Scissors Congruence and K-theory by Inna Zakharevich Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2012 c Inna Zakharevich, MMXII. All rights reserved. The author hereby grants to MIT permission to reproduce and distribute publicly paper and electronic copies of this thesis document in whole or in part. Author................................................................................... Department of Mathematics March 19, 2012 Certified by............................................................................... Michael J. Hopkins Professor, Harvard University Thesis Supervisor Accepted by.............................................................................. Bjorn Poonen Chairman, Department Committee on Graduate Theses 2 Scissors Congruence and K-theory by Inna Zakharevich Submitted to the Department of Mathematics on March 19, 2012, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Abstract In this thesis we develop a version of classical scissors congruence theory from the perspective of algebraic K-theory. Classically, two polytopes in a manifold X are defined to be scissors congruent if they can be decomposed into finite sets of pairwise-congruent polytopes. We generalize this notion to an abstract problem: given a set of objects and decomposition and congruence relations between them, when are two objects in the set scissors congruent? By packaging the scissors congruence information in a Waldhausen category we construct a spectrum whose homotopy groups include information about the scissors congruence problem. We then turn our attention to generalizing constructions from the classical case to these Waldhausen categories, and find constructions for cofibers, suspensions, and products of scissors congruence problems. Thesis Supervisor: Michael J. Hopkins Title: Professor, Harvard University 3 4 Acknowledgments The number of people to whom I owe a debt of gratitude for this dissertation is very large, and while I will attempt to enumerate them all in this section I am well aware that my debt is far greater than a mere acknowledgements section (no matter how eloquent) can discharge. This work would not have happened at all without the guidance of my advisor, Michael Hopkins. His ideas and support have been invaluable, and I do not know how to express my thanks to him. Without the weekly conversations we had about geometry, category theory, algebra, topology, television and video games this work would likely be a mass of random letters and gibberish. Mike, I am extremely grateful for all of your guidance and teaching, and especially for your friendship. I would like to thank Clark Barwick for the many hours he spent teaching me|especially in my second and third year|and for founding the Bourbon Seminar. The effects of your help and guidance on my work cannot be overstated. I would also like to thank Andrew Blumberg for his help on many technical aspects of this project, and for his perspectives on K-theory in general. I would like to thank the MIT mathematics department. I owe the mathematics department for many things, from large things like providing me with an amazing mathematical community to small details like afternoon tea and cookies. I have attended many wonderful lectures and discussed mathematics with many interesting people, and I am grateful that I could be part of it all. I would like to specifically thank, in alphabetical order, Ricardo Andrade, Reid Barton, Mark Behrens, Dustin Clausen, Jennifer French, Phil Hirschorn, Sam Isaacson, Jacob Lurie, Haynes Miller, Angelica Osorno, Luis Alexandre Pereira, and Olga Stroilova. These thanks also extend to the greater mathematical community in the Boston area, especially to the members of the topology group. Of course, it is impossible to complete any research without support from many people outside of the research community. I would like to thank all of the organizations that supported me fincancially during the five years of my Ph.D.: the MIT mathematics department, the NSF GRFP, and Cabot House at Harvard. Although it is a clich´eto thank my friends in a context such as this, it is no less true that I could not have finished this dissertation without them. I am grateful to all of you. Alya Asarina and Adi Greif have been my friends for over fifteen years. Although thanking them in the context of this dissertation is a great understatement of their influence in my life, I want to thank them for their unwavering love, for all of the time that we spent together, and for all of the cups of tea that we shared. Lastly, and most importantly, I would like to thank my family. I want to thank all of you for your love, for your encouragement, and for your advice|even (and maybe especially) when I didn't listen to it. You have made my life richer and more beautiful. I especially want to thank my husband, Tom, for taking care of me and letting me take care of him, and for all of his infinite supply of love, even when I woke him up at two in the morning. 