Algebraic and Combinatorial Codimension–1 Transversality Andrew Ranicki
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On the Semicontinuity of the Mod 2 Spectrum of Hypersurface Singularities
1 ON THE SEMICONTINUITY OF THE MOD 2 SPECTRUM OF HYPERSURFACE SINGULARITIES Andrew Ranicki (Edinburgh) http://www.maths.ed.ac.uk/eaar Bill Bruce 60 and Terry Wall 75 Liverpool, 18-22 June, 2012 2 A mathematical family tree Christopher Zeeman Frank Adams Terry Wall Andrew Casson Andrew Ranicki 3 The BNR project on singularities and surgery I Since 2011 have joined Andr´asN´emethi(Budapest) and Maciej Borodzik (Warsaw) in a project on the topological properties of the singularities of complex hypersurfaces. I The aim of the project is to study the topological properties of the singularity spectrum, defined using refinements of the eigenvalues of the monodromy of the Milnor fibre. I The project combines singularity techniques with algebraic surgery theory to study the behaviour of the spectrum under deformations. I Morse theory decomposes cobordisms of manifolds into elementary operations called surgeries. I Algebraic surgery does the same for cobordisms of chain complexes with Poincar´eduality { generalized quadratic forms. I The applications to singularities need a relative Morse theory, for cobordisms of manifolds with boundary and the algebraic analogues. 4 Fibred links I A link is a codimension 2 submanifold Lm ⊂ Sm+2 with neighbourhood L × D2 ⊂ Sm+2. I The complement of the link is the (m + 2)-dimensional manifold with boundary (C;@C) = (cl.(Sm+2nL × D2); L × S1) such that m+2 2 S = L × D [L×S1 C : I The link is fibred if the projection @C = L × S1 ! S1 can be extended to the projection of a fibre bundle p : C ! S1, and there is given a particular choice of extension. -
The Fundamental Group
The Fundamental Group Tyrone Cutler July 9, 2020 Contents 1 Where do Homotopy Groups Come From? 1 2 The Fundamental Group 3 3 Methods of Computation 8 3.1 Covering Spaces . 8 3.2 The Seifert-van Kampen Theorem . 10 1 Where do Homotopy Groups Come From? 0 Working in the based category T op∗, a `point' of a space X is a map S ! X. Unfortunately, 0 the set T op∗(S ;X) of points of X determines no topological information about the space. The same is true in the homotopy category. The set of `points' of X in this case is the set 0 π0X = [S ;X] = [∗;X]0 (1.1) of its path components. As expected, this pointed set is a very coarse invariant of the pointed homotopy type of X. How might we squeeze out some more useful information from it? 0 One approach is to back up a step and return to the set T op∗(S ;X) before quotienting out the homotopy relation. As we saw in the first lecture, there is extra information in this set in the form of track homotopies which is discarded upon passage to [S0;X]. Recall our slogan: it matters not only that a map is null homotopic, but also the manner in which it becomes so. So, taking a cue from algebraic geometry, let us try to understand the automorphism group of the zero map S0 ! ∗ ! X with regards to this extra structure. If we vary the basepoint of X across all its points, maybe it could be possible to detect information not visible on the level of π0. -
1. CW Complex 1.1. Cellular Complex. Given a Space X, We Call N-Cell a Subspace That Is Home- Omorphic to the Interior of an N-Disk
