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CERF THEORY for GRAPHS Allen Hatcher and Karen Vogtmann
CERF THEORY FOR GRAPHS Allen Hatcher and Karen Vogtmann ABSTRACT. We develop a deformation theory for k-parameter families of pointed marked graphs with xed fundamental group Fn. Applications include a simple geometric proof of stability of the rational homology of Aut(Fn), compu- tations of the rational homology in small dimensions, proofs that various natural complexes of free factorizations of Fn are highly connected, and an improvement on the stability range for the integral homology of Aut(Fn). §1. Introduction It is standard practice to study groups by studying their actions on well-chosen spaces. In the case of the outer automorphism group Out(Fn) of the free group Fn on n generators, a good space on which this group acts was constructed in [3]. In the present paper we study an analogous space on which the automorphism group Aut(Fn) acts. This space, which we call An, is a sort of Teichmuller space of nite basepointed graphs with fundamental group Fn. As often happens, specifying basepoints has certain technical advantages. In particu- lar, basepoints give rise to a nice ltration An,0 An,1 of An by degree, where we dene the degree of a nite basepointed graph to be the number of non-basepoint vertices, counted “with multiplicity” (see section 3). The main technical result of the paper, the Degree Theorem, implies that the pair (An, An,k)isk-connected, so that An,k behaves something like the k-skeleton of An. From the Degree Theorem we derive these applications: (1) We show the natural stabilization map Hi(Aut(Fn); Z) → Hi(Aut(Fn+1); Z)isan isomorphism for n 2i + 3, a signicant improvement on the quadratic dimension range obtained in [5]. -
Homological Algebra
Homological Algebra Donu Arapura April 1, 2020 Contents 1 Some module theory3 1.1 Modules................................3 1.6 Projective modules..........................5 1.12 Projective modules versus free modules..............7 1.15 Injective modules...........................8 1.21 Tensor products............................9 2 Homology 13 2.1 Simplicial complexes......................... 13 2.8 Complexes............................... 15 2.15 Homotopy............................... 18 2.23 Mapping cones............................ 19 3 Ext groups 21 3.1 Extensions............................... 21 3.11 Projective resolutions........................ 24 3.16 Higher Ext groups.......................... 26 3.22 Characterization of projectives and injectives........... 28 4 Cohomology of groups 32 4.1 Group cohomology.......................... 32 4.6 Bar resolution............................. 33 4.11 Low degree cohomology....................... 34 4.16 Applications to finite groups..................... 36 4.20 Topological interpretation...................... 38 5 Derived Functors and Tor 39 5.1 Abelian categories.......................... 39 5.13 Derived functors........................... 41 5.23 Tor functors.............................. 44 5.28 Homology of a group......................... 45 1 6 Further techniques 47 6.1 Double complexes........................... 47 6.7 Koszul complexes........................... 49 7 Applications to commutative algebra 52 7.1 Global dimensions.......................... 52 7.9 Global dimension of -
Stable Homology of Automorphism Groups of Free Groups
Annals of Mathematics 173 (2011), 705{768 doi: 10.4007/annals.2011.173.2.3 Stable homology of automorphism groups of free groups By Søren Galatius Abstract Homology of the group Aut(Fn) of automorphisms of a free group on n generators is known to be independent of n in a certain stable range. Using tools from homotopy theory, we prove that in this range it agrees with homology of symmetric groups. In particular we confirm the conjecture that stable rational homology of Aut(Fn) vanishes. Contents 1. Introduction 706 1.1. Results 706 1.2. Outline of proof 708 1.3. Acknowledgements 710 2. The sheaf of graphs 710 2.1. Definitions 710 2.2. Point-set topological properties 714 3. Homotopy types of graph spaces 718 3.1. Graphs in compact sets 718 3.2. Spaces of graph embeddings 721 3.3. Abstract graphs 727 3.4. BOut(Fn) and the graph spectrum 730 4. The graph cobordism category 733 4.1. Poset model of the graph cobordism category 735 4.2. The positive boundary subcategory 743 5. Homotopy type of the graph spectrum 750 5.1. A pushout diagram 751 5.2. A homotopy colimit decomposition 755 5.3. Hatcher's splitting 762 6. Remarks on manifolds 764 References 766 S. Galatius was supported by NSF grants DMS-0505740 and DMS-0805843, and the Clay Mathematics Institute. 705 706 SØREN GALATIUS 1. Introduction 1.1. Results. Let F = x ; : : : ; x be the free group on n generators n h 1 ni and let Aut(Fn) be its automorphism group. -
Unknot Recognition Through Quantifier Elimination
