On the $K$-theory and homotopy theory of the Klein bottle group

Abstract

We construct infinitely many chain homotopically distinct algebraic 2-complexes for the Klein bottle group and give various topological applications. We compare our examples to other examples in the literature and address the question of geometric realizability.

Division in group rings of surface groups

“…Claim 1: Any boundary point of ax = −by appears in a unique x-translate (that is, γx with a γ = 0) and also in a unique y-translate (ρy with b ρ = 0.) Therefore, the extremal points of x can all be canceled by y-translates (weighted with appropriate coefficients) 9 to obtain an element…”

“…whose support does not contain any of the extremal points from the support of x. 9 The coefficients are cγ = −bγ/a1 if γy contains an extremal point of x, and cγ = 0 otherwise.…”

“…6 Torsion free 1-relator groups have the stronger property of having aspherical presentation complexes ( [4]). 7 The Klein bottle group K = a, b | a 2 b 2 = 1 also has such stably free but not free ZK-modules generated by a pair of elements, but they have not yet been geometrically realized as π2-modules of 2-complexes ( [9]). 8 Another way to show π2X is free, which also works for the groups in Corollary 5, is given in section 6.…”

Two remarks on Wall's D2 problem

“…If G is a polycyclic-by-finite group, the group ring ZG is again noetherian and d G = h G + 1, where h G denotes the Hirsch length of G (see [27, 6•6•1]). The examples of [9,15,16,17] show that for general infinite fundamental groups (for example, the fundamental group of the trefoil knot), there can be (infinitely) many distinct 2-complexes with the same Euler characteristic.…”

A fake Klein bottle with bubble

“…Motivated by the work of Harlander and his student Misseldine [5, 6, section 1.8], an open question in low‐dimensional topology for the last 14 years has been as follows: Is there a finite 2‐complex, $$X$$ with the same fundamental group and Euler characteristic as $$K^\circ$$, but with $$H_2(\widetilde{X})$$ not freely generated as a $$\mathbb {Z}[\pi _1(X)]$$ module? We resolve this: Theorem There exists a finite 2‐complex $$X$$, with $$\pi _1(X)\cong \pi _1(K^\circ)$$ and $$\chi (X)=\chi (K^\circ)$$, but with $$H_2(\widetilde{X})$$ not freely generated as a $$\mathbb {Z}[\pi _1(X)]$$ module.…”