Cobordism Categories and Parametrized Morse Theory

Fix a tangential structure $\theta: B \longrightarrow BO(d+1)$ and an integer $k < d/2$. In this paper we determine the homotopy type of a cobordism category $\mathbf{Cob}^{\text{mf}, k}_{\theta}$, where morphisms are given by $\theta$-cobordisms $W: P \rightsquigarrow Q$ equipped with a choice of proper Morse function $h_{W}: W \longrightarrow [0, 1]$, with the property that all critical points $c \in W$ of $h_{W}$ satisfy the condition: $k < \text{index}(c) < d-k+1$. In particular, we prove that there is a weak homotopy equivalence $B\mathbf{Cob}^{\text{mf}, k}_{\theta} \simeq\Omega^{\infty}\mathbf{hW}^{k}_{\theta}$, where $\mathbf{hW}^{k}_{\theta}$ is a Thom spectrum associated to the space of Morse jets on $\mathbb{R}^{d+1}$. In the special case that $k = -1$, the equivalence $B\mathbf{Cob}^{\text{mf}, -1}_{\theta} \simeq\Omega^{\infty}\mathbf{hW}^{-1}_{\theta}$ follows from the work of Madsen and Weiss used in their celebrated proof of the Mumford conjecture. Following the methods of Madsen and Weiss we use the weak equivalence $B\mathbf{Cob}^{\text{mf}, k}_{\theta} \simeq\Omega^{\infty}\mathbf{hW}^{k}_{\theta}$ to give an alternative proof the "high-dimensional Madsen-Weiss theorem" of Galatius and Randal-Williams which identifies the homology of the moduli spaces, $BDiff((S^{n}\times S^{n})^{\# g}, D^{2n})$, in the limit $g \to \infty$.