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THE ORIGIN OF PICTURES: NEAR-RINGS AND ALGEBRAIC K-THEORY

KIYOSHI IGUSA

Abstract. These are lecture notes for a talk I gave at the Seminar at the University of Connecticut on March 21, 2018. I explain the origin of “pictures” which played a large part in many of my earlier papers in topology including my PhD thesis. Since details on the combinatorics of near-rings are already published in my 1982 paper (in which they are called “quasi-rings”), I presented only the ideas and an open question about near-rings which has an important application in topology.

Introduction Gordana Todorov, Jerzy Weyman, Kent Orr and I are working on a book about “pic- tures” which have gained renewed attention because they are equivalent to “scattering diagrams”. This talk is about very old work that I did to introduce pictures and their re- lation to the cohomology of GL(n, Z). In particular, I will discuss the following pictures (from our book!) and their relationship to H3(GL(n, Z), Z/2).

The algebraic side of this story is a fun topic. The cohomology which detects the “exotic element” of K3Z is the degree 3 class which counts the number of times (modulo 2) that commutativity of is used to prove that multiplication is associative! This is an old result (about the obstruction to right distributivity in left near-rings) which I am happy to present in a new light.

1 2 KIYOSHI IGUSA Although I want to avoid the topology, a little bit of history is in order. The focus is the following exact for any π.

f χπ g h π3A(Bπ) −→ K3Zπ −→ H1(Zπ, Z2π) −→ π2A(Bπ) −→ K2Zπ → 0 (1) g, h were defined by A. Hatcher and J. Wagoner [3]. (2) I defined f in my PhD thesis [5]. (3) I showed that elements of K3 of any can be represented by 2-dimensional “pictures” [4]. (4) H1(Zπ, Z2π) is Z2π (group ring of π over the field F2 = Z2 with two elements) modulo the conjugation action of π. Thus

H1(Zπ, Z2π) = Z2π when π is abelian. (5) This talk is about the mapping χπ. For the π = 1 we have:

χ1 π3A(∗) → K3Z −→ Z2 → π2A(∗) → K2Z → 0

χ1 Z2 ⊕ Z24 → Z48 −→ Z2 → Z2 → Z2 → 0

• Milnor [9]: K2Z = Z2. • Lee-Szczarba [8]: K3Z = Z48. • Waldhausen-Cerf (combining results of [11],[1]): π2A(∗) = Z2. Thus, we already know that χ1 : K3Z → Z2 is surjective. In [4] I called this map the “Grassmann invariant”. This talk has three parts:

(1) K3R is the of deformation classes of “pictures”. (2) χπ is the obstruction to matrix associativity for “near-rings”. (3) Morse theory on circle bundles gives picture for K3Z having χ1 6= 0 (from [7]).

1. Pictures and H3(G) Definition 1.1. Let G = hX | Yi be a group with generating set X = {x, y, z, ···} and relation set Y ⊂ F where F is the 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 the vertex. Example: THE ORIGIN OF PICTURES: NEAR-RINGS AND ALGEBRAIC K-THEORY 3 y

yzx−1 ∈ Y ` Y−1 * x

z

Theorem 1.2. Elements of H3(G) are given by “deformation classes” of pictures for G.

3 Example 1.3. For G = Z3 = hx | x i, the generator of H3(Z3) = Z3 is:

x x−3 *

x3 x *

x

u Definition 1.4. The Steinberg group of the group ring Zπ has generators xij where −u 1 ≤ i 6= j ≤ n and u ∈ π (with inverse written xij ) modulo the relations: u v uv (1)[ xij, xjk] = xik in the word order: u v −u −uv −v xijxjkxij xik xjk u v (2)[ xij, xk`] = 1 when j 6= k and i 6= `. Example 1.5. An example of a picture for the Steinberg group. All arcs oriented clockwise. u x12

xv uv 23 x13

w x14 4 KIYOSHI IGUSA

Theorem 1.6 (Gersten). H3(St∞(R)) = K3R.

Theorem 1.7 (I-Klein). The following picture represents the generator of K3Z = Z48.

Proof. Reducing mod 2, this picture and Example 1.3 map to the same element of K3Z2 which is Z3 by [10]. (The Z3 → St(Z2) is given by sending x to x31x13.) So, the 3-torsion part of this element is nonzero. The near-ring associator χ1 : K3Z  Z2 is nontrivial on this element. (It is the sum over values at the vertices. Only the blue circled vertex contributes 1.) Therefore, this picture also maps to the generator of the 2-torsion part of Z48.  THE ORIGIN OF PICTURES: NEAR-RINGS AND ALGEBRAIC K-THEORY 5 2. Left near-rings Definition 2.1. A left-near-ring (with unity) is structure (R, +, ×, 0, 1) so that (1)( R, +, 0) is a group. (2)( R, ×, 1) is a . (3) (left distributive) a(b + c) = ab + ac (⇒ x0 = 0). (4)0 x = 0. In other words, R satisfies all the axioms of a ring with unity except commutativity of addition and right distributivity. In a left near-semi-ring we also drop the axiom of additive inverses. Example: 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 := (sgn a)(sgn b)|a||b|. Elements of ±M will be called mono- mials. P P (2) If a = aj, b = bk then the product ab is the sum of the monomials ajbk in according to the pair (k, (sgn bk)j). For example,

(a1 + a2)(b − c) = (a1 + a2)b + (a1 + a2)(−c) = a1b + a2b −a2c − a1c | {z } reversed We use the notation: (bk) X X ab = ajbk k j

