The $Delta$-twisted homology and fiber bundle structure of twisted simplicial sets

Time:    15:30-17:00    2022/11/21

Venue:   BIMSA 1129B

Zoom:    537 192 5549 （PW: BIMSA）

Abtract：

In this talk, I will give a brief introduction for simplicial set theory. Then I will report our recent progress for unifying $delta$-homology, introduced by Alexander Grigor'yan, Yuri Muranov and Shing-Tung Yau, and twisted Cartesian product, introduced by Barratt, Gugenheim and Moore. 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.