1962/07/01 by John Milnor · 264 citations
Mathematics · #Homotopy and Cohomology in Algebraic Topology #Geometric and Algebraic Topology #Algebraic Geometry and Number Theory #Mathematics #Torsion (gastropod) #Duality (order theory) #Pure mathematics #Anatomy
paper · doi:10.2307/1970268
published in Annals of Mathematics 76(1), 137 (Princeton University)
openalex publication_date 1962/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/02
[a, pf] = [a, f]P, where [a, f ] denotes the value of the function f at a. If A is free and finitely generated, then clearly A* is free and finitely generated, and A* * can be identified with A. Note that any homomorphism h: A, -* A2 gives rise to a dual homomorphism h*: A* -+ A*. As an example consider the following geometrical situation. Let M be a simplical complex whose underlying space is an oriented n-manifold without boundary. Let 11 be a group of fixed point free simplicial automorphisms of M. Then the chain group Cq(M; Z) can be considered as a free left module over the integral group ring Z[LI]. Now suppose that M has a dual cell subdivision M'. Then the chain group Cn-q(M'; Z) is also a free left Z[fl]-module. We will assume that the quotient space MIR is compact, so that these modules are finitely generated. There is a canonical anti-automorphism p p of Z[lI] which takes each group element w into s'l.