# Colloque 2016 du GDR 2875, Topologie Algébrique et Applications

11-14 October 2016
Europe/Paris timezone

## A small projective resolution of complex K-theory

12 Oct 2016, 11:40
50m
Amphi Lavoisier ()

### Amphi Lavoisier

Invited speaker Topologie algébrique et applications

Prof. Greg Arone

### Description

Around 1982 Nick Kuhn proved that the symmetric powers of the sphere spectrum give rise to a minimal projective resolution of $HZ$. He then asked if there were other interesting examples of small projective resolutions of spectra, in particular of spectra like $bo$ or $bu$. In this talk I will show how to construct a small projective resolution of the connective K-theory spectrum $bu$. Our resolution has many similarities to the classic one that arises from the symmetric powers filtration. We give a unified proof of exactness of both resolutions, that is different from Kuhn’s proof. A key ingredient in our proof is a vanishing result for the Bredon homology of the complex of partitions and the complex of direct-sum decompositions. Joint work with Kathryn Lesh.

### Primary author

Greg Arone (Stockholm)

### Presentation Materials

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