The origin of pictures: near rings and K 3 Z

Transcription

1 Brandeis University March 21, 2018

2 Disclaimer I did not invent pictures. They existed before I was born. were sometimes called Peiffer diagrams. They were used in combinatorial group theory. This talk is about my use of cluster pictures to study algebraic K-theory.

3 The exact sequence We have an exact sequence for any abelian group π: π 3 A(Bπ) f K 3 Zπ χπ Z 2 π g π 2 A(Bπ) h K 2 Zπ 0 1 g, h were defined by Hatcher and Wagoner (1973). 2 f was defined in my PhD thesis (1979). 3 I showed that elements of K 3 of any ring are represented by 2-dimensional pictures. 4 This talk is about χ π : K 3 Zπ Z 2 π = Zπ/2Zπ.

4 Special case: π = 1 For the trivial group π = 1 we have: π 3 A( ) K 3 Z χ 1 Z 2 π 2 A( ) K 2 Z 0 χ Z 2 Z 24 Z 1 48 Z2 Z 2 Z 2 0 using the result K 3 Z = Z 48 by R.Lee and Szczarba (1976). A( ) is Waldhausen A-theory of one point.

5 Definition and examples Steinberg group Definition Let G = X Y be a group with generating set X = {x, y, z, } and relation set Y F where F is the free group generated by X. Then a picture for G is an oriented planar graph L with 1 Edges labeled with elements of X. 2 Vertices labeled with elements of Y Y 1. 3 Base point angle at each vertex. so that the labels of the edges coming to a vertex, read counterclockwise starting at the basepoint angle is the relation at that vertex.

6 Example 1 Definition and examples Steinberg group y yzx 1 Y Y 1 * x z (composing left-to-right)

7 Example 2 Definition and examples Steinberg group Theorem Elements of H 3 (G) are given by deformation classes of pictures for G. Eg: For G = Z 3 = x x 3, the generator of H 3 (Z 3 ) = Z 3 is: x * x 3 x 3 * x x

8 Definition and examples Steinberg group Definition The Steinberg group of the group ring Zπ has generators xij u where 1 i j n and u π (with inverse written x u ij ) modulo the relations: 1 [xij u, x jk v ] = x ik uv in the word order: x u ij x v jk x u ij x uv ik x v jk 2 [xij u, x kl v ] = 1 when j k and i l.

9 Definition and examples Steinberg group An example of a picture for the Steinberg group. All arcs oriented clockwise. x u 12 x v 23 x uv 13 x w 14 This is also the cluster picture for the quiver

10 Definition and examples Steinberg group Another picture for the Steinberg group. x u 12 x v 23 x uv 13 x vw 24 x uvw 14 x w 34 This is also the cluster picture for the quiver

11 Definition and examples Steinberg group Theorem (Gersten: 1973) H 3 (St(R)) = K 3 R. (Elements are given by pictures for St(R).) Theorem (I-Klein: 1993) The generator of K 3 Z = Z 48 is given by the picture:

12 Definition and examples Steinberg group Since K 3 Zπ = H 3 (St(Zπ)), χ π : K 3 Zπ Z 2 π is given by a cohomology class χ π H 3 (St(Zπ); Z 2 π) This cohomology class is pulled back from a cohomology class [ε Tr α] H 3 (GL(Zπ); Z 2 π) which is obtained from the near-ring version of Zπ.

13 Definition and main example Right associativity Matrix multiplication Definition A left-near-ring (with unity) is structure (R, +,, 0, 1) so that 1 (R, +, 0) is a group. 2 (R,, 1) is a monoid. 3 (left distributive) a(b + c) = ab + ac ( x0 = 0). 4 0x = 0. In other words, R satisfies all the axioms of a ring with unity except commutativity of addition and right distributivity.

14 Main Example Definition and main example Right associativity Matrix multiplication Let M = (M,, 1) be a monoid and let G(M) be the free group generated by M. The group law will be written as addition. Thus, the inverse of x M is written x. G(M) is a left near-ring with multiplication defined as follows. 1 For a, b ±M, ab := (sgna)(sgnb) a b. Elements of ±M will be called monomials. 2 If a = a j, b = b k then the product ab is the sum of the monomials a j b k in lexicographic order according to the pair (k, (sgnb k )j).

15 Definition and main example Right associativity Matrix multiplication For example, (a 1 + a 2 )(b c) = (a 1 + a 2 )b + (a 1 + a 2 )( c) We use the notation: = a 1 b + a 2 b a 2 c a 1 c }{{} reversed ab = k (b k ) a j b k j If we had right distributivity we would get: (a 1 + a 2 )(b c) = a 1 (b c) + a 2 (b c) = a 1 b a 1 c + a 2 b a 2 c

16 Definition and main example Right associativity Matrix multiplication In general (a 1 + a 2 )b a 1 b + a 2 b. These are the sums of the terms a ij b k in a different order: a 1 b + a 2 b = i (a 1 + a 2 )b = k (b k ) a ij b k (1) k j (b k ) (b k ) a ij b k (2) i j

17 Definition and main example Right associativity Matrix multiplication The obstruction to right distributivity is the symmetric tensor: ρ : G(M) G(M) G(M) S 2 Z 2 M = Z 2 M Z 2 M ρ(a 1, a 2 ; b) = a ij b k a i j b k where the sum is over all pairs of terms which switch order from (1) to (2). In the example, ρ(a 1, a 2 ; b c) = a 1 c a 2 b + a 1 c a 2 c

18 Definition and main example Right associativity Matrix multiplication Now compose with the intersection pairing ε =, : Z 2 M Z 2 M Z 2 M (Consider Z 2 M as the set of all finite subsets of M and take intersection.) Then: ερ(a 1, a 2 ; b) = aij b k, a i j b k Z2 M is the set of elements of M which are commuted with themselves an odd number of times under right distribution.

19 Definition and main example Right associativity Matrix multiplication Let M n (G(M)) be the set of n n matrices with entries in G(M). If A, B, C M n (G(M)) we compare A(BC) and (AB)C. Then entries of A(BC) are: For (AB)C we have: A(BC) pq = k A pi B ik C kq i (AB)C pq = k ( Ap1 B 1k + A p2 B 2k + + A pn B nk ) Ckq

20 Definition and main example Right associativity Matrix multiplication To get from one to the other, we need to distribute the product, then permute the terms. We get the associator α(a, B, C) pq = ρ(a pi B ik, A pj B jk ; C kq ) + A pi B ik C kq A pi B i k C k q i<j Take trace followed by the intersection pairing to get ε Tr α(a, B, C) Z 2 M This is the set of elements of M commuted with themselves an odd number of times under matrix association.

21 Definition and main example Right associativity Matrix multiplication Theorem (I: 1982) ε Tr α is a 3-cocycle representing a class in H 3 (GL n (ZM); Z 2 M) which restricts to χ M H 3 (St n (ZM); Z 2 M) the morphism in the algebraic K-theory exact sequence. Conjecture (Hatcher) For M = π any abelian group, the image of χ π : K 3 Zπ Z 2 π has at most 2 elements. It is sufficient to prove this for the Klein 4-group π = Z 2 Z 2. This conjecture is needed to answer a basic question about diffeomorphisms of smooth manifolds: What is π 0 Diff (M)?

22 Definition and main example Right associativity Matrix multiplication THANK YOU!