SOME REMARKS ON PL COLLAPSIBLE COVERS OF 2-DIMENSIONAL POLYHEDRA

概要

cardinality of a cover of X by open sets which are contractible in the space. It is a classical
homotopy invariant of a space, introduced in [12], which has became over the years an
important tool in homotopy theory (see [5] for a good account on the subject). A natural
upper bound for the L-S category of a space X is provided by its geometric category gcat(X),
defined as the minimum number of open contractible sets that cover X. For a polyhedron P
(i.e. the underlying topological space of some simplicial complex), the geometric category
coincides with the minimum number of contractible subpolyhedra that cover P.
In this note we propose to study a variant of the geometric category in the context of compact connected polyhedra, which we call PL geometric category and denote it by plgcat. For
this invariant, we replace the purely topological notion of “contractible” in the definition
of geometric category by the more geometrically flavored notion of “PL collapsible” (refer
to Section 2 for precise definitions). This point of view allows to exploit certain combinatorial properties of a space that admits triangulations while at the same time accounts for
its inherent geometry and topology. We show in first place that the PL geometric category
of a polyhedron of dimension n is bounded by n + 1, thus generalizing the corresponding
result for geometric category (cf. [5, Proposition 3.2]). This implies that the PL geometric
category of a non PL collapsible polyhedron of dimension 2 may only be 2 or 3. One of our
main objectives is to understand the topological and geometrical properties that distinguish
2-dimensional polyhedra P with plgcat(P) = 2 from those with plgcat(P) = 3. In this direction, we find that the condition of having PL geometric category 2 is fairly restrictive. In
particular, it determines the simple homotopy type of the polyhedron: by Proposition 2.10
below, such a polyhedron is simple homotopy equivalent to a wedge sum of spheres of dimension 1 and 2. Moreover, it is not difficult to verify that a contractible polyhedron P of
dimension 2 with plgcat(P) = 2 satisfies the Andrews-Curtis conjecture [1], which states
2010 Mathematics Subject Classification. 55M30, 57Q05, 57M20, 52B99, 20F05. ...

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