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Math and Biology Seminar
The $Delta$-twisted homology and fiber bundle structure of twisted simplicial sets
The $Delta$-twisted homology and fiber bundle structure of twisted simplicial sets
Organizers
Speaker
Time
Sunday, December 4, 2022 10:30 AM - 11:30 AM
Venue
Online
Online
Zoom 293 812 9202
(BIMSA)
Abstract
Different from classical homology theory, Alexander Grigor'yan, Yuri Muranov and Shing-Tung Yau recently introduced $delta$-(co)homology, taking the (co)boundary homomorphisms as $\delta$-weighted alternating sum of (co)faces. For understanding the ideas of $delta$-homology, Li, Vershinin and Wu introduced $delta$-twisted homology and homotopy in 2017. On the other hand, the twisted Cartesian product of simplicial sets was introduced by Barratt, Gugenheim and Moore in 1959, playing a key role for establishing the simplicial theory of fibre bundles and fibrations. The corresponding chain version is twisted tensor product introduced by Brown in 1959. In this talk, I will report our recent progress for unifying $delta$-homology and twisted Cartesian product. We introduce $\Delta$-twisted Carlsson construction of $\Delta$-groups and simplicial groups, whose abelianization gives a twisted chain complex generalizeing the $delta$-homology, called $\Delta$-twisted homology. We show that Mayer-Vietoris sequence theorem holds for $\Delta$-twisted homology. Moreover, we introduce the concept of $\Delta$-twisted Cartesian product as a generalization of the twisted Cartesian product, and explore the fiber bundle structure. The notion of $\Delta$-twisted smash product, which is a canonical quotient of $\Delta$-twisted Cartesian product, is used for determining the homotopy type of $\Delta$-twisted Carlsson construction of simplicial groups.