Division Algebras and Parallelizable Spheres III
|
|
- Michael Clark
- 5 years ago
- Views:
Transcription
1 Division Algebras and Parallelizable Spheres III Seminar on Vectorbundles in Algebraic Topology ETH Zürich Ramon Braunwarth May 8, 2018 These are the notes to the talk given on April 23rd 2018 in the Vector Bundles in Algebraic Topology Seminar. In the two talks before, we discussed Adam s Theorem and we used the Splitting Principle to prove it. What remains is proving the Splitting Principle. For this, we first state and prove the Leray-Hirsch theorem for K-theory, which will be useful for proving the Splitting Principle. Theorem 1 (Leray-Hirsch). Let p : E B be a fiber bundle with E and B compact Hausdorff spaces and with fiber F such that K (F ) is free. Suppose that there exist classes c 1,..., c k K (E) that restrict to a basis for K (F ) in each fiber F. If either (a) B is a finite cell complex, or (b) F is a finite cell complex with all cells having even dimension, then K (E) as a module over K (B) is free with basis {c 1,..., c k }. Remark 2. Before we prove this theorem, let us make some preliminary remarks: (i) The module multiplication of two elements β K (B) and γ K (E) is defined by β γ = p (β)γ. (ii) We notice that the conclusion of the theorem is equivalent to saying that for the inclusion map i : F E, the following map is an isomorphism: Φ : K (B) K (F ) K (E), b j i (c j ) p (b j )c j j j (iii) In the case of a trivial bundle E = F B, we can choose the classes c i as the pullbacks of a basis of K (F ) under the projection E F. The proof of the Leray-Hirsch theorem now consists of four parts: First, we define a commutative diagram from two long exact sequences. Second, we use this diagram and the five-lemma to prove that the map Φ defined above is an isomorphism if (a) 1
2 holds. Third, we prove the same statement in case (b) for product bundles. Finally, in the fourth part, we prove (b) in the general case. Proof of Leray-Hirsch Theorem. Part 1: The Commutative Diagramm Consider a subspace B B. Let E := p 1 (B ) be the pullback bundle of B under p. Consider the following diagramm, where Φ is the map from Remark 2: K (B, B ) K (F ) K (B) K (F ) K (B ) K (F ) Diagram ( ) The map Φ rel is defined by the same formula as Φ with the small change that p (b i )c i is now the relative product K (E, E ) K (E) K (E, E ). The right hand side map Φ is defined using the restrictions of the c i to E. Claim: The above diagram commutes and the rows are exact. The bottom row is of course exact and the top row is exact since tensoring with the free module K (F ) replaces an exact sequence by the direct sum of several copies of itself. We show that the diagram commutes by factoring Φ as the composition Φ = Θ Ψ with b i i (c i ) i Ψ i which allows us to add a middle row to ( ): p (b i ) i (c i ) Θ i p (b i )c i, K (B, B ) K (F ) K (B) K (F ) K (B ) K (F ) Ψ rel Ψ Ψ K (E, E ) K (F ) K (E) K (F ) K (E ) K (F ) Θ rel Θ Θ The upper squares commute, because inclusion and pullback commute and the lower squares commute by Proposition 2.15 in [1]. We have thus proven the claim. Part 2: Proof of case (a) We will prove that the map Φ is an isomorphism by showing that both Φ rel and Φ in ( ) are isomorphisms and invoking the five-lemma. The general strategy will be very similar in the following two parts of the proof. The reader may want to look out for the following steps every time: (i) We define a commutative diagram with the appropriate spaces, where we want to use the five-lemma (ii) We define a second diagram sharing a map with the first that can be shown to be an isomorphism this 2
