Spin structures on almost-flat manifolds

Abstract

We give a necessary and suffcient condition for almost-flat manifolds with cyclic holonomy to admit a Spin structure. Using this condition we find all 4-dimensional orientable almost- flat manifolds with cyclic holonomy that do not admit a Spin structure

Classification of Spin Structures on Four‐dimensional Almost‐flat Manifolds

“…We recall a key result from [12] which states that the classifying map of the tangent bundle of an almost-flat manifold M factors through the classifying space of the holonomy group F . We denote by…”

“…We recall a key result from [12] which states that the classifying map of the tangent bundle of an almost-flat manifold M factors through the classifying space of the holonomy group F . We denote by [7], we have that the isolator Λ γ i (Λ) = Λ ∩ γ i (N), where γ i (N), 1 ≤ i ≤ c + 1, form the lower central series of N. By Lemmas 1.1.2-3 of [7], the resulting adapted lower central series…”

“…It is well-known that all closed orientable manifolds of dimension at most three have a spin structure (see [15, p. 35], [17,Exercise 12.B and VII, Theorem 2]). In [12], a complete list of non-spin 4-dimensional orientable almost-flat manifolds with cyclic 2-group holonomy was given. In the current paper, we introduce an algorithm to determine the existence of a spin structure on an almost-flat manifold using the classifying representation of its fundamental group.…”

“…We should also point out that we find one family (no. 80) of non-spin manifolds with holonomy group C 4 which was inadvertently omitted in [12].…”

“…[12, Lemma 2.6]). Let M be an orientable almost-flat manifold with the fundamental group Γ satisfying (3).…”