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From manifolds to invariants of En-algebras MASSACHUSETTS INSTITUTE by OF TECHNIOLOGY Ricardo Andrade APR 08 2011 Licenciatura, Instituto Superior Tecnico (2005) LjBRARIES Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy ARCHIVEs at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY September 2010 @ Ricardo Andrade, MMX. This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License. To view a copy of this license, visit http: //creativecommons .org/licenses/by-sa/3.0/legalcode 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 August 24, 2010 Certified by.... Haynes R. Miller Professor of Mathematics Thesis Supervisor 7) Accepted by ... 6$, Bjorn Poonen Chairman, Department Committee on Graduate Students From manifolds to invariants of En-algebras by Ricardo Andrade Submitted to the Department of Mathematics on August 24, 2010, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract This thesis is the first step in an investigation of an interesting class of invariants of En-algebras which generalize topological Hochschild homology. The main goal of this thesis is to simply give a definition of those invariants. We define PROPs EG, for G a structure group sitting over GL(n, R). Given a manifold with a (tangential) G-structure, we define functors 0 EG[M]: (EG) -+ Top constructed out of spaces of G-augmented embeddings of disjoint unions of euclidean spaces into M. These spaces are modifications to the usual spaces of embeddings of manifolds. Taking G - 1, El is equivalent to the n-little discs PROP, and El [M] is defined for any parallelized n-dimensional manifold M. The invariant we define for a Es-algebra A is morally defined by a derived coend TG(A; M) := EG[M] 9 A n EG for any n-manifold M with a G-structure. The case T' (A; Sl) recovers the topological Hochschild homology of an associative ring spectrum A. These invariants also appear in the work of Jacob Lurie and Paolo Sal- vatore, where they are involved in a sort of non-abelian Poincare duality. Thesis Supervisor: Haynes R. Miller Title: Professor of Mathematics ACKNOWLEDGEMENTS Acknowledgements This is a personal note, attempting to recall some of the people who I feel have significantly enriched my life as a graduate student at MIT, academic and otherwise. It is marked by my current perspective, with all its prejudices and faulty memory. I must start with my advisor, Haynes Miller. I want to thank him not just for the vast mathematical knowledge he shared with me. He also showed me unending patience, and I learned about the value and difficulty of finding a subject I care about. In my biased opinion, I cannot think of a better choice for my Ph.D. advisor. On the academic side, I cannot possibly begin to comprehend the great network of people, events, and discussions which have directed my current ideas and research. This is compounded by my ephemeral memory, and my tendency to keep academic discussions to a minimum. Still, I would like to mention, in no particular order, Mark Behrens, Tyler Lawson, Mike Hill, Teena Gerhardt, Josh Nichols-Barrer, Gustavo Granja, Vigleik An- geltveit, John Francis, Matthew Gelvin, Jacob Lurie, Angdlica Osorno, Jen- nifer French, Sam Isaacson, and Nick Rozenblyum. Many more are certainly missing, and the details of our interactions are lost. I would also like to thank Mark Behrens and Clark Barwick for taking part in my thesis committee. On a more personal note, some in the group of graduate students at MIT have been like a family away from home to me. A few were there at the beginning and left shortly. Teena Gerhardt, Mike Hill, Josh Nichols-Barrer, Vigleik Angeltveit, and Max Lipyanskiy made me feel welcome from the start, even if I came off as annoying or derisive at times. Others have been there with me for most of my stay at MIT. My friendships with Matthew Gelvin, Ana Rita Pires, Angelica Osorno, Amanda Redlich, Olga Stroilova, and Jennifer French are priceless. I could go on thanking more people, or analyzing the discussions which influenced me most... Instead, I will simply state my appreciation of my parents, my brother, and the beautiful complexity of the universe. Contents Acknowledgements 5 Introduction 11 Bibliographic references and influences 12 Summary 15 Chapter I. Basic terminology 17 Introduction 17 Summary 17 1. Categories of sets and categories 18 2. Categories 19 3. Categories of topological spaces 21 4. Principal bundles 22 5. Homotopy theory of topological spaces 23 6. Moore path space 24 7. Enriched categories 26 8. Properties of enriched functors 28 9. Model categories 29 10. PROPs 31 11. Operads and categories of operators 33 12. Examples of PROPs in Set and Top 35 Chapter II. Internal categories 39 Introduction 39 Summary 39 1. Internal categories 40 2. Categories of internal categories 