Sibe Mardešić
University of Zagreb, Croatia
Functoriality of the standard resolution of the Cartesian
product
of a compactum and a polyhedron
To study the shape of the Cartesian product \(X\times P\) of a compact
Hausdorff space \(X\) and a polyhedron \(P\), the author has introduced in [2] a resolution \({\bsq}\colon X\times P\to\bsY\), here called the standard resolution of \(X\times P\). It consists of paracompact spaces having the homotopy type of polyhedra and is completely determined by the limit \({\bsp}\colon X\to{\bsX}\) of a cofinite inverse system \(\bsX\) of compact polyhedra and by a triangulation \(K\) of \(P\). Now the construction is considerably enriched by showing that the standard resolution is a functor. More precisely, with every homotopy class \([{\bsf}]\) of coherent mappings \({\bsf}\colon {\bsX}\to{\bsX}'\), one can associate a homotopy class \([{\bsg}]\) of homotopy mappings \({\bsg}\colon{\bsY}\to{\bsY}'\) between the corresponding standard resolutions such that \([{\bsf}]=1\) implies \([{\bsg}]=1\) and \([{\bsf}'']=[{\bsf}'][{\bsf}]\) implies \([{\bsg}'']=[{\bsg}'][{\bsg}]\). The proof uses in an essential way particular cellular decompositions of simplicial complexes and their properties.
Among the consequences of the functoriality of the standard resolution is the existence of a functor \(R\) from the strong shape category of compact Hausdorff spaces to the shape category Sh of topological spaces such that \(R(X)=X\times P\). This resut is nontrivial, because \(X\times P\) need not be the product of \(X\) and \(P\) in the shape category Sh, as demonstrated in [1]. The functor \(R\) plays an essential role in proving the theorem that, for compact Hausdorff spaces \(X,X'\) such that \(X\) is strong shape dominated by \(X'\), \(X\times P\) is a product in Sh whenever \(X'\times P\) is a product in Sh. An easy consequence of the latter result and a result from [3] is Kodama's theorem from 1973 that, for \(X\) an FANR, \(X\times Y\) is a product in Sh, for every topological space \(Y\).
REFERENCES
[1] J. Dydak and S. Mardešić. A counterexample concerning products in the shape category, Fund. Math., 186 (2005), 39–54.
[2] S. Mardešić. A resolution for the product of a compactum with a polyhedron, Topology and its Appl., 133 (2003), 37–63.
[3] S. Mardešić. Products of compacta with polyhedra and topological spaces in the shape category, Mediterr. J. Math., 1 (2004), 43–49.