UNKNOT RECOGNITION THROUGH QUANTIFIER ELIMINATION SYED M. MEESUM AND T. V. H. PRATHAMESH Abstract. Unknot recognition is one of the fundamental questions in low dimensional topology. In this work, we show that this problem can be encoded as a validity problem in the existential fragment of the first-order theory of real closed fields. This encoding is derived using a well-known result on SU(2) representations of knot groups by Kronheimer-Mrowka. We further show that applying existential quantifier elimination to the encoding enables an UnKnot Recogntion algorithm with a complexity of the order 2O(n), where n is the number of crossings in the given knot diagram. Our algorithm is simple to describe and has the same runtime as the currently best known unknot recognition algorithms. 1. Introduction In mathematics, a knot refers to an entangled loop. The fundamental problem in the study of knots is the question of knot recognition: can two given knots be transformed to each other without involving any cutting and pasting? This problem was shown to be decidable by Haken in 1962 [6] using the theory of normal surfaces. We study the special case in which we ask if it is possible to untangle a given knot to an unknot. The UnKnot Recogntion recognition algorithm takes a knot presentation as an input and answers Yes if and only if the given knot can be untangled to an unknot. The best known complexity of UnKnot Recogntion recognition is 2O(n), where n is the number of crossings in a knot diagram [2, 6]. More recent developments show that the UnKnot Recogntion recogni- tion is in NP \ co-NP. -
Computational Topology and the Unique Games Conjecture
Computational Topology and the Unique Games Conjecture Joshua A. Grochow1 Department of Computer Science & Department of Mathematics University of Colorado, Boulder, CO, USA [email protected] https://orcid.org/0000-0002-6466-0476 Jamie Tucker-Foltz2 Amherst College, Amherst, MA, USA [email protected] Abstract Covering spaces of graphs have long been useful for studying expanders (as “graph lifts”) and unique games (as the “label-extended graph”). In this paper we advocate for the thesis that there is a much deeper relationship between computational topology and the Unique Games Conjecture. Our starting point is Linial’s 2005 observation that the only known problems whose inapproximability is equivalent to the Unique Games Conjecture – Unique Games and Max-2Lin – are instances of Maximum Section of a Covering Space on graphs. We then observe that the reduction between these two problems (Khot–Kindler–Mossel–O’Donnell, FOCS ’04; SICOMP ’07) gives a well-defined map of covering spaces. We further prove that inapproximability for Maximum Section of a Covering Space on (cell decompositions of) closed 2-manifolds is also equivalent to the Unique Games Conjecture. This gives the first new “Unique Games-complete” problem in over a decade. Our results partially settle an open question of Chen and Freedman (SODA, 2010; Disc. Comput. Geom., 2011) from computational topology, by showing that their question is almost equivalent to the Unique Games Conjecture. (The main difference is that they ask for inapproxim- ability over Z2, and we show Unique Games-completeness over Zk for large k.) This equivalence comes from the fact that when the structure group G of the covering space is Abelian – or more generally for principal G-bundles – Maximum Section of a G-Covering Space is the same as the well-studied problem of 1-Homology Localization. -
Topology and Data
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 46, Number 2, April 2009, Pages 255–308 S 0273-0979(09)01249-X Article electronically published on January 29, 2009 TOPOLOGY AND DATA GUNNAR CARLSSON 1. Introduction An important feature of modern science and engineering is that data of various kinds is being produced at an unprecedented rate. This is so in part because of new experimental methods, and in part because of the increase in the availability of high powered computing technology. It is also clear that the nature of the data we are obtaining is significantly different. For example, it is now often the case that we are given data in the form of very long vectors, where all but a few of the coordinates turn out to be irrelevant to the questions of interest, and further that we don’t necessarily know which coordinates are the interesting ones. A related fact is that the data is often very high-dimensional, which severely restricts our ability to visualize it. The data obtained is also often much noisier than in the past and has more missing information (missing data). This is particularly so in the case of biological data, particularly high throughput data from microarray or other sources. Our ability to analyze this data, both in terms of quantity and the nature of the data, is clearly not keeping pace with the data being produced. In this paper, we will discuss how geometry and topology can be applied to make useful contributions to the analysis of various kinds of data. -
11/29/2011 Principal Investigator: Margalit, Dan