1. CW Complex 1.1. Cellular Complex. Given a space X, we call n-cell a subspace that is home- omorphic to the interior of an n-disk. Definition 1.1. Given a Hausdorff space X, we call cellular structure on X a n collection of disjoint cells of X, say ffeαgα2An gn2N (where the An are indexing sets), together with a collection of continuous maps called characteristic maps, say n n n k ffΦα : Dα ! Xgα2An gn2N, such that, if X := feαj0 ≤ k ≤ ng (for all n), then : S S n (1) X = ( eα), n2N α2An n n n (2)Φ α Int(Dn) is a homeomorphism of Int(D ) onto eα, n n F k (3) and Φα(@D ) ⊂ eα. k n−1 eα2X We shall call X together with a cellular structure a cellular complex. n n n Remark 1.2. We call X ⊂ feαg the n-skeleton of X. Also, we shall identify X S k n n with eα and vice versa for simplicity of notations. So (3) becomes Φα(@Dα) ⊂ k n eα⊂X S Xn−1. k k n Since X is a Hausdorff space (in the definition above), we have that eα = Φα(D ) k n k (where the closure is taken in X). Indeed, we easily have that Φα(D ) ⊂ eα by n n continuity of Φα. Moreover, since D is compact, we have that its image under k k n k Φα also is, and since X is Hausdorff, Φα(D ) is a closed subset containing eα. Hence we have the other inclusion (Remember that the closure of a subset is the k intersection of all closed subset containing it). -
Homotopy Theory of Diagrams and Cw-Complexes Over a Category
Can. J. Math.Vol. 43 (4), 1991 pp. 814-824 HOMOTOPY THEORY OF DIAGRAMS AND CW-COMPLEXES OVER A CATEGORY ROBERT J. PIACENZA Introduction. The purpose of this paper is to introduce the notion of a CW com plex over a topological category. The main theorem of this paper gives an equivalence between the homotopy theory of diagrams of spaces based on a topological category and the homotopy theory of CW complexes over the same base category. A brief description of the paper goes as follows: in Section 1 we introduce the homo topy category of diagrams of spaces based on a fixed topological category. In Section 2 homotopy groups for diagrams are defined. These are used to define the concept of weak equivalence and J-n equivalence that generalize the classical definition. In Section 3 we adapt the classical theory of CW complexes to develop a cellular theory for diagrams. In Section 4 we use sheaf theory to define a reasonable cohomology theory of diagrams and compare it to previously defined theories. In Section 5 we define a closed model category structure for the homotopy theory of diagrams. We show this Quillen type ho motopy theory is equivalent to the homotopy theory of J-CW complexes. In Section 6 we apply our constructions and results to prove a useful result in equivariant homotopy theory originally proved by Elmendorf by a different method. 1. Homotopy theory of diagrams. Throughout this paper we let Top be the carte sian closed category of compactly generated spaces in the sense of Vogt [10]. -
Noncommutative Localization in Algebra and Topology
Noncommutative localization in algebra and topology ICMS Edinburgh 2002 Edited by Andrew Ranicki Electronic version of London Mathematical Society Lecture Note Series 330 Cambridge University Press (2006) Contents Dedication . vii Preface . ix Historical Perspective . x Conference Participants . xi Conference Photo . .xii Conference Timetable . xiii On atness and the Ore condition J. A. Beachy ......................................................1 Localization in general rings, a historical survey P. M. Cohn .......................................................5 Noncommutative localization in homotopy theory W. G. Dwyer . 24 Noncommutative localization in group rings P. A. Linnell . 40 A non-commutative generalisation of Thomason's localisation theorem A. Neeman . 60 Noncommutative localization in topology A. A. Ranicki . 81 v L2-Betti numbers, Isomorphism Conjectures and Noncommutative Lo- calization H. Reich . 103 Invariants of boundary link cobordism II. The Blanch¯eld-Duval form D. Sheiham . 