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MASSEY PRODUCT and ITS APPLICATIONS 1. Massey Products 1.1. Cup Products Review. Given a Topological Space X and a Commutative R

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MASSEY PRODUCT and ITS APPLICATIONS 1. Massey Products 1.1. Cup Products Review. Given a Topological Space X and a Commutative R MASSEY PRODUCT AND ITS APPLICATIONS HE WANG Abstract. This is the note for my talk in Graduate Student Seminar NEU. W. Massey defined Massey product as a higher order cohomology operation, which is a generalization of cup product. A first application of Massey product is in Knot theory showing that Borromean rings are linked with zero linking numbers. As another application, people can use Massey product describe the differentials in spectral sequences. There are many other important applications, such as obstruction to the formality of a space, which may be omitted in this talk. 1. Massey Products 1.1. Cup Products review. Given a topological space X and a commutative ring R, we can define the cohomology groups H∗(X; R): There is also a product structure on H∗(X; R), called cup product [ : Hp(X; R) ^ Hq(X; R) ! Hp+q(X; R): Example 1.1. Let T = S1 × S1 be the torus. Then H0(T ; R) = R, H1(T ; R) = R2 generated by fa; bg, H2(T ; R) = R generated by fγg: Then, we have a [ b = γ = −b [ a and the other cup products are zeros. In fact, the cohomology ring H∗(T ; R) of torus T V ∗ n V is the exterior algebra R[a; b]: In general, H (T ; R) = R[a1; ··· ; an]: Cup product is not easy to compute. However, many methods can be used to compute cup product, definition combined simplicial homology, method in Hatcher's book, inter- section method combined Poincare duality, product formula, even (Leray-Serre) spectral sequence etc.. Remark 1.2. Cup products are great because rings have more properties, and cup products can be used to distinguish between spaces that might have the same cohomology groups. In above example, the torus T = S1 × S1 can be distinguished from the the wedge sum of two circles and one spheres S1 _ S1 _ S2, even though they have the same homology groups, since all cup products in S1 _ S1 _ S2 are zeros. 1.2. Triple Massey Products. If x 2 Ci(X; R), the symbol x will denote (−1)1+ix. We first define the Massey triple product. Let x1, x2 and x3 be cocycle of degrees r1, r2 and r3 with cohomology classes [x1] [ [x2] = 0 and [x2] [ [x3] = 0 . Thus, there are cochains x12 of degree r1 + r2 − 1 and x23 of degree r2 + r3 − 1 such that dx12 =x ¯1 [ x2 and dx23 =x ¯2 [ x3. Define the cochain ! of degree (r1 + r2 + r3 − 1) by ! = x12 [ x3 + x1 [ x23; Date: Oct. 4, 2012. 1 2 HE WANG Then ! satisfies r +r r d(!) = (−1) 1 2 dx12 ^ x3 + (−1) 1 x¯1 ^ dx23 r +r r +r +1 = (−1) 1 2 x¯1 ^ x2 ^ x3 + (−1) 1 2 x¯1 ^ x2 ^ x3 = 0: So, ! is in fact a cocycle. Definition 1.3. A set of all the cohomology classes [!] obtained by the above procedure is defined to be the Massey triple product h[x1]; [x2]; [x3]i of [x1]; [x2] and [x3]. Due to the ambiguity of x12 and x23, the Massey triple product h[x1]; [x2]; [x3]i is an element of the quotient group r1+r2+r3−1 r2+r3−1 r1+r2−1 H (X) ([x1]H (X) + H (X)[x3]): For example, we consider the Massey products of 1 dimensional classes u1; u2; u3 2 1 2 H (X; R): Hence, hu1; u2; u3i ⊂ H (X; R), and the indeterminacy Inhu1; u2; u3i of triple Massey product hu1; u2; u3i is 1 1 2 Inhu1; u2; u3i = u1H (X) + H (X)u3 ⊂ H (X; R): 2 hu1; u2; u3i can be seen as an element in H (G)=Inhu1; u2; u3i: We say hu1; u2; u3i is vanishing if this element is 0. The Massey product hu1; u2; u3i is said to be decomposable if it contains a cohomology class that can be written as a product αβ of two elements in H>0(G); otherwise, it is called indecomposable. Example 1.4. Let T = S1 × S1 be the torus. Then H0(T ; R) = R, H1(T ; R) = R2 generated by fa; bg, H2(T ; R) = R generated by fγg: Since a [ a = 0, we can define ha; a; ai: However, it is vanishing by the reason of definition. In fact, all Massey products in T are vanishing. 