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Universal Simplicial Monoid Constructions on Simplicial Categories and Their Associated Spectral Sequences

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Universal Simplicial Monoid Constructions on Simplicial Categories and Their Associated Spectral Sequences View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by ScholarBank@NUS UNIVERSAL SIMPLICIAL MONOID CONSTRUCTIONS ON SIMPLICIAL CATEGORIES AND THEIR ASSOCIATED SPECTRAL SEQUENCES GAO MAN (M.Sc., Nankai University) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE 2012 ii to my parents DECLARATION I hereby declare that this thesis is my original work and it has been written by me in its entirety. I have duly acknowledged all the sources of information which have been used in the thesis. This thesis has also not been submitted for any degree in any university previously. Gao Man 27 September 2012 Acknowledgements Many people have played important roles in the past five years of my life. Here I hope to express my gratitude towards them. I would like to put on record my deepest respect and heartfelt gratitude to Professor Wu Jie. As my thesis advisor, he patiently helped clarify my thoughts and gave me invaluable guidance and utmost support. He provided many wonderful mathematical insights to affect me to perform my research in a most independent fashion. I will always remember the encouragement and advice he extended to further my education and research ability. I wish to thank Professor A. J. Berrick for his brilliant teaching and amazing insights. From taking his courses on topology and algebra, I was privileged to see and learn how to be a great teacher, especially how to improve the relationship between teacher and students. I will cherish the memories of the time. I gratefully thank my master advisor Professor Lin Jinkun(Nankai University), Assistant Professor Han Fei(NUS) and Professor L¨uZhi(Fudan University) for the generous help. They wrote the references for my Research Assistant application. I would like to thank National University of Singapore for providing me a full research scholarship and a excellent study environment. I was a Research iii Acknowledgements iv Assistant for the project \Structures of Braid and Mapping Class Groups(R-146- 001-137-112)". This project was funded by Ministry of Education of Singapore. I acknowledge their support. I would like to acknowledge MathOverflow. I learnt a lot from the mathemati- cians there. I am greatly indebted to my parents for their endless love and unconditional support in my research and daily life. I am deeply thankful to my siblings for their sincere love and constant encouragement. I am also indebted to my husband, Colin, not only for his unwavering love but also for his patience and understanding. I give sincere thanks to my parents-in-law for their care and support. Finally, I wish to deeply thank all my friends. They are too many to list. I feel very lucky to have been their friend. I will always remember the happy times with them. Gao Man May 2012 The examiners gave very helpful comments. Several of the results in this Thesis are sharper than the original version. I would like to thank the examiners, Prof Jon Berrick, Prof L¨uZhi and Prof Tan Kai Meng, for their careful reading of my Thesis. Gao Man September 2012 Contents Acknowledgements iii Summary viii List of Categories ix 1 Introduction 1 1.1 Applications of Carlsson's Construction . .2 1.2 Historical Context . .3 1.3 Main Results: Universal Simplicial Monoid Constructions . .4 1.4 Main Results: Spectral Sequences . 11 1.5 Future Problems . 17 1.6 Organization of The Thesis . 17 2 Category Theory 19 2.1 Categories . 19 2.2 Colimits . 21 2.3 Quotient Objects Defined by a Relation . 23 v Contents vi 2.4 Adjoint Functors . 24 2.5 The Free Monoid and Free Group on A Set . 25 2.6 The Group Completion and The Groupoid Completion . 29 3 Simplicial Homotopy Theory 31 3.1 Simplicial Sets . 31 3.2 Simplicial Monoid Actions . 40 3.3 Bisimplicial Sets . 41 3.4 The Nerve . 42 4 A General Cofiber Sequence 47 4.1 Universal Monoids and Universal Groups . 47 4.2 Equivalence to a One-Object Category . 50 4.3 Nerves of Sets . 55 4.4 The Main Theorem . 56 4.5 Translation Categories and Homotopy Colimits . 62 5 A General Cofiber Sequence: Reduced Version 68 5.1 Reduced Universal Monoids and Reduced Universal Groups . 68 5.2 Necessary Changes for Reduced Version . 71 6 Generalization of the Constructions of Carlsson and Wu 75 6.1 Action Categories . 75 6.2 The Borel Construction . 77 6.3 A Generalized Simplicial Monoid Construction . 79 6.4 A Cofiber Sequence Involving the Borel Construction . 84 7 The Homology of Carlsson's Construction and Its Nerve 90 7.1 The Long Exact Sequence in Cohomology . 90 7.2 Sections of the Orbit Projection . 93 Contents vii 7.3 Actions Free Away from the Basepoint . 95 7.4 Models for the Group Ring of Carlsson's Construction . 