Poincaré Homology Spheres and Exotic Spheres from Links of Hypersurface Singularities

Singularities arise naturally in algebraic geometry when trying to classify algebraic varieties. However, they are also a rich source of examples in other fields, particularly in topology. We will explain how complex hypersurface singularities give rise to knots and links, some of which are manifolds that are homologically spheres but not topologically (Poincaré homology spheres), or manifolds that are homeomorphic to spheres but not diffeomorphic to them (exotic spheres). The talk should be accessible to anyone with some basic knowledge of differential and algebraic topology, although even that is not strictly necessary. We list some references: (1) For a general overview on exotic spheres and the technical background necessary for some proofs, see [Ran12], who also lists many useful references. (2) For plane curve singularities, see [BK12]. (3) The speaker first learned this material from attending a course on the topology of algebraic singularities [Ném14]. (4) For an introduction on singularities in higher dimensions, see [Kau87, Ch. XIX]. (5) For a technical reference on singularities in higher dimensions, see [Mil68]. 1. Motivation Our goal for today is to see why singularities are interesting. In geometry, we are mostly interested in smooth objects, such as smooth manifolds or smooth varieties. However, there are two ways in which singularities naturally arise in algebraic geometry: (1) In moduli theory, to obtain a compact moduli space, you want to include “limits” of spaces, which inevitably become singular. (2) In the minimal model program, performing various surgery operations leads to spaces with terminal singularities. One might ask why one should care about singularities for reasons external to algebraic geometry, and indeed, we have the following: (3) Complex algebraic singularities give rise to interesting smooth manifolds. For example, (a) Some curve singularities in C2 give rise to algebraic knots K1, (b) A surface singularity in C3 gives rise to the Poincaré homology sphere K3, and (c) A fourfold singularity in C5 gives rise to an exotic sphere K7, which satisfy the following properties: K1 ∼= diffeo S1 but (K1 ⊂ S3) 6∼= isotopy (S1 ⊂ S3), H∗(K 3,Z) ∼= groups H∗(S 3,Z) but K3 6∼= homeo S3, K7 ∼= homeo S7 but K7 6∼= diffeo S7. We will spend today constructing these examples. We will also explain some of the ideas and constructions used in showing the claims in both columns, although we cannot give full proofs. Date: September 22, 2017 at the University of Michigan Student Algebraic Geometry Seminar.