# Semisimplicial spaces

@article{Ebert2019SemisimplicialS, title={Semisimplicial spaces}, author={Johannes Ebert and Oscar Randal-Williams}, journal={Algebraic \& Geometric Topology}, year={2019} }

This is an exposition of homotopical results on the geometric realization of semisimplicial spaces. We then use these to derive basic foundational results about classifying spaces of topological categories, possibly without units. The topics considered include: fibrancy conditions on topological categories; the effect on classifying spaces of freely adjoining units; approximate notions of units; Quillen’s Theorems A and B for non-unital topological categories; the effect on classifying spaces…

## 21 Citations

Some finiteness results for groups of automorphisms of manifolds

- MathematicsGeometry & Topology
- 2019

We prove that in dimensions not equal to 4, 5, or 7, the homology and homotopy groups of the classifying space of the topological group of diffeomorphisms of a disk fixing the boundary are finitely…

The homotopy type of the topological cobordism category

- Mathematics
- 2018

We define a cobordism category of topological manifolds and prove that if $d \neq 4$ its classifying space is weakly equivalent to $\Omega^{\infty -1} MTTop(d)$, where $MTTop(d)$ is the Thom spectrum…

Homological stability for moduli spaces of disconnected submanifolds, I

- MathematicsAlgebraic & Geometric Topology
- 2021

A well-known property of unordered configuration spaces of points (in an open, connected manifold) is that their homology stabilises as the number of points increases. We generalise this result to…

Homological stability of topological moduli spaces

- MathematicsGeometry & Topology
- 2019

Given a graded $E_1$-module over an $E_2$-algebra in spaces, we construct an augmented semi-simplicial space up to higher coherent homotopy over it, called its canonical resolution, whose graded…

Dynamical and cohomological obstructions to extending group actions

- Mathematics
- 2020

Motivated by a question of Ghys, we study obstructions to extending group actions on the boundary ∂M of a 3-manifold to a C-action on M . Among other results, we show that for a 3-manifold M , the…

A new approach to twisted homological stability, with applications to congruence subgroups

- Mathematics
- 2021

We introduce a new method for proving twisted homological stability, and use it to prove such results for symmetric groups and general linear groups. In addition to sometimes slightly improving the…

On flat manifold bundles and the connectivity of Haefliger's classifying spaces

- Mathematics
- 2022

We investigate low homological consequences of a conjecture due to Haefliger and Thurston in the context of foliated manifold bundles. In particular, Haefliger-Thurston’s conjecture predicts that…

On the h-Cobordism Category I

- MathematicsInternational Mathematics Research Notices
- 2019

We consider the topological category of $h$-cobordisms between smooth manifolds with boundary and compare its homotopy type with the standard $h$-cobordism space of a compact smooth manifold.

On the edge of the stable range

- MathematicsMathematische Annalen
- 2020

We prove a general homological stability theorem for certain families of groups equipped with product maps, followed by two theorems of a new kind that give information about the last two homology…

Homological stability for Artin monoids

- MathematicsProceedings of the London Mathematical Society
- 2020

We prove that certain sequences of Artin monoids containing the braid monoid as a submonoid satisfy homological stability. When the K(π,1) conjecture holds for the associated family of Artin groups,…

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