Classifying spaces of low-dimensional bordism categories

Thesis

The truncated bordism category Cob_d is the symmetric monoidal category whose objects are closed oriented (d-1)-manifolds and whose morphisms are diffeomorphism classes of compact oriented d-dimensional bordisms. In this thesis we study the classifying spaces of Cob₁, Cob₂, and several related categories.

In the first part we show that the conditions in Steimle’s "additivity theorem for cobordism categories" can be weakened to only require locally (co)Cartesian fibrations, making it applicable to a larger class of functors. As an application, we construct a fiber sequence that relates the classifying space of the (2,1)-category Csp of cospans of finite sets to the classifying space of its homotopy category Csp = h(Csp), where isomorphic cospans are identified.

In the second part we compute the classifying space of Cob₁ by exhibiting it as a circle bundle over Ω^∞−2 CP_-1^∞. As part of our proof we construct a quotient Cob₁ → Cob₁^red where circles are deleted and we show that its classifying space is Ω^∞−2 CP_-1^∞. Based on this we construct a bordism model for the topological cyclic homology of simply connected spaces. We also explicitly describe an infinite loop space map B(Cob₁^red) → Q(Σ² CP^∞₊) and use it to derive combinatorial formulas for rational cocycles on Cob₁^red representing shifted Miller-Morita-Mumford classes κi ϵ H²ⁱ⁺²(B(Cob₁); Q)

The third part concerns the classifying space of the surface category Cob₂ and its subcategory Cob₂^χ≤0 subset of Cob₂ that contains all bordisms without disks or spheres. We show that, after passing to a refinement Cob₂ → Cob₂ , the classifying space of the surface category is BCob₂ ≃ S¹ - proving that a conjecture of Tillmann becomes true in this setting. We also compute the classifying space of Cob₂^χ≤0 and show that its rational homotopy groups contain the homology of the tropical moduli spaces ∆_g, which is known to grow exponentially. The tools we develop can be applied to a large class of symmetric monoidal categories, which we call labelled cospan categories. In particular, we show that B(Csp) is contractible, whereas the classifying space of its homotopy category is B(Csp) ≃ τ≥3Q(S²).

Authors: Steinebrunner, J

• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: Bordism categories; Topological cyclic homology; Miller-Morita-Mumford classes; Tropical moduli spaces; Category theory; Quillen's theorem B
• Subjects: Algebraic topology; Cobordism theory; Categories (Mathematics); Classifying spaces