Cut and paste invariants of manifolds and relations to cobordism
Moduli and Friends seminar
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Monday 25 March 2024

Carmen Rovi (Loyola University Chicago)Cut and paste invariants of manifolds and relations to cobordism

The classical problem of scissor's congruence asks whether two polytopes can be obtained from one another through a process of cutting and pasting. In the 1970s this question was posed instead for smooth manifolds: which manifolds M and N can be related to one another by cutting M into pieces and gluing them back together to obtain N? In recent work with Renee Hoekzema, Mona Merling, Laura Murray, and Julia Semikina, we upgraded the group of cut-and-paste invariants of manifolds with boundary to an algebraic K-theory spectrum and lifted the Euler characteristic to a map of spectra. I will discuss how cut-and-paste invariants relate to cobordism of manifolds and how the novel construction categorifies these invariants. I will also discuss new results on the categorification of cobordism cut-and-paste invariants: the group of invariants preserved by both cobordism and cut-and-paste equivalence.

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