2.1. Right associativity. In general (a1 + a2)b 6= a1b + a2b. These are the sums of the terms aijbk in a different order:

(bk) X X X (2.1) a1b + a2b = aijbk i k j

(bk) (bk) X X X (2.2) (a1 + a2)b = aijbk k i j The (reduced) obstruction to right distributivity is the symmetric tensor: 2 ρ : G(M) × G(M) × G(M) → S Z2M = Z2M⊗eZ2M X ρ(a1, a2; b) = aijbk⊗eai0j0 bk0 where the sum is over all pairs of terms which switch order from (2.1) to (2.2). Now compose with the intersection pairing ε = h·, ·i : Z2M⊗eZ2M → Z2M (Con- sider Z2M as the set of all finite of M and take intersection.) Then: X ερ(a1, a2; b) = haijbk, ai0j0 bk0 i ∈ Z2M is the set of elements of M which are commuted with themselves an odd number of times. 6 KIYOSHI IGUSA

2.2. . Let Mn(G(M)) be the set of n × n matrices with entries in G(M). If A, B, C ∈ Mn(G(M)) we compare A(BC) and (AB)C. Then entries of A(BC) are: X X A(BC)pq = ApiBikCkq k i For (AB)C we have: X (AB)Cpq = (Ap1B1k + Ap2B2k + ··· + ApnBnk) Bkq k To get from one to the other, we need to distribute the product, then permute the terms. We get a (reduced) associator X X α(A, B, C)pq = ρ(ApiBik,ApjBjk; Ckq) + ApiBikCkq⊗eApi0 Bi0k0 Ck0q i

ε ◦ T r ◦ α(A, B, C) ∈ Z2M 3 Theorem 2.2 (I: 1982). This is a 3-cocycle representing a class in H (GLn(ZM); Z2M) 3 which restricts to χM ∈ H (Stn(ZM); Z2M) coming from algebraic K-theory.

Conjecture 2.3 (Hatcher). For M = π any , the image of χπ : K3Zπ → Z2π has order 2.

It is enough to prove this for the Klein 4-group π = Z2 × Z2. This significance of this statement is that the cokernel of the map χπ : K3Zπ → Z2π is a subgroup of π0Diff(M × I rel M × 0) for π1M = π.

3. Calculation

The formula for χπ is given as follows.

(1) Label each region of the picture L with an element r ∈ GLn(Zπ) starting with the unique unbounded region which we label with the matrix In and, when we pass a wall from left to right, change the label by right multiplication by the label of the wall: u r ↑ u rx xij ij

This is well-defined since the labels around each vertex multiply to In. u v (2) χπ(L) is the sum over all vertices carrying commutators of the form [xij, xik] of the following element of Z2π: X (3.1) rpi husjp, vskpi p

where s = (sqp) is the inverse of r = (rpq) and r is the label of any one of the regions adjacent to the vertex. THE ORIGIN OF PICTURES: NEAR-RINGS AND ALGEBRAIC K-THEORY 7

In the example on page 4, the blue circle has x12 coming into the lower right. However, it should be coming into the upper left and, therefore, when drawn properly, x12 should cross x13. So, (i, j, k) = (1, 2, 3). The matrices r, s are 1 0 0 1 0 0 r = 1 1 1 , s = 0 1 1 1 0 1 1 0 1

The only value of p for which s2p, s3p are both nonzero is p = 3 and

rp1 hs2p, s3pi = 1

So, this vertex contributes 1 to the value of χ1(L). Each of the other vertices contribute 0. So, χ1(L) = 1. In Example 1.5 there are three intersections which contribute a total of 0. The introduction had another of the five pictures on which (3.1) needs to be zero in order to define a mapping χπ : K3Zπ → Z2π. References [1] Jean Cerf, “La stratification naturelle des espaces de fonctions diff´erentiables r´eelles et le th´eoreme de la pseudo-isotopie.” Publications Math´ematiquesde l’Institut des Hautes Etudes´ Scientifiques 39.1 (1970): 7–170. [2] S.M. Gersten, K3 of a ring is H3 of the Steinberg group, Proceedings of the American Mathematical Society (1973): 366–368. [3] Allen Hatcher and John Wagoner, “Pseudo-isotopies of compact manifolds.” (1973). [4] Kiyoshi Igusa, The generalized Grassmann invariant, preprint. [5] ,“The W h3(π) obstruction for pseudoisotopy.” PhD diss., Princeton, 1979. [6] , On the algebraic K-theory of A∞-ring spaces, Algebraic K-Theory, Part II (Oberwolfach, 1980), Springer, Berlin, 1982, pp. 146–194. [7] Kiyoshi Igusa and John Klein, The Borel regulator map on pictures. II. An example from Morse theory, K-Theory 7 (1993), no. 3, 225–267. [8] Ronnie Lee, and R. H. Szczarba, The group K3Z is cyclic of order forty-eight, Annals of Mathe- matics (1976): 31–60. [9] John Milnor, “Introduction to Algebraic K-Theory” (AM-72). Vol. 72. Princeton University Press, 2016. [10] Daniel Quillen, On the cohomology and K-theory of the general linear groups over a finite field, Annals of Mathematics (1972): 552–586. [11] Friedhelm Waldhausen, Algebraic K-theory of spaces, Algebraic and geometric topology. Springer, Berlin, Heidelberg, 1985. 318–419.

Department of Mathematics, Brandeis University, Waltham, MA 02454 E-mail address: [email protected]