# TTT65: Sheffield

Department of Pure Mathematics, the University of Sheffield.

## Date

4 June 2008

## Speakers

Ieke Moerdijk (Sheffield and Utrecht)

An extension of the notion of Reedy category

Richard Garner (Uppsala and Cambridge)

Universal cofibrant replacements

Ran Levi (Aberdeen)

Some observations on a conjecture of Friedlander and Milnor

## Abstracts

### Ieke Moerdijk: An extension of the notion of Reedy category

I will present an extension of the notion of 'Reedy category' having possibly nontrivial automorphism groups. Like the classical notion, this extension has the property that diagrams indexed by a 'generalised Reedy category' still carry a closed model structure. Unlike the classical notion, the new notion includes important examples like Segal's category Gamma, Connes's category Lambda, and the indexing category Omega for dendroidal sets. Joint work with Clemens Berger.

### Richard Garner: Universal cofibrant replacements

The work of Jeff Smith makes it relatively straightforward to construct model structures on categories of algebraic entities. It is considerably less straightforward to get out of these model structures things that you can compute with. However, this need not be so. We explain how any cofibrantly generated model structure gives rise to a canonically and universally determined notion of cofibrant replacement, and by looking at some examples, see that this frequently gives us something both tractable and useful.

### Ran Levi: Some observations on a conjecture of Friedlander and Milnor

Let $G$ be a Lie group with finitely many components, and let $G^\delta$ be the group $G$ considered as a discrete group. Friedlander and Milnor conjectured that the obvious map $BG^\delta\to BG$ induces an isomorphism on homology with any finite coefficients.

Milnor showed that the conjecture holds whenever the identity component of $G$ is solvable, and that the map above induces a split epimorphism on homology with any finite coefficients. Friedlander and Mislin generalised the Isomorphism Conjecture, and showed that it is equivalent to another conjecture they named the Finite Subgroup Conjecture.

In this lecture we utilise homology decomposition techniques to show that Milnor's homological splitting result follows in fact from a topological splitting, which holds after $p$-completion at any prime $p$. We use this result to obtain an easy proof of the equivalence of the Isomorphism Conjecture and the Finite Subgroup Conjecture.

## Further information

The meeting was partially supported by a Scheme 3 grant from the London Mathematical Society.