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The Twisted Homology of Simplicial Set

Acta Mathematica Sinica,English Series 2022年第10期38卷 1781-1802页

作者： Meng Meng ZHANG Jing Yan LI Jie WU School of Mathematical Sciences Hebei Normal UniversityShijiazhuang 050024P.R.China Yanqi Lake Beijing Institute of Mathematical Sciences and Applications Beijing 101408P.R.China

In this article,we give a generalization of δ-twisted homology introduced by Jingyan Li,Vladimir Vershinin and Jie Wu,called Δ-twisted homology,which enriches the theory of δ-(co)homology introduced by Alexander Grigor’yan,Yuri Muranov and Shing-Tung Yau.We show that the Mayer-Vietoris sequence theorem holds for Δ-twisted homology.Applying the Δ-twisted ideas to Cartesian products,we introduce the notion of Δ-twisted Cartesian product on simplicial sets,which generalizes the classical work of Barratt,Gugenheim and Moore on twisted Cartesian products of simplicial sets.Under certain hypothesis,we show that the coordinate projection of Δ-twisted Cartesian product admits a fibre bundle structure.

关键词： Simplicial set Δ-Twisted homology homotopy type Fibre bundle

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Orbit configuration spaces of small covers and quasi-toric manifolds

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Science China Mathematics 2021年第1期64卷 167-196页

作者： Junda Chen Zhi Lü Jie Wu School of Mathematical Sciences Fudan UniversityShanghai200433China Department of Mathematics National University of SingaporeSingapore119260Republic of Singapore

In this article,we investigate the orbit configuration spaces of some equivariant closed manifolds over simple convex polytopes in toric topology,such as small covers,quasi-toric manifolds and(real)moment-angle manifolds;especially for the cases of small covers and quasi-toric manifolds.These kinds of orbit configuration spaces have non-free group actions,and they are all noncompact,but still built via simple convex polytopes.We obtain an explicit formula of the Euler characteristic for orbit configuration spaces of small covers and quasi-toric manifolds in terms of the h-vector of a simple convex polytope.As a by-product of our method,we also obtain a formula of the Euler characteristic for the classical configuration space,which generalizes the Félix-Thomas formula.In addition,we also study the homotopy type of such orbit configuration spaces.In particular,we determine an equivariant strong deformation retraction of the orbit configuration space of 2 distinct orbit-points in a small cover or a quasi-toric manifold,which allows to further study the algebraic topology of such an orbit configuration space by using the Mayer-Vietoris spectral sequence.

关键词： orbit configuration space small cover quasi-toric manifold (real)moment-angle manifold Euler characteristic homotopy type

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