A Koszul complex of unstables modules and cohomotopy of a Thom spectrum

作者：Nguyen Dang Ho Hai^*

来源：Bulletin de la Societe Mathematique de France, 2012, 140(2): 257-308.

(A Koszul complex of unstables modules and cohomotopy of a Thom spectrum) We constructed in [8] a minimal injective resolution of an unstable module over the modulo 2 Steenrod algebra. From this resolution, a Segal conjecture-type result was obtained for a certain Thom spectrum. In this paper we propose to study similar problems modulo odd primes. Given p an odd prime, we construct in this work a Koszul complex in the category of unstable modules over the mod p Steenrod algebra. An injective resolution of an interesting unstable module is obtained as a special case of this Koszul complex. This unstable module is the mod p cohomology of a Thom spectrum used in the description of the layers of the Goodwillie tower of the identity functor evaluated on the sphere S-3. As an application of the injective resolution, we compute some cohomotopy groups of the Thom spectrum using work of S. Zarati [241 on the derived functors of the destabilisation functor.

• 出版日期2012