The homotopy type of the cobordism category

TL;DR

This work identifies the homotopy type of the classifying space of the embedded d-dimensional cobordism category by proving a weak equivalence $B\mathscr{C}_d \simeq \Omega^{\infty-1}\mathit{MT}(d)$, unifying cobordism theory with Thom spectra. The authors develop a robust sheaf-theoretic framework, construct cocycle models, and use Phillips' submersion theorem to relate cobordism data to infinite loop spaces, with extensions to tangential structures via $\mathit{MT}(\theta)$. For $d=2$, the approach yields a new proof of the generalized Mumford conjecture and delivers Harer-type stability results in the cobordism setting. The results provide a coherent, flexible method to study oriented and nonoriented cobordisms, tangential structures, and their associated generalized homology theories through Thom spectra and cobordism categories, with potential applications in topological quantum field theories and moduli problems.

Abstract

The embedded cobordism category under study in this paper generalizes the category of conformal surfaces, introduced by G. Segal in order to formalize the concept of field theories. Our main result identifies the homotopy type of the classifying space of the embedded d-dimensional cobordism category for all d. For d=2, our results lead to a new proof of the generalized Mumford conjecture, somewhat different in spirit from the original one.

Figures (4)

• Figure 1:
• Figure 2:
• Figure 3: Image of $(\tilde{\pi},\tilde{j}): \tilde{L} \to [0,4]\times[0,\infty)$.
• Figure 4: $(f_t,j_t)(K_t)$ for $d=1$ and various values of $\tau = \sigma(4\lambda(t))\in[0,4]$.

Theorems & Definitions (46)

• Remark 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.6
• Definition 2.7
• Definition 2.8
• Proposition 2.9
• proof
• Remark 2.10