143 Noncommutative localization in noncommutative geometry Z. Skoda· ........................................................220 vi Dedicated to the memory of Desmond Sheiham (13th November 1974 ¡ 25th March 2005) ² Cambridge University (Trinity College), 1993{1997 B.A. Hons. Mathematics 1st Class, 1996 Part III Mathematics, Passed with Distinction, 1997 ² University of Edinburgh, 1997{2001 Ph.D. Invariants of Boundary Link Cobordism, 2001 ² Visiting Assistant Professor, Mathematics Department, University of California at Riverside, 2001{2003 ² Research Instructor, International University Bremen (IUB), 2003{2005 vii Publications: 1. Non-commutative Characteristic Polynomials and Cohn Localization Journal of the London Mathematical Society (2) Vol. 64, 13{28 (2001) http://arXiv.org/abs/math.RA/0104158 2. Invariants of Boundary Link Cobordism Memoirs of the American Mathematical Society, Vol. 165 (2003) http://arXiv.org/abs/math.AT/0110249 3. Whitehead Groups of Localizations and the Endomorphism Class Group Journal of Algebra, Vol. -
1 CW Complex, Cellular Homology/Cohomology
Our goal is to develop a method to compute cohomology algebra and rational homotopy group of fiber bundles. 1 CW complex, cellular homology/cohomology Definition 1. (Attaching space with maps) Given topological spaces X; Y , closed subset A ⊂ X, and continuous map f : A ! y. We define X [f Y , X t Y / ∼ n n n−1 n where x ∼ y if x 2 A and f(x) = y. In the case X = D , A = @D = S , D [f X is said to be obtained by attaching to X the cell (Dn; f). n−1 n n Proposition 1. If f; g : S ! X are homotopic, then D [f X and D [g X are homotopic. Proof. Let F : Sn−1 × I ! X be the homotopy between f; g. Then in fact n n n D [f X ∼ (D × I) [F X ∼ D [g X Definition 2. (Cell space, cell complex, cellular map) 1. A cell space is a topological space obtained from a finite set of points by iterating the procedure of attaching cells of arbitrary dimension, with the condition that only finitely many cells of each dimension are attached. 2. If each cell is attached to cells of lower dimension, then the cell space X is called a cell complex. Define the n−skeleton of X to be the subcomplex consisting of cells of dimension less than n, denoted by Xn. 3. A continuous map f between cell complexes X; Y is called cellular if it sends Xk to Yk for all k. Proposition 2. 1. Every cell space is homotopic to a cell complex. -
INTRODUCTION to ALGEBRAIC TOPOLOGY 1 Category And
INTRODUCTION TO ALGEBRAIC TOPOLOGY (UPDATED June 2, 2020) SI LI AND YU QIU CONTENTS 1 Category and Functor 2 Fundamental Groupoid 3 Covering and fibration 4 Classification of covering 5 Limit and colimit 6 Seifert-van Kampen Theorem 7 A Convenient category of spaces 8 Group object and Loop space 9 Fiber homotopy and homotopy fiber 10 Exact Puppe sequence 11 Cofibration 12 CW complex 13 Whitehead Theorem and CW Approximation 14 Eilenberg-MacLane Space 15 Singular Homology 16 Exact homology sequence 17 Barycentric Subdivision and Excision 18 Cellular homology 19 Cohomology and Universal Coefficient Theorem 20 Hurewicz Theorem 21 Spectral sequence 22 Eilenberg-Zilber Theorem and Kunneth¨ formula 23 Cup and Cap product 24 Poincare´ duality 25 Lefschetz Fixed Point Theorem 1 1 CATEGORY AND FUNCTOR 1 CATEGORY AND FUNCTOR Category In category theory, we will encounter many presentations in terms of diagrams. Roughly speaking, a diagram is a collection of ‘objects’ denoted by A, B, C, X, Y, ··· , and ‘arrows‘ between them denoted by f , g, ··· , as in the examples f f1 A / B X / Y g g1 f2 h g2 C Z / W We will always have an operation ◦ to compose arrows. The diagram is called commutative if all the composite paths between two objects ultimately compose to give the same arrow. For the above examples, they are commutative if h = g ◦ f f2 ◦ f1 = g2 ◦ g1. Definition 1.1. A category C consists of 1◦. A class of objects: Obj(C) (a category is called small if its objects form a set). We will write both A 2 Obj(C) and A 2 C for an object A in C. -
MY MOTHER TEOFILA REICH-RANICKI Andrew Ranicki Edinburgh German Circle, 28Th February, 2012 2 Timeline