5 6 Contents 1 Introduction 9 1.0.1 A quick introduction to scissors congruence . .9 1.0.2 Connections to K-theory . 11 2 Preliminaries 15 2.1 Notation . 15 2.2 Categorical Preliminaries . 15 2.2.1 Grothendieck Twists . 15 2.2.2 Double Categories . 17 2.2.3 Multicategories . 18 2.3 Waldhausen K-theory . 19 2.3.1 The K-theory of a Waldhausen category . 19 2.3.2 K-Theory as a multifunctor . 23 3 Scissors congruence spectra 27 3.1 Abstract Scissors Congruence . 27 3.2 Waldhausen Category Structure . 31 3.3 Examples . 32 3.3.1 The Sphere Spectrum . 32 3.3.2 G-analogs of Spheres . 33 3.3.3 Classical Scissors Congruence . 33 3.3.4 Sums of Polytope Complexes . 33 3.3.5 Numerical Scissors Congruence . 33 3.4 Proof of theorem 3.2.2 . 34 3.4.1 Some technicalities about sub-maps and shuffles . 34 3.4.2 Checking the axioms . 37 4 Suspensions, cofibers, and approximation 43 4.1 Thickenings . 43 4.2 Filtered Polytopes . 45 4.3 Combing . 48 4.4 Simplicial Polytope Complexes . 52 4.5 Cofibers . 55 4.6 Wide and tall subcategories . 57 4.7 Proof of lemma 4.3.3 . 61 7 5 Further structures on PolyCpx 65 5.1 Two Symmetric Monoidal Structures . 65 5.1.1 Cartesian Product . 65 5.1.2 Initial Subcomplexes and Smash Products . 66 5.2 The K-theory functor is monoidal . 69 5.2.1 Waldhausen categories . 69 5.2.2 Γ-spaces . 70 5.3 Examples . 74 5.3.1 Euclidean Ring Structure . 74 5.3.2 Spherical Ring Structure . 75 5.3.3 The Burnside Category . 75 5.4 Simplicial Enrichment . 76 8 Chapter 1 Introduction 1.0.1 A quick introduction to scissors congruence The classical question of scissors congruence concerns the subdivisions of polyhedra. Given a polyhedron, when is it possible to dissect it into smaller polyhedra and rearrange the pieces into a rectangular prism? Or, more generally, given two polyhedra when is it possible to dissect one and rearrange it into the other? In more modern language, we can express scissors congruence as a question about groups. Let P(E3) be the free abelian group generated by polyhedra P , quotiented out by the two relations [P ] = [Q] if P =∼ Q, and [P [ P 0] = [P ] + [P 0] if P \ P 0 has measure 0. Can we compute P(E3)? Can we construct a full set of invariants on it? More generally, let X be En (Euclidean space), Sn, or Hn (hyperbolic space). We define a simplex of X to be the convex hull of n + 1 points in X (restricted to an open hemisphere if X = Sn), and a polytope of X to be a finite union of simplices. Let G be any subgroup of the group of isometries of X. Then we can define a group P(X; G) to be the free abelian group generated by polytopes P of X, modulo the two relations [P ] = [g · P ] for any polytope P and g 2 G, and [P [ P 0] = [P ] + [P 0] for any two polytopes P; P 0 such that P \ P 0 has measure 0. The goal of Hilbert's third problem is to classify these groups. n Consider the example when X = E and Gn is the group of Euclidean isometries. We have a homomorphism n m n+m P(E ;Gn) ⊗ P(E ;Gm) P(E ;Gn+m) given on generators by [P ] ⊗ [Q] [P × Q]; where P × Q is the subset of En × Em =∼ En+m which projects to P in En and Q in Em. Similarly, for X = Sn and G = O(n) we can construct a homomorphism P(Sn;O(n)) ⊗ P(Sm;O(m)) P(Sn+m+1;O(n + m + 1)) n n+1 in the following manner. Consider S to be embedded in R , and let Pe be the solid cone gen- n n+1+m+1 n+m+2 erated by all points inside P ⊆ S . Then Pe × Qe ⊆ R is a solid cone in R , and thus spans a polytope in Sn+m+1. We call this polytope P ∗ Q, as it can also be constructed as the orthogonal join (inside Sn+m+1) of P and Q. Then the homomorphism P(Sn;O(n)) ⊗ P(Sm;O(m)) P(Sn+m+1;O(n + m + 1)) given by [P ] ⊗ [Q] [P ∗ Q] is the desired homomor- phism. 1 L n 1 L n Letting P(E ) = n≥0 P(E ;Gn) and P(S ) = Z⊕ n≥0 P(S ;O(n)) we see that the above homomorphisms assemble into graded homomorphisms P(E1) ⊗ P(E1) P(E1) and P(S1) ⊗ P(S1) P(S1): 9 In order for the grading to work properly we need to give P(Sn;O(n)) the grading n + 1; the extra Z we added on will have grading 0 and should be considered to be the scissors congruence group of the empty polytope.