On the infinite loop space structure of the cobordism category
type: Item
instance of: scholarly article
title: On the infinite loop space structure of the cobordism category
zbMATH Open document ID: 1380.55010
author: Hoang Kim Nguyen
published in: Algebraic \& Geometric Topology
publication date: 2017-03-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/1505.03490
review text: Given a tangential structure \(\theta : X \longrightarrow BO(d)\), one can construct the cobordism category \(\mathrm{Cob}_\theta(d)\), with objects roughly given by a manifold together with a \(\theta\)-structure. This is a symmetric monoidal category using disjoint union. In [Acta Math. 202, No. 2, 195--239, (2009; Zbl 1221.57039)], \textit{S. Galatius} et al. proved that there is an equivalence \[ B\mathrm{Cob}_\theta(d) \simeq MT\theta(d)[1] \] between the classifying space of this category and the zeroth space of the shifted Madsen-Tillman spectrum. Since both of these spaces have infinite loop space structures (the former from being the classifying space of a symmetric monoidal category and the latter from being the zeroth space of an \(\Omega\)-spectrum), it is natural to ask whether this is an equivalence of infinite loop spaces. This is the question that the paper answers in the positive. To prove that there is an equivalence of infinite loop spaces, the author uses the model of \(\Gamma\)-spaces developed by \textit{G. Segal} [Topology 13, 293--312 (1974; Zbl 0284.55016)], and he uses an intermediary spectrum, \(\psi_\theta\), developed by \textit{S. Galatius} and \textit{O. Randal-Williams} [Geom. Topol. 14, No. 3, 1243--1302 (2010; Zbl 1205.55007)]. The three main \(\Gamma\)-spaces used in the comparison are all constructed similarly, using the following idea. Let \(m\in \mathbb{N}\). Then we can consider the tangential structure \[ \theta(m_+) : \coprod_{m} X \longrightarrow BO(d) \] given by the coproduct of \(m\) copies of \(\theta\). Using this map one can construct the category \(\mathrm{Cob}_{\theta(m_+)}(d)\), and the spectra \(MT\theta(m_+)(d)\) and \(\psi_{\theta(m_+)}\), and show that these assemble into a \(\Gamma\)-category \(\Gamma \mathrm{Cob}_{\theta}(d)\), and two \(\Gamma\)-spectra \(\Gamma MT\theta(d)\) and \(\Gamma \psi_{\theta}\), respectively, using disjoint union of manifolds appropriately in each case. The author then proves that t…
Mathematics Subject Classification ID: 55P47
Mathematics Subject Classification ID: 57R56
Mathematics Subject Classification ID: 57R90
zbMATH DE Number: 6698207
zbMATH Keywords: cobordism category
zbMATH Keywords: Madsen-Tillmann spectrum
zbMATH Keywords: infinite loop spaces
zbMATH Keywords: Gamma-spaces
reviewed by: Angélica M. Osorno
MaRDI profile type: Publication
OpenAlex ID: W1832629750
arXiv ID: 1505.03490
DOI: 10.2140/AGT.2017.17.1021
description: scientific article