3 way, (iii) We follow our construction backwards to obtain the desired statement. Recall that we assume that B is a finite cell complex. We prove (a) by a double induction on the dimension of B and, with given dimension, on the number of cells in B. For n = 0, B is a finite discrete set, so every vector bundle is trivial and hence the K-rings consist of only one equivalence class. For the induction step, we assume that B is n-dimensional and obtained from a complex B by attaching a n-cell e n. As above, let E := p 1 (B ). By induction on the number of cells in B we may assume that Φ is an isomorphism. In order to show that also Φ rel is an isomorphism, we consider a characteristic map ϕ : (D n, S n 1 ) (B, B ) for the attached cell e n. By Corollary 1.8 in [1] we know that since D n is contractible, the pullback bundle ϕ (E) is trivial, so we obtain the following commutative diagram: K (B, B ) K (F ) K (D n, S n 1 ) K (F ) Φ rel Φ 1 Φ 2 K (E, E ) K (ϕ (E), ϕ (E )) K (D n F, S n 1 F ) The horizontal maps to the left are isomorphisms, since ϕ restricts to a homeomorphism in the interior of D n, hence it induces homeomorphisms B/B D n /S n 1 and E/E ϕ (E)/ϕ (E ). We consider the diagram ( ) with (B, B ) replaced by (D n, S n 1 ): K (D n, S n 1 ) K (F ) K (D n ) K (F ) K (S n 1 ) K (F ) Commutative diagram for (B, B ) = (D n, S n 1 ) ( ) Here, we may assume by induction that Φ is an isomorphism, since S n 1 is of dimension n 1. Since D n deformation retracts to a point, the map Φ is an isomorphism by the zero-dimensional case and Corollary 1.8 in [1]. Therefore we can apply the five lemma to this diagram and we obtain that Φ rel is an isomorphism. Now notice that Φ rel in ( ) is exactly the right hand side Φ 2 in the diagram before. It follows that Φ rel in ( ) is an isomorphism and thus by the five-lemma, also Φ is an isomorphism, which is what we wanted to prove. Part 3: Proof of part (b) for product bundles We first show the statement for the case of a product bundle E = F B. This will then be the basis for the argument in the general case using the local triviality condition. By comparing the formulas from the definition of the external product and Remark 2(ii), we observe that for product bundles Φ is the external product µ : K (B) K (F ) K (E), so we can exchange the roles of B and F in the above diagrams and use a similar rationale as above: We consider the diagram ( ) for an arbitrary compact Hausdorff space F, a finite cell complex B having all cells of even dimension and B being constructed by attaching an n-cell e n to a subcomplex 3
4 B. We again have exact rows in the diagram. Now to the five-lemma part of the proof: We want to show that Φ is an isomorphism. If we can show that Φ rel is an isomorphism in this situation, then by induction we find that Φ is an isomorphism as well. With the five-lemma we obtain that Φ is an isomorphism. Φ rel iso. Ind. = Φ iso. 5 lem. = Φ iso. = Success! We prove that Φ rel is an isomorphism: We note that B/B = S n. Hence, we can replace the pair (B, B ) by the pair (D n, S n 1 ) and use the diagram ( ). The map Φ is an isomorphism since D n deformation retracts to a point. Next, we explain why Φ in the same diagram is an isomorphism: Recall the consequences of Bott Periodicity on page 60 in [1], where we proved that the external product K(S n ) K(X) K(S n X) is an isomorphism for even n, thus also Φ is an isomorphism for even n. For odd n, we replace K 0 (S n ) by K 1 (S n 1 ) and vice versa, so we have an isomorphism by the same argument. It follows that Φ rel is an isomorphism by the five-lemma, which finishes the proof of the product bundle case. Part 4: Proof of part (b) for nonproducts The proof will essentially follow along the lines of the previous two parts with the important difference that B is just a compact Hausdorff space and not a cell complex. We therefore need a more subtle statement for the induction. Let us for this purpose define the following condition (in [1], its name is good, which is really boring): Let U B. If for all compact V U the map Φ : K (V ) K (F ) K (p 1 (V )) is an isomorphism, then U is called sparkly. Since by the local triviality condition each point has a trivial neighbourhood, Part 3 shows that we obtain a covering of B with sparkly sets. As B is compact, finitely many sparkly sets suffice. We thus want to show that for two sparkly sets U 1 and U 2, their union U 1 U 2 is still sparkly. Consider a set V U 1 U 2. We have V = V 1 V 2, where V i = V U i for i {1, 2}. We prove that such a set V is sparkly by using again the commutative diagram from above and the five-lemma. The diagram ( ) for (B, B ) = (V, V 2 ) looks like this: K (V, V 2 ) K (F ) K (V ) K (F ) K (V 2 ) K (F ) Commutative diagram for (V, V 2 ) ( ) Since V 2 U 2 and we assumed U 2 to be sparkly 1, we have that Φ in ( ) is an isomorphism. In order to invoke the five-lemma, we now show that Φ rel in ( ) is an isomorphism as well: The quotient space V/V 2 is homeomorphic to V 1 /(V 1 V 2 ), so we can equvalently show that the map Φ rel for V 1 /(V 1 V 2 ) is an isomorphism. One last time, we need the commutative diagram: 1 Note: We are not claiming that Bono is a vampire! 4
