Infinitely many pairwise homotopy inequivalent 2-complexes \(K_ i\) with fixed \(\pi_ 1 K_ i\) and \(\chi (K_ i)\)
The examples of Dunwoody and the reviewer, that the homotopy type of a finite 2-complex is not determined by the fundamental group and the Euler characteristic, resulted in finitely many distinct classes for a given \(\pi_ 1\) and \(\chi\). In the present paper, an infinite such family is constructed: The idea takes up Dunwoody's result that for any \(i\geq 0\), \(s^{2i+1}\) and \(t^{3i+1}\) form a generating set for the trefoil group \(G= \langle s,t\mid s^ 2= t^ 3\rangle\), and that the corresponding relation modules are pairwise non-isomorphic for distinct primes \(i\). The essential step then is a nice commutator trick that realizes these modules as \(H_ 2\)-modules of covering spaces of complexes which are extensions of the standard complex of \(\langle s,t\mid s^ 2= t^ 3\rangle\).
scientific article; zbMATH DE number 427666