97 8 The Lower Central Series Spectral Sequence 102 8.1 The Spectral Sequence Associated to a Filtration . 103 8.2 Residual Nilpotence . 104 8.3 Proof of Main Theorem . 109 9 The Augmentation Ideal Filtration Spectral Sequence 112 9.1 The Augmentation Ideal . 113 9.2 Algebra Models for the E0 Term . 114 9.3 Collapse at the E1 Term . 119 9.4 The Mayer-Vietoris Spectral Sequence . 123 9.5 Homology Decomposition of the Pinched Subset . 125 9.6 An Example . 133 10 The Word Length Filtration Spectral Sequence 137 10.1 James' Construction . 137 10.2 Wu's Construction . 139 Bibliography 141 Summary We explore a simplicial group construction of Carlsson [Car84] in this Thesis. This exploration takes both a macroscopic and a microscopic character. On one hand, we provide a deep conceptual explanation of Carlsson's construction and its geometric realization. On the other hand, we compute in detail the mod 2 homology of Carlsson's construction in the case of actions by Z2, the discrete group with two elements. In order to understand Carlsson's construction conceptually, we use the machinery of category theory and adjoint functors. We describe Carlsson's construction as the universal monoid on the action category. To compute the mod 2 homology of Carlsson's construction in detail, we rely on the machinery of spectral sequences. We utilize the algebraic nature of Carlsson's group construction to create filtrations. These filtrations in turn define spectral sequences which we analyze to obtain information about the homology of Carlsson's construction. Throughout this thesis, we emphasize the applications of our work to equiv- ariant homology. We hope to indicate how category theory, simplicial homotopy theory and spectral sequences can yield fruitful applications to the field of equiv- ariant homology. viii List of Categories Category Description ∆ Finite ordered sets and order-preserving maps ∆ # X Simplex category of a simplicial set X Cat Small categories and functors Grpd Small groupoids and functors Mon Monoids and monoid homomorphisms Grp Groups and group homomorphisms MonAct Monoid actions and morphisms of monoid actions Cop Opposite category of C PtC Pointed objects in a category C and basepoint-preserving maps sC Category of simplicial objects in C and simplicial maps s2Set Bisimplicial sets and bisimplicial maps ix Chapter 1 Introduction \Live as if you were to die tomorrow. Learn as if you were to live forever."|Gandhi. This thesis begins with a simplicial group construction of Carlsson. In the pa- per [Car84], Carlsson introduced a simplicial group whose homotype type is the loop space of a certain cofiber. This is the cofiber of the inclusion of a simpli- cial G-set into its Borel construction. Carlsson understood his construction as an equivariant generalization of James' construction, as his notation JG(X) indicates. However, this is wrong as Peter J. Eccles pointed out that Carlsson's construction actually generalizes Milnor's construction in his review of [Car84] (see MR0721454 (85e:55019)). In Example 6.3.7, we will see that for the trivial action of the discrete group of integers, then Carlsson construction degenerates into Milnor's construc- tion. Carlsson stated his results for actions of a discrete simplicial group G. However, his methods actually work for actions of a general simplicial group. 1 1.1 Applications of Carlsson's Construction 2 1.1 Applications of Carlsson's Construction There are several applications of Carlsson's construction. Firstly, Carlsson's con- struction is a simplicial group that models certain spaces of interest in homotopy theory. For instance, to justify his construction, Carlsson himself noted that his construction models certain quotients of a Thom complex by its lowest cells. In particular, all stunted projective spaces are modeled. Carlsson had hoped that the spectral sequence associated to the restricted mod p lower central series of his con- struction can give some information about the homotopy of these spaces. However studying this spectral sequence is not easy as Carlsson's construction is not a free simplicial group. We have some partial results on the weak convergence of lower central series spectral sequence of Carlsson's construction in Section 2 Chapter 8. Using the fact that Milnor's construction is the Kan construction on the suspension space, we also obtain some generic results regarding this spectral sequence. See Section 3 Chapter 8. Secondly, Carlsson's construction has applications to other areas of simplicial homotopy theory. For instance, Carlsson's construction has been applied to give combinatorial descriptions of homotopy groups and to minimal simplicial sets (See [Wu00], [MW09], [MW11] and [MPW11]). However, mainly only Carlsson's con- structions of trivial actions are used in such applications. We would like to point out a third way in which Carlsson's construction is inter- esting.
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