution if for every finite group G , the equivariant Bredon cohomology group H 1 G (|Δ p (G)|;A 1 ) vanishes, where |Δ p (G)| is the simplicial complex of the non-trivial p -subgroups of G and A 1 is the coefficient functor G/H↦Hom(H,k × ) . The purpose of this note is to show that this group does not vanish if G=Σ p 2 where p≥5 .

AB - Given a block b in kG where k is an algebraically closed field of characteristic p, Given a block b in kG where k is an algebraically closed field of characteristic p , there are classes α Q ∈H 2 (Aut F (Q);k × ) , constructed by Külshammer and Puig, where F is the fusion system associated to b and Q is an F -centric subgroup. The gluing problem in F has a solution if these classes are the restriction of a class α∈H 2 (F c ;k × ) . Linckelmann showed that a solution to the gluing problem gives rise to a reformulation of Alperin’s weight conjecture. He then showed that the gluing problem has a solution if for every finite group G , the equivariant Bredon cohomology group H 1 G (|Δ p (G)|;A 1 ) vanishes, where |Δ p (G)| is the simplicial complex of the non-trivial p -subgroups of G and A 1 is the coefficient functor G/H↦Hom(H,k × ) . The purpose of this note is to show that this group does not vanish if G=Σ p 2 where p≥5 .

KW - gluing problem

U2 - 10.4310/HHA.2010.v12.n1.a1

DO - 10.4310/HHA.2010.v12.n1.a1

M3 - Article

VL - 12

SP - 1

EP - 10

JO - Homology, Homotopy and Applications

JF - Homology, Homotopy and Applications

SN - 1532-0073

IS - 1

ER -