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Strong homotopy types, nerves and collapses. (English) Zbl 1242.57019

The simple homotopy type of a simplicial complex is defined in terms of elementary collapses. In the paper under review the authors suggest a modification called strong homotopy type. In this case an elementary strong collapse from \(K\) to \(K\setminus v\) is possible if \(v\) is a vertex of \(K\) such that \(lk_K(v)\) is a simplicial cone over another vertex \(v'\). Here \(K\setminus v\) denotes the full subcomplex spanned by all vertices of \(K\) distinct from \(v\), and \(v\) is said to be dominated by \(v'\). A strong collapse is a sequence of elementary strong collapses, and the strong homotopy type is defined accordingly. The strong collapsing procedure terminates at a so-called minimal complex, that is one without a dominated vertex. Such a minimal subcomplex of \(K\) is called a core, and a core is shown to be unique up to isomorphism.
Furthermore, two complexes have the same strong homotopy type if and only if their cores are isomorphic. A theory of strong collapses is developed in view of barycentric subdivisions, nerves and evasiveness. In particular a complex is strong collapsible to a vertex if and only if its first barycentric subdivision is. At the end an evasiveness conjecture is stated as follows: If \(K\) is a minimal, vertex-homogeneous, non-evasive complex, then it is a point.

MSC:

57Q10 Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc.
57Q91 Equivariant PL-topology
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