1 MY MOTHER TEOFILA REICH-RANICKI Andrew Ranicki Edinburgh German Circle, 28th February, 2012 2 Timeline I 12th March, 1920. Born inL´od´z,Poland. I 22nd July, 1942. Married to Marcel in the Warsaw Ghetto. I 3rd February, 1943. Marcel and Tosia escape from Ghetto. I 1943-1944 Hidden by Polish family near Warsaw. I 7th September, 1944. Liberation by Red Army. I 1948-1949 London: Marcel is a Polish diplomat. I 30th December, 1948. Son Andrew born in London. I 1949-1958 Warsaw: Marcel writes about German literature. I 1958. Move from Poland to Germany. I 1959-1973 Hamburg: Marcel at Die Zeit. I 1973- Frankfurt a.M.: Marcel at the Frankfurter Allgemeine Zeitung. "Pope of German literature". I 1999 Exhibition of the Warsaw Ghetto drawings. I 29th April, 2011. Tosia dies in Frankfurt a.M. 3 Tosia's names I Tosia = Polish diminutive of Teofila. I 1920-1942 Teofila Langnas, maiden name. I 1942-1945 Teofila Reich, on marriage to Marcel Reich. I 1945-1958 Teofila Ranicki, after Marcel changes name to Ranicki, as more suitable for a Polish diplomat than Reich. I 1958- Teofila Reich-Ranicki, after Marcel changes name to Reich-Ranicki on return to Germany (where he had gone to school). 4 Tosia's parents Father: Pawe l Langnas, 1885-1940 Mother: Emilia Langnas, 1886-1942 5 L´od´z,1927-1933 I German school. I 6 L´od´z,1933-1939 I Polish school. I Interested in art, cinema, literature { but not mathematics! I Graduation photo 7 21st January, 1940 I Tosia was accepted for studying art at an Ecole´ des Beaux Arts, Paris, to start on 1st September, 1939. -
Generalized Cohomology Theories
Lecture 4: Generalized cohomology theories 1/12/14 We've now defined spectra and the stable homotopy category. They arise naturally when considering cohomology. Proposition 1.1. For X a finite CW-complex, there is a natural isomorphism 1 ∼ r [Σ X; HZ]−r = H (X; Z). The assumption that X is a finite CW-complex is not necessary, but here is a proof in this case. We use the following Lemma. Lemma 1.2. ([A, III Prop 2.8]) Let F be any spectrum. For X a finite CW- 1 n+r complex there is a natural identification [Σ X; F ]r = colimn!1[Σ X; Fn] n+r On the right hand side the colimit is taken over maps [Σ X; Fn] ! n+r+1 n+r [Σ X; Fn+1] which are the composition of the suspension [Σ X; Fn] ! n+r+1 n+r+1 n+r+1 [Σ X; ΣFn] with the map [Σ X; ΣFn] ! [Σ X; Fn+1] induced by the structure map of F ΣFn ! Fn+1. n+r Proof. For a map fn+r :Σ X ! Fn, there is a pmap of degree r of spectra Σ1X ! F defined on the cofinal subspectrum whose mth space is ΣmX for m−n−r m ≥ n+r and ∗ for m < n+r. This pmap is given by Σ fn+r for m ≥ n+r 0 n+r and is the unique map from ∗ for m < n+r. Moreover, if fn+r; fn+r :Σ X ! 1 Fn are homotopic, we may likewise construct a pmap Cyl(Σ X) ! F of degree n+r 1 r. -
Homology of Cell Complexes
Geometry and Topology of Manifolds 2015{2016 Homology of Cell Complexes Cell Complexes Let us denote by En the closed n-ball, whose boundary is the sphere Sn−1.A cell complex (or CW-complex) is a topological space X equipped with a filtration by subspaces X(0) ⊆ X(1) ⊆ X(2) ⊆ · · · ⊆ X(n) ⊆ · · · (1) (0) 1 (n) where X is discrete (i.e., has the discrete topology), such that X = [n=0X as a topological space, and, for all n, the space X(n) is obtained from X(n−1) by attaching n−1 (n−1) a set of n-cells by means of a family of maps f'j : S ! X gj2Jn . Here Jn is any set of indices and X(n) is therefore a quotient of the disjoint union X(n−1) ` En j2Jn n−1 (n−1) where we identify, for each point x 2 S , the image 'j(x) 2 X with the point x in the j-th copy of En. It is customary to denote (n) (n−1) n X = X [f'j g fej g n (n) and view each ej as an \open n-cell". The space X is called the n-skeleton of X, and the filtration (1) together with the attaching maps f'jgj2Jn for all n is called a cell decomposition or CW-decomposition of X. Thus a topological space may admit many distinct cell decompositions (or none). Cellular Chain Complexes (n) Let X be a cell complex with n-skeleton X and attaching maps f'jgj2Jn for all n. -
