Real algebraic structures on topological spaces. (English) Zbl 0531.57019

A real algebraic set is a set of the form \(p^{-1}(0)\) for some real polynomial map p: \({\mathbb{R}}^ n\to {\mathbb{R}}^ k\). The purpose of this paper is a part of an attempt to characterize real algebraic sets topologically. Main Theorem: The interior of any compact A-space, which is defined below, is homeomorphic to a real algebraic set. In fact, this homeomorphism is a stratified set isomorphism between the singular stratification of the real algebraic set and the A-space. Since PL manifolds are A-spaces [see the first author and L. Taylor, ibid. 53, 163-195 (1981; Zbl 0476.57008)] the following theorem holds: The interior of any compact PL manifold is PL homeomorphic to a real algebraic set. The definition of an A-space is as follows: An \(A_ 0\)- space is a smooth manifold. Inductively, an \(A_ k\)-space Y (\(k\geq 1)\) and its boundary \(\partial Y\) are given as follows: \(Y=Y_ 0\cup_{\phi}\amalg^{t}_{i=1}(N_ i\times c\Sigma_ i)\) where \(Y_ 0\) is an \(A_{k-1}\)-space with boundary, \(N_ i's\) are smooth manifolds and each \(\Sigma_ i\) is an \(A_{k-1}\)-space which is the boundary of a compact \(A_{k-1}\)-space, \(c\Sigma_ i=\Sigma_ i\times [0,1]/\Sigma_ i\times 0\) is a cone over \(\Sigma_ i\), \(\phi =\{\phi_ i\}\), and each \(\phi_ i: N_ i\times \Sigma_ i\to \partial Y_ 0\) is an \(A_{k-1}\)-embedding (i.e. a piecewise differentiable embedding preserving and respecting all the strata and the links of the strata). Also, the boundary of Y is defined by \[ \partial Y=(\partial Y_ 0-\cup \phi_ i(N_ i\times \Sigma_ i))\cup_{\phi}\amalg^{t}_{i=1}\partial N_ i\times c\Sigma_ i. \] An A-space is an \(A_ k\)-space for some k. Roughly speaking, A-spaces are topological spaces built up from smooth manifolds by the operations of coning over boundaries, crossing with smooth manifolds and taking unions along the boundary. It is known that not every real algebraic set is homeomorphic to an A-space. But the authors are hopeful in obtaining a topological characterization of real algebraic sets.
Reviewer: K.Shiraiwa


57Q99 PL-topology
32C05 Real-analytic manifolds, real-analytic spaces
57N80 Stratifications in topological manifolds


Zbl 0476.57008
Full Text: DOI Numdam EuDML


[1] S. Akbulut andH. King, The topology of real algebraic sets with isolated singularities, to appear inAnnals of Math. · Zbl 0494.57004
[2] S. Akbulut andH. King, A topological characterization of two dimensional real algebraic sets, to appear.
[3] S. Akbulut andL. Taylor, A topological resolution theorem,Publ. Math. I.H.E.S.,53 (1981), 163–196. · Zbl 0476.57008
[4] P. Conner andE. Floyd,Differentiable Periodic Maps, Ergebnisse der Mathematik, vol. 33, Springer, Berlin (1964). · Zbl 0125.40103
[5] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero,Annals of Math.,79 (1964), 109–326. · Zbl 0122.38603
[6] E. Spanier,Algebraic topology, McGraw-Hill, 1966. · Zbl 0145.43303
[7] D. Sullivan, Combinatorial Invariants of Analytic Spaces, Proceedings of Liverpool Singularities Symposium 1,Lecture notes in Mathematics, Vol. 192, Springer (1971), 165–168. · Zbl 0227.32005
[8] S. Akbulut andH. King, A relative Nash Theorem, to appear inTrans. Amer. Math. Soc.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.