Final Report: 1122020 Final Report for Period: 09/2010 - 08/2011 Submitted on: 11/29/2011 Principal Investigator: Margalit, Dan . Award ID: 1122020 Organization: Georgia Tech Research Corp Submitted By: Margalit, Dan - Principal Investigator Title: Algebra and topology of the Johnson filtration Project Participants Senior Personnel Name: Margalit, Dan Worked for more than 160 Hours: Yes Contribution to Project: Post-doc Graduate Student Undergraduate Student Technician, Programmer Other Participant Research Experience for Undergraduates Organizational Partners Other Collaborators or Contacts Mladen Bestvina, University of Utah Kai-Uwe Bux, Bielefeld University Tara Brendle, University of Glasgow Benson Farb, University of Chicago Chris Leininger, University of Illinois at Urbana-Champaign Saul Schleimer, University of Warwick Andy Putman, Rice University Leah Childers, Pittsburg State University Allen Hatcher, Cornell University Activities and Findings Research and Education Activities: Proved various theorems, listed below. Wrote papers explaining the results. Wrote involved computer program to help explore the quotient of Outer space by the Torelli group for Out(F_n). Gave lectures on the subject of this Project at a conference for graduate students called 'Examples of Groups' at Ohio State University, June Page 1 of 5 Final Report: 1122020 2008. Mentored graduate students at the University of Utah in the subject of mapping class groups. Interacted with many students about the book I am writing with Benson Farb on mapping class groups. Gave many talks on this work. Taught a graduate course on my research area, using a book that I translated with Djun Kim. Organized a special session on Geometric Group Theory at MathFest. Finished book with Benson Farb, published by Princeton University Press. -
Math 601 Algebraic Topology Hw 4 Selected Solutions Sketch/Hint
MATH 601 ALGEBRAIC TOPOLOGY HW 4 SELECTED SOLUTIONS SKETCH/HINT QINGYUN ZENG 1. The Seifert-van Kampen theorem 1.1. A refinement of the Seifert-van Kampen theorem. We are going to make a refinement of the theorem so that we don't have to worry about that openness problem. We first start with a definition. Definition 1.1 (Neighbourhood deformation retract). A subset A ⊆ X is a neighbourhood defor- mation retract if there is an open set A ⊂ U ⊂ X such that A is a strong deformation retract of U, i.e. there exists a retraction r : U ! A and r ' IdU relA. This is something that is true most of the time, in sufficiently sane spaces. Example 1.2. If Y is a subcomplex of a cell complex, then Y is a neighbourhood deformation retract. Theorem 1.3. Let X be a space, A; B ⊆ X closed subspaces. Suppose that A, B and A \ B are path connected, and A \ B is a neighbourhood deformation retract of A and B. Then for any x0 2 A \ B. π1(X; x0) = π1(A; x0) ∗ π1(B; x0): π1(A\B;x0) This is just like Seifert-van Kampen theorem, but usually easier to apply, since we no longer have to \fatten up" our A and B to make them open. If you know some sheaf theory, then what Seifert-van Kampen theorem really says is that the fundamental groupoid Π1(X) is a cosheaf on X. Here Π1(X) is a category with object pints in X and morphisms as homotopy classes of path in X, which can be regard as a global version of π1(X). -
The Fundamental Group and Seifert-Van Kampen's
THE FUNDAMENTAL GROUP AND SEIFERT-VAN KAMPEN'S THEOREM KATHERINE GALLAGHER Abstract. The fundamental group is an essential tool for studying a topo- logical space since it provides us with information about the basic shape of the space. In this paper, we will introduce the notion of free products and free groups in order to understand Seifert-van Kampen's Theorem, which will prove to be a useful tool in computing fundamental groups. Contents 1. Introduction 1 2. Background Definitions and Facts 2 3. Free Groups and Free Products 4 4. Seifert-van Kampen Theorem 6 Acknowledgments 12 References 12 1. Introduction One of the fundamental questions in topology is whether two topological spaces are homeomorphic or not. To show that two topological spaces are homeomorphic, one must construct a continuous function from one space to the other having a continuous inverse. To show that two topological spaces are not homeomorphic, one must show there does not exist a continuous function with a continuous inverse. Both of these tasks can be quite difficult as the recently proved Poincar´econjecture suggests. The conjecture is about the existence of a homeomorphism between two spaces, and it took over 100 years to prove. Since the task of showing whether or not two spaces are homeomorphic can be difficult, mathematicians have developed other ways to solve this problem. One way to solve this problem is to find a topological property that holds for one space but not the other, e.g. the first space is metrizable but the second is not. Since many spaces are similar in many ways but not homeomorphic, mathematicians use a weaker notion of equivalence between spaces { that of homotopy equivalence. -
A TEXTBOOK of TOPOLOGY Lltld
