拟同构的挠
Torsion of Quasi-Isomorphisms
摘要：
本文对域上基链复形的拟同构引入了Reidemeister挠的概念。如果链映射的同调之间的诱导同态是同构，则称链映射为拟同构。我们的挠的概念推广了无环基链复形的挠，并且是从一个基链复形到另一个基链复形的所有拟同构的集合上的链同伦不变量。它与基于无圈的链复形的挠具有良好的性质，如乘法性和对偶性。在较弱的条件下，我们将挠进一步推广到环上自由链复形之间的拟同构。我们预期，拟同构的挠的研究将在许多方面取得丰硕的成果，特别是在三维流形中的链的研究方面。
英文摘要：
In this paper, we introduce the notion of Reidemeister torsion for quasi-isomorphisms of based chain complexes over a field. We call a chain map a quasi-isomorphism if its induced homomorphism between homology is an isomorphism. Our notion of torsion generalizes the torsion of acyclic based chain complexes, and is a chain homotopy invariant on the collection of all quasi-isomorphisms from a based chain complex to another. It shares nice properties with torsion of acyclic based chain complexes, like multiplicativity and duality. We will further generalize our torsion to quasi-isomorphisms between free chain complexes over a ring under some mild condition. We anticipate that the study of torsion of quasi-isomorphisms will be fruitful in many directions, and in particular, in the study of links in 3-manifolds.
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