Contemporary Mathematics 288
CONTEMPORARY MATHEMATICS 288 Global Differential Geometry: The Mathematical Legacy ·of Alfred Gray International Congress on Differential Geometry September 18-23, 2000 Bilbao, Spain Marisa Fernandez Joseph A. Wolf Editors http://dx.doi.org/10.1090/conm/288 Selected Titles In This Series 288 Marisa Fernandez and Joseph A. Wolf, Editors, Global differential geometry: The mathematical legacy of Alfred Gray, 2001 287 Marlos A. G. Viana and Donald St. P. Richards, Editors, Algebraic methods in statistics and probability, 2001 286 Edward L. Green, Serkan HO!~ten, Reinhard C. Laubenbacher, and Victoria Ann Powers, Editors, Symbolic computation: Solving equations in algebra, geometry, and engineering, 2001 285 Joshua A. Leslie and Thierry P. Robart, Editors, The geometrical study of differential equations, 2001 284 Gaston M. N'Guerekata and Asamoah Nkwanta, Editors, Council for African American researchers in the mathematical sciences: Volume IV, 2001 283 Paul A. Milewski, Leslie M. Smith, Fabian Waleffe, and Esteban G. Tabak, Editors, Advances in wave interaction and turbulence, 2001 282 Arlan Ramsay and Jean Renault, Editors, Gruupoids in analysiR, gcnmetry, and physics, 2001 281 Vadim Olshevsky, Editor, Structured matrices in mathematics, computer science, and engineering II, 2001 280 Vadim Olshevsky, Editor, Structured matrices in mathematics, computer science, and engineering I, 2001 279 Alejandro Adem, Gunnar Carlsson, and Ralph Cohen, Editors, Topology, geometry, and algebra: Interactions and new directions, 2001 278 Eric Todd Quinto, Leon Ehrenpreis, Adel Faridani, Fulton Gonzalez, and Eric Grinberg, Editors, Radon transforms and tomography, 2001 277 Luca Capogna and Loredana Lanzani, Editors, Harmonic analysis and boundary value problems, 2001 276 Emma Previato, Editor, Advances in algebraic geometry motivated by physics, 2001 275 Alfred G. -
Curriculum Vitae of Andrew Ranicki Born
Curriculum vitae of Andrew Ranicki Born: London, 30th December, 1948 Dual UK / German citizen Cambridge University, B. A. 1969 Ph. D. 1973 (supervised by Prof. J. F. Adams and Mr. A. J. Casson) Trinity College Yeats Prize, 1970 Cambridge University Smith Prize, 1972 Research Fellow of Trinity College, Cambridge, 1972{1977 Visiting Member, Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette, France, 1973{1974 Princeton University : Instructor, 1977{1978 Assistant Professor, 1978{1982 Visiting Member, Institute for Advanced Studies, Princeton, 1981{1982 Edinburgh University : Lecturer, 1982{1987 Reader, 1987{1995 Professor of Algebraic Surgery, 1995{2016 Honorary Professorial Fellow, 2017{ Junior Whitehead Prize of London Mathematical Society, 1983 Visiting Professor, University of Kentucky, Lexington, 1985{1986 Visiting Member, SFB 170, G¨ottingenUniversity, 1987{1988 Fellow, Royal Society of Edinburgh, 1992 Senior Berwick Prize of London Mathematical Society, 1994 Leverhulme Trust Fellowship, 2001 Co-Organizer, Oberwolfach Conference on Topology, 1984{1991 British Topology Meeting, Edinburgh, 1985, 2001, 2011 Oberwolfach Conference on Surgery and L-theory, 1985, 1990 Oberwolfach Conference on the Novikov conjecture, 1993 Conference on Geometric Topology and Surgery, Josai, Japan, 1996 Special Session at AMS Regional Meeting, Memphis, Tennessee, 1997 Conference on Quadratic Forms and Their Applications, Dublin, 1999 Conference on Noncommutative Localization, ICMS, Edinburgh, 2002 Special Session at AMS Regional Meeting, Bloomington,