ABSTRACT
If M is a manifold, S ( M ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq443.tif"/> , the structure set of M is defined, following Sullivan and Wall [11], as the set of pairs (N, f) consisting of a manifold N and a simple homotopy equivalence f : N → M that restricts to a homeomorphism on the boundary. Two pairs, (N1, f1) and (N2, f2), represent the same element if there is a homeomorphism H : N1 → N2 such that f ~ f2H rel ∂. One of the most beautiful results in the theory is Siebenmann’s periodicity theorem that S ( M ) ≅ S ( M × D 4 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq444.tif"/> for most M, e.g. all M with nonempty boundary (see [7] and the paper of Nicas [4] for a correction). Siebenmann’s proof was rather indirect; it proceeded by constructing a simplicial set whose π0 was S ( M ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq445.tif"/> and which was the fiber of a fibration which had periodicity properties. While this is enough for Siebenmann’s (and others') applications, (such as providing a group structure on S ( M ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq446.tif"/> via the obvious one on S ( M × D 4 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq447.tif"/> analogous to the definition of π4), its indirectness is mysterious (see [7]). (This map is not just crossing with D4; one does not then get a homeomorphism on the boundary.) In this paper we shall give a geometrically defined map S ( M ) → S ( M × D 4 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq448.tif"/> (or actually to S ( E ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq449.tif"/> for a class of total spaces of four-plane bundles) that has all the properties of Siebenmann periodicity, by means of embedding theory. In addition to whatever aesthetic advantages there may be in the gained geometricity, there is also at least one practical pay-off. As an application we shall show that many homotopy ℂPn's have locally smooth S1-actions.
