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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

MSC:

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

Citations:

Zbl 0476.57008
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References:

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[2] S. Akbulut andH. King, A topological characterization of two dimensional real algebraic sets, to appear.
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