SEIFERT AND THRELFALL: A TEXTBOOK OF TOPOLOGY lltld SEI FER T: 7'0PO 1.OG 1' 0 I.' 3- Dl M E N SI 0 N A I. FIRERED SPACES This is a volume in PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks Editors: SAMUELEILENBERG AND HYMANBASS A list of recent titles in this series appears at the end of this volunie. SEIFERT AND THRELFALL: A TEXTBOOK OF TOPOLOGY H. SEIFERT and W. THRELFALL Translated by Michael A. Goldman und S E I FE R T: TOPOLOGY OF 3-DIMENSIONAL FIBERED SPACES H. SEIFERT Translated by Wolfgang Heil Edited by Joan S. Birman and Julian Eisner @ 1980 ACADEMIC PRESS A Subsidiary of Harcourr Brace Jovanovich, Publishers NEW YORK LONDON TORONTO SYDNEY SAN FRANCISCO COPYRIGHT@ 1980, BY ACADEMICPRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER. ACADEMIC PRESS, INC. 11 1 Fifth Avenue, New York. New York 10003 United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London NWI 7DX Mit Genehmigung des Verlager B. G. Teubner, Stuttgart, veranstaltete, akin autorisierte englische Ubersetzung, der deutschen Originalausgdbe. Library of Congress Cataloging in Publication Data Seifert, Herbert, 1897- Seifert and Threlfall: A textbook of topology. Seifert: Topology of 3-dimensional fibered spaces. (Pure and applied mathematics, a series of mono- graphs and textbooks ; ) Translation of Lehrbuch der Topologic. Bibliography: p. Includes index. 1. -
NECESSARY CONDITIONS for DISCONTINUITIES of MULTIDIMENSIONAL SIZE FUNCTIONS Introduction Topological Persistence Is Devoted to T
NECESSARY CONDITIONS FOR DISCONTINUITIES OF MULTIDIMENSIONAL SIZE FUNCTIONS A. CERRI AND P. FROSINI Abstract. Some new results about multidimensional Topological Persistence are presented, proving that the discontinuity points of a k-dimensional size function are necessarily related to the pseudocritical or special values of the associated measuring function. Introduction Topological Persistence is devoted to the study of stable properties of sublevel sets of topological spaces and, in the course of its development, has revealed itself to be a suitable framework when dealing with applications in the field of Shape Analysis and Comparison. Since the beginning of the 1990s research on this sub- ject has been carried out under the name of Size Theory, studying the concept of size function, a mathematical tool able to describe the qualitative properties of a shape in a quantitative way. More precisely, the main idea is to model a shape by a topological space endowed with a continuous function ϕ, called measuring function. Such a functionM is chosen according to applications and can be seen as a descriptor of the features considered relevant for shape characterization. Under these assumptions, the size function ℓ( ,ϕ) associated with the pair ( , ϕ) is a de- scriptor of the topological attributes thatM persist in the sublevel setsM of induced by the variation of ϕ. According to this approach, the problem of comparingM two shapes can be reduced to the simpler comparison of the related size functions. Since their introduction, these shape descriptors have been widely studied and applied in quite a lot of concrete applications concerning Shape Comparison and Pattern Recognition (cf., e.g., [4, 8, 15, 34, 35, 36]). -
New Ideas in Algebraic Topology (K-Theory and Its Applications)
NEW IDEAS IN ALGEBRAIC TOPOLOGY (K-THEORY AND ITS APPLICATIONS) S.P. NOVIKOV Contents Introduction 1 Chapter I. CLASSICAL CONCEPTS AND RESULTS 2 § 1. The concept of a fibre bundle 2 § 2. A general description of fibre bundles 4 § 3. Operations on fibre bundles 5 Chapter II. CHARACTERISTIC CLASSES AND COBORDISMS 5 § 4. The cohomological invariants of a fibre bundle. The characteristic classes of Stiefel–Whitney, Pontryagin and Chern 5 § 5. The characteristic numbers of Pontryagin, Chern and Stiefel. Cobordisms 7 § 6. The Hirzebruch genera. Theorems of Riemann–Roch type 8 § 7. Bott periodicity 9 § 8. Thom complexes 10 § 9. Notes on the invariance of the classes 10 Chapter III. GENERALIZED COHOMOLOGIES. THE K-FUNCTOR AND THE THEORY OF BORDISMS. MICROBUNDLES. 11 § 10. Generalized cohomologies. Examples. 11 Chapter IV. SOME APPLICATIONS OF THE K- AND J-FUNCTORS AND BORDISM THEORIES 16 § 11. Strict application of K-theory 16 § 12. Simultaneous applications of the K- and J-functors. Cohomology operation in K-theory 17 § 13. Bordism theory 19 APPENDIX 21 The Hirzebruch formula and coverings 21 Some pointers to the literature 22 References 22 Introduction In recent years there has been a widespread development in topology of the so-called generalized homology theories. Of these perhaps the most striking are K-theory and the bordism and cobordism theories. The term homology theory is used here, because these objects, often very different in their geometric meaning, Russian Math. Surveys. Volume 20, Number 3, May–June 1965. Translated by I.R. Porteous. 1 2 S.P. NOVIKOV share many of the properties of ordinary homology and cohomology, the analogy being extremely useful in solving concrete problems.