On the infinite loop space structure of the cobordism category
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 the \(\Gamma\)-space \(\Gamma^{(1)} MT\theta(d)\) obtained from evaluating the \(\Gamma\)-spectrum at the first level of the spectrum is a model for the connective cover if \(MT\theta(d)[1]\), and similarly, that \(\Gamma^{(0)} \psi_{\theta}\) is a model for the connective cover of \(\psi_\theta\). This is done by proving some general results about equivalences of spectra arising from \(\Gamma\)-spectra that are of independent interest (see for example Proposition 5.2). The proof of the main theorem is completed by constructing maps of \(\Gamma\)-spaces which are levelwise equivalences, \[ B\Gamma \mathrm{Cob}_{\theta}(d) \overset{\simeq}{\rightarrow} \Gamma^{(0)} \psi_{\theta} \overset{\simeq}{\leftarrow} \Gamma^{(1)} MT\theta(d), \] thus giving the promised equivalence of the infinite loop spaces. One thing that is not resolved is whether the infinite loop space structure on \(B\mathrm{Cob}_\theta(d)\) obtained from the \(\Gamma\)-space defined above coincides with the one given by the symmetric monoidal structure on the category. We believe that to be the case, but as we just said, this is not explored in the paper.
scientific article