Generalized bias-invariants and their calculation (Q2702054)

The paper concerns the distinction of homotopy types of 2-complexes with the same fundamental group \(\pi_1\) and (minimal) Euler characteristic. The problem whether an isomorphism of the second homology \(H_2(\pi_1)\) can be lifted to a geometrically induced isomorphism of the second homology of the complexes leads to the so-called bias invariant. To compute it one has to determine the second homotopy groups and their images under the Hurewicz map, a problem which is undecidable in general. In the present paper it is shown that, by considering coverings of a complex associated to characteristic subgroups of the fundamental group (the commutator subgroup in the applications given in the paper), simplified but computable invariants can be obtained (which give only necessary conditions for homotopy equivalence, in general). Using these techniques, examples of 2-complexes as above are discussed which can be distinguished homotopically by these modified invariants.NEWLINENEWLINEFor the entire collection see [Zbl 0940.00028].