42 3. Examples of internal categories 44 4. Coproducts in finitely complete categories 46 5. Relation between internal and enriched categories 48 6. Internal presheaves 52 CONTENTS 7. Relation between external presheaves and internal presheaves 56 8. Internal presheaves of categories 58 9. Grothendieck construction 63 10. Variation on Grothendieck construction 67 11. Homotopical properties of Grothendieck construction 70 Chapter III. Categories of sticky configurations 77 Introduction 77 Summary 77 1. Sticky homotopies 78 2. Functoriality of sticky homotopies 80 3. Categories of sticky homotopies 83 4. Category of sticky configurations 86 5. Generalities on G-equivariance 87 6. G-objects in Set and Top 88 7. G-equivariant sticky configurations 90 8. From MG to M 91 9. Sticky configurations and covering spaces 95 Chapter IV. Sticky configurations in S1 103 Introduction 103 Summary 103 1. Homotopical discreteness 103 2. Z-equivariant sticky configurations in R 104 3. A category of Elmendorf 107 4. Equivalence with Elmendorf's category 108 5. Relation to associative PROP 112 6. Relation to A P 112 7. Cyclic bar construction 114 Chapter V. Spaces of embeddings of manifolds 117 Introduction 117 Summary 117 1. Spaces of embeddings and categories of manifolds 118 2. Simple PROPs of embeddings 119 3. Right modules over PROPs of embeddings 121 4. Homotopy type of the right modules over En 121 5. G-structures on manifolds 125 6. Constructions on G-structures 127 CONTENTS 7. Examples of G-structures 129 8. Augmented embedding spaces 130 9. Interlude: homotopy pullbacks over a space 132 10. Categories of augmented embeddings 134 11. PROPs of augmented embeddings 136 12. Right modules over PROPs of augmented embeddings 137 13. Internal presheaves on EG 138 14. Homotopy type of the right modules over EG 140 Chapter VI. Stratified spaces 145 Introduction 145 Summary 145 1. Stratified spaces 146 2. Spaces and categories of filtered paths 149 3. Strong spaces of filtered paths 152 4. Application: sticky homotopies from filtered paths 153 5. Homotopy link spaces 155 6. Tameness and homotopy links 159 7. Homotopically stratified spaces 163 8. Application: spaces related to M(M) 167 Chapter VII. Sticky configurations and embedding spaces 171 Introduction 171 Summary 171 1. The Grothendieck construction of embeddings 172 2. Homotopy invariance of total category 173 3. Analysis of morphisms of Tn[M] 177 4. Connecting Tn[M] and M(M) 179 5. Composition in ZM 185 6. Equivalence between Tn[M] and M(M) 189 Chapter VIII. Homotopical properties of enriched categories 201 Introduction 201 Summary 201 1. Categories of intervals 202 2. Monads and categories of intervals 208 3. Bar constructions for enriched categories 209 4. Derived enriched colimits 212 5. Homotopy colimits of enriched functors 217 10 CONTENTS 6. Weak equivalence of enriched categories 220 7. Grothendieck constructions 224 Chapter IX. Invariants of Es-algebras 229 Introduction 229 Summary 229 1. The invariants 229 2. Classifying spaces of path categories 232 3. Relation to M(M) 237 4. Relation to topological Hochschild homology 239 Bibliography 241 Introduction This introduction will describe a bit of the short history leading to the research presented in this text. In a nutshell, the material stems from an investigation of a sufficiently natural diagrammatic interpretation of topo- logical Hochschild homology (THH). This naturality shows the way towards generalizations of THH for En-algebras. The author's gateway into this problem was an apparent lack of nat- urality of the usual definition of topological Hochschild homology, as the geometric realization of the cyclic bar construction of an associative ring spectrum. A telling sign is that the indexing category A P for the cyclic bar construction does not reflect the full rotational symmetry of S'. Con- sequently, it is insufficient to recover the action of S' on THH. In order to repair this state of affairs, the author conceived of the cat- egory E, here called Elmendorf's category. This category is essentially a combinatorial description of the spaces of configurations in S'. Topological Hochschild homology can be recovered as a homotopy colimit along E. This draws an analogy with the well-known result that THH of a commutative ring spectrum is given by tensoring with S1. A more natural, yet equivalent, amalgamation of the spaces of configu- rations of S1 is given by the topologically enriched category M(SI) of sticky configurations in S'. The advantage of M(Sl) is that it generalizes promptly to a category M(X) for any space X. The significance of M(X) is most ap- parent for the case of a manifold, where it carries homotopical information about the tangent space of the manifold, and about embeddings into other manifolds.
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