A K-theory spectrum for cobordism cut and paste groups

2022/10/03 by Hoekzema, Renee S., Rovi, Carmen, Semikina, Julia
#18D99 #18F25 #18F30 #19D99 #55P15 #57N40 #57N70 #57R75 #57R90 #Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics #Geometric Topology (math.GT) #K-Theory and Homology (math.KT)

paper · doi:10.48550/arxiv.2210.00682

Abstract

Cobordism groups and cut-and-paste groups of manifolds arise from imposing two different relations on the monoid of manifolds under disjoint union. By imposing both relations simultaneously, a cobordism cut and paste group SK_n is defined. In this paper, we extend this definition to manifolds with boundary obtaining SK^∂_n and study the relationship of this group to an appropriately defined cobordism group of manifolds with boundary. The main results are the construction of a spectrum that recovers on π₀ the cobordism cut and paste groups of manifolds with boundary, SK^∂_n, and a map of spectra that lifts the canonical quotient map SK^∂_n → SK^∂_n.

Related

• Parametrized scissors congruence K-theory of manifolds and cobordism categories
• Cobordism groups of simple branched coverings
• Cut and paste invariants of manifolds via algebraic K-theory
• SKK groups of manifolds and non-unitary invertible TQFTs
• Contact monoids and Stein cobordisms