5 K (V 1, V 1 V 2 ) K (F ) K (V 1 ) K (F ) K (V 1 V 2 ) K (F ) Commutative diagram for (V 1, (V 1 V 2 ) ( ) Since we assume that U 1 is sparkly and both V 1 and V 1 V 2 are compact subsets of U 1, we have that both Φ and Φ in ( ) are isomorphisms. Hence, by the five-lemma so is Φ rel in ( ) and thus in ( ). Applying the five-lemma one last time in ( ) yields that U 1 U 2 is sparkly. Thus, by induction we obtain an isomorphism ( k ) Φ : K (B) K (F ) = K U i K (F ) K (E), i=1 which finishes the proof of the Leray-Hirsch theorem. With the Leray-Hirsch theorem proven, we can now focus on proving the splitting principle. Before we restate the Splitting Principle and start its proof, we show an application of the Leray-Hirsch theorem, which will be very usefull in the proof below. Example 3. Let E X be a vector bundle with fibers C n and compact X. We then have an associated projective bundle p : P (E) X with fibers CP n 1, where P (E) is the space of lines in E, i.e. one-dimensional linear subspaces of fibers of E. If we now consider P (E) as our new base space, there is the canonical line bundle L P (E). This L consists of the vectors in the lines in P (E). By Proposition 2.24 in [1], in each fiber CP n 1 of P (E) the classes 1, L,..., L n 1 restrict to a basis for K (CP n 1 ). It now follows from the Leray-Hirsch theorem that K (P (E)) is a free K (X)-module with basis 1, L,..., L n 1. p (E) E P (E) X Pullback in Example 3 Let us remind ourselves of the Splitting Principle: Theorem 4 (Splitting Principle). Given a vector bundle π : E X with X compact Hausdorff, there is a compact Hausdorff space F (E) and a map p : F (E) X such that the induced map p : K (X) K (F (E)) is injective and p splits as a sum of line bundles. The strategy of the following proof is using the projective bundle and the basis for K (P (E)) from Example 3 to iteratively split off orthogonal line bundles from the pullback bundle p (E). From this process we will obtain a flag bundle that will fulfill the conclusion of the theorem. Recall that for an n-dimensional vector bundle E B, the associated flag bundle F (E) B is the n-fold product of P (E) and has fibers consisting of n-tuples of orthogonal lines through the origin in R n. 5 p
6 Proof of the Splitting Principle. Consider again the situation from Example 3. The fact that 1 is an element of the basis implies that p : K (X) K (P (E)) is injective. Claim: The pullback bundle p (E) over P (E) contains the line bundle L P (E) as subbundle. We have the formulas L = {(l, v) P (E) E v l}, p (E) = {(l, v) P (E) E p(l) = π(v)}. It is easy to see that (l, v) L : p(l) = π(v). Hence L p (E) and we have the following diagram: L p (E) E π P (E) p X It follows that the pullback bundle splits as p (E) = L E for E P (E) the subbundle orthogonal to L with respect to some inner product. We repeat this process by forming P (E ) (the space of pairs of orthogonal lines in fibers in E. We can again split off a line bundle and continue this process a finite number of times until we obtain the flag bundle F (E) X, whose points are n-tuples of orthogonal lines in each fiber. If fibers of E have different dimensions over different connected components of X, we do this separately on each component. By construction, the pullback of E over F (E) splits as a sum of line bundles and the map F (E) X induces an injective map on K, since the latter is a composition of injective maps by the very first statement in this proof. References [1] A. Hatcher. Vector Bundles and K-Theory. Version 2.2, November
Division Algebras and Parallelizable Spheres, Part II
Division Algebras and Parallelizable Spheres, Part II Seminartalk by Jerome Wettstein April 5, 2018 1 A quick Recap of Part I We are working on proving the following Theorem: Theorem 1.1. The following
More informationALGEBRAICALLY TRIVIAL, BUT TOPOