The basic point is that sphere complements determine the homology, but not the homotopy type, in general. What is determined, however, is the stable homotopy type, which was conceived as a first approximation to homotopy type. Thus Spanier–Whitehead duality fits into stable homotopy theory.
Statementedit
Let X be a compact neighborhood retract in . Then and are dual objects in the category of pointed spectra with the smash product as a monoidal structure. Here is the union of and a point, and are reduced and unreduced suspensions respectively.
spanier, whitehead, duality, mathematics, duality, theory, homotopy, theory, based, geometrical, idea, that, topological, space, considered, dual, complement, sphere, where, large, enough, origins, alexander, duality, theory, homology, theory, concerning, comp. In mathematics Spanier Whitehead duality is a duality theory in homotopy theory based on a geometrical idea that a topological space X may be considered as dual to its complement in the n sphere where n is large enough Its origins lie in Alexander duality theory in homology theory concerning complements in manifolds The theory is also referred to as S duality but this can now cause possible confusion with the S duality of string theory It is named for Edwin Spanier and J H C Whitehead who developed it in papers from 1955 The basic point is that sphere complements determine the homology but not the homotopy type in general What is determined however is the stable homotopy type which was conceived as a first approximation to homotopy type Thus Spanier Whitehead duality fits into stable homotopy theory Statement editLet X be a compact neighborhood retract in R n displaystyle mathbb R n nbsp Then X displaystyle X nbsp and S n S R n X displaystyle Sigma n Sigma mathbb R n setminus X nbsp are dual objects in the category of pointed spectra with the smash product as a monoidal structure Here X displaystyle X nbsp is the union of X displaystyle X nbsp and a point S displaystyle Sigma nbsp and S displaystyle Sigma nbsp are reduced and unreduced suspensions respectively Taking homology and cohomology with respect to an Eilenberg MacLane spectrum recovers Alexander duality formally References editSpanier Edwin H Whitehead J H C 1953 A first approximation to homotopy theory Proceedings of the National Academy of Sciences of the United States of America 39 7 655 660 Bibcode 1953PNAS 39 655S doi 10 1073 pnas 39 7 655 MR 0056290 PMC 1063840 PMID 16589320 Spanier Edwin H Whitehead J H C 1955 Duality in homotopy theory Mathematika 2 56 80 doi 10 1112 s002557930000070x MR 0074823 tom Dieck Tammo 2008 Algebraic topology European Mathematical Society Publishing House ISBN 978 3 03719 048 7 Retrieved from https en wikipedia org w index php title Spanier Whitehead duality amp oldid 1045392984, wikipedia, wiki, book, books, library,