1.3. Quadruple Massey Products. We can generalize the triple Massey products to the 4th order Massey product. Let u; v; w and x be homogeneous elements from H∗(X; R) or degree p; q; r and s respectively. Assume hu; v; wi and hv; w; xi exist and vanish simul- taneously, then we can define the fourth order Massey product hu; v; w; xi: Let x1; x2; x3; x4 be the representative cocycles for u; v; w; x respectively. Because of the existence of hu; v; wi and hv; w; xi, we have cochains x12; x23; x34 such that dx12 =x ¯1 [ x2; dx23 =x ¯2 [ x3; dx34 =x ¯3 [ x4: Since both hu; v; wi and hv; w; xi vanish, there exist cochains x13 and x24 such that d(x13) = x1 [ x23 + x12 [ x3; d(x24) = x2 [ x34 + x23 [ x4: One can show that the cochain ! = x1 [ x24 + x12 [ x34 + x13 [ x4 is actually a cocycle of degree p + q + r + s − 2: The quadruple Massey product hu; v; w; xi is defined to be the set of all cohomology class [!] 2 Hp+q+r+s−2(X; R): MASSEY PRODUCT AND ITS APPLICATIONS 3 1.4. Higher Order Massey Products. Let's use the following convention to describe higher Massey product. It is better to look at the data for triple Massey product (and quadruple Massey product) in matrix 0 1 x x x ∗ 1 0 1 x x ∗ 1 1 12 13 1 12 1 x x x 1 x x B 2 23 24 C B 2 23 C ; B 1 x x C B 1 x C B 3 34 C @ 3 A B 1 x C 1 @ 4 A 1 1 First, we consider the Massey products of 1 dimensional classes u1; u2; ··· ; uk 2 H (X; C): 1 Definition 1.5. Given u1; ··· ; uk 2 H (G; C). We call a defining system for the k-fold 1 Massey product hu1; ··· ; uki an array of 1-cochains M = fmi;j 2 C (G; C) j 1 ≤ i < j ≤ k + 1; (i; j) 6= (1; k + 1)g such that (1) mi;i+1 is a cocycle representing ui; j−1 P (2) δmi;j = mi;s [ ms;j: s=i+1 The value of the product at M, denoted hu1; ··· ; ukiM is defined to be the cohomology 2 class in H (G; C) represented by the cocycle k X ! = m1;s [ ms;k+1: s=2 The product hu1; ··· ; uki is defined only if such an M exists, and is taken to be the subset 2 of H (G; C) consisting of all elements of the form hu1; ··· ; ukiM : hu1; ··· ; uki = fhu1; ··· ; ukiM j M a defining systemg: The indeterminacy of Massey product hu1; ··· ; uki is 2 Inhu1; ··· ; uki = fa − b j a; b 2 hu1; ··· ; ukig ⊂ H (G; C): Remark 1.6. There is a unique matrix associated to each defining system M as follows. 0 1 1 m1;1 m1;2 m1;3 ··· m1;n−1 B 1 m2;2 m2;3 ··· m2;n−1 m2;n C B C B 1 m3;3 ··· m3;n−1 m3;n C B C B .. C B 1 . C B C B mn−1;n−1 mn−1;n C B C @ 1 mn;n A 1 The index of the input is a little different from the triple and quadruple Massey products. The reason is that people want to use this as a subgroup of matrix group. Massey product has the following properties. ([7]) 4 HE WANG ∗ Proposition 1.7. ([7]) Let u1; u2; ··· ; uk 2 H (X; C): Then, 1.(Linearity) If k 2 R, then for all 1 ≤ i ≤ n; khu1; ··· ; uni ⊆ hu1; ··· ; kui; ··· ; uni: 2.(Naturality) If f : XiY is a continuous map, then ∗ ∗ ∗ f (hu1; ··· ; uni) ⊆ hf (u1); ··· ; f (un)i: 3.(Associativity) For v 2 H∗(X); hu1; ··· ; univ ⊆ hu1; ··· ; unvi; deg v : vhu1; ··· ; uni ⊆ (−1) hvu1; ··· ; uni n P P 4.(Symmetry) If pj = deg uj and l = prps + (n − 1) pr + (n − 1)(n − 2)=2; 1≤r<s≤n r=1 then l hun; ··· ; u1i = (−1) hu1; ··· ; uni: The last property in ([7]) maybe not precisely, we will see that in the examples of Bor- romean ring. I change it by David Kraines's paper "Massey Higher Products" 2. Classical Application: Borromean rings 2.1. Some Knot Theory. Let L be an oriented tame ordered link in S3, with com- 3 S ponents L1; ··· ;Ln: Its complement, S n i Li; has the homotopy type of a connected 2-dimensional finite CW-complex. More precisely, let nS1 be the disjoint union of n cir- 1 3 2 cles S1;S2; ··· Sn. A link is a embedding l : nS ! S : Denote Ti be Si × D ; we can 3 3 S extend l to an embedding T1 [···[ Tn ! S : Let X be the closure of S − Ti, a S 1 ∼ 3 compact 3-manifold with boundary Si × S : In G = π1(X) = π1(S nL); the images of 1 the zi × S (zi 2 Si) are called the meridians mi. The images of the Si are called the longitudes li: The linking number lij = lk(Li;Lj) is the image of the i-th longitude in the ∼ Z = H1(X(Lj); Z): In particular, lii = 0 and lij = lji.
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