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Poincaré Homology Spheres and Exotic Spheres

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Poincaré Homology Spheres and Exotic Spheres POINCARE´ HOMOLOGY SPHERES AND EXOTIC SPHERES FROM LINKS OF HYPERSURFACE SINGULARITIES TAKUMI MURAYAMA Abstract. 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 (Poincar´ehomology 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 [N´em14]. (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 Poincar´ehomology sphere K3, and (c) A fourfold singularity in C5 gives rise to an exotic sphere K7, which satisfy the following properties: K1 =∼ S1 but (K1 ⊂ S3) =6∼ (S1 ⊂ S3); diffeo isotopy 3 ∼ 3 3 ∼ 3 H∗(K ; Z) = H∗(S ; Z) but K =6 S ; groups homeo K7 =∼ S7 but K7 =6∼ S7: homeo diffeo 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. 1 2 TAKUMI MURAYAMA 2. Hypersurface singularities and their associated links All of our examples today will be hypersurface singularities, i.e., singularities defined by one equation in complex affine space. Before we begin, we recall what a singularity is in the first place. Definition 2.1. Let f 2 C[z0; z2; : : : ; zn], which defines a function f : Cn+1 −! C: Then, the vanishing locus of f is V (f) := f −1(0) ⊂ Cn+1: The vanishing locus V (f) is singular at P 2 V (f) if the (complex) gradient @f @f @f rf := ; ; ··· ; @z0 @z1 @zn vanishes at P , and is smooth at P if rf(P ) 6= 0. Remark 2.2. This condition on the gradient is known as the Jacobian criteron. In higher codimension, the condition is that the matrix (@fi=@zj) has full rank. In differential topology, the fact that V (f) is a smooth manifold is sometimes known as the preimage theorem [GP10, p. 21]. We will be interested in the following special case: Setup 2.3. Suppose V (f) has an isolated singularity at ~0, i.e., suppose ~0 is a singular point of V (f) and V (f) is smooth in a punctured neighborhood of ~0. Given such an isolated singularity, we can associate a smooth manifold to it that can be used to study the singularity. This was first done for curve singularities by Brauner in 1928 [BK12, p. 223]. Definition 2.4. Given an isolated singularity V (f), the link of f is the intersection 2n+1 K(f) := V (f) \ S" ; 2n+1 ~ n+1 where S" is the (2n + 1)-dimensional sphere of radius " > 0 centered at 0 2 C . Figure 1. The link of an isolated singularity (from [Kau96, Fig. 14.1]). 8.3]. x , BK12 [ see knots; torus iterated to rise give equations complicated n a hwta more that show can one pairs, Puiseux of theory the and diagrams Newton . 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Example 5 projections projections 3 two 1,2 1,2 = i , 1 s • 1 s x example. 1 s : w. 1-dimensional a studying start now We l. l. 1 1 s x 1 s c K to restriction the Then . 1 s x 1 s of factors the of nt rmpaecresingularities curve plane from Knots 3. l. osntdpn on depend not does ) f ( K ,adi (2 a is and 1, " < 0 for " )dmninlsot manifold. smooth 1)-dimensional − for for m degree of K le rc ci he t n by 1 s rcle i c e th of ngs i r cove lds e yi at2.5 Fact sa sltdsnuaiy hntedffoopimtp of type diffeomorphism the then singularity, isolated an is ) f ( V If . 2.9]) , Cor. Mil68 are are numbers winding the way this in and 2, = i for n resp. 1 = i [ (see define define can one sl, X sl KC circle any for general, In . characterised with with and definition: our in projections, the of theorems restriction by Ehresmann's sl and + K : Sard of corollary mappings two following the used implicitly We . l. ppings. ppings. a m these of degrees e h t are h c whi n, and m numbers, o w t them OOOYSHRSADEOI SPHERES EXOTIC AND 3 SPHERES HOMOLOGY E POINCAR of of morphism eo hom a is re e h t t o K kn a h c su for that ´ prove can One standard standard the into K carries which , y identit the to motopic o h , 1 s x 1 s 4 TAKUMI MURAYAMA 4. The Alexander polynomial To finish Example 3.1, we want to show that the (3; 5)-torus knot is not isotopic to the unknot S1 ⊂ R3. The invariant we will use to distinguish these is the Alexander polynomial, which has a purely knot-theoretic definition in terms of skein relations when n = 1 [Lic97, Ch. 6]. We will not define the Alexander polynomial, but we list the properties that we will need: Theorem 4.1 (see [Lic97, Ch. 6]; [Mil68, xx8,10]). There is a Laurent polynomial ∆f (t) associated to the link K(f) of an isolated singularity, such that (i) If n = 1 and if K(f) is the unknot, then ∆f (t) = 1; (ii)∆ f (t) is invariant under isotopy; (iii)∆ f (1) = ±1 if and only K(f) is a homology sphere. Definition 4.2. The Laurent polynomial in Theorem 4.1 is (the higher-dimensional version of) the Alexander polynomial. Remark 4.3. We give a brief description of how to construct the Alexander polynomial. First, the mapping f : Cn+1 ! C gives rise to a map 2n+1 1 φ: S" r K(f) −! S f(z) z 7−! jf(z)j This is a smooth fiber bundle [Mil68, Thm. 4.8] with fiber F , called the Milnor fiber; see Figure4 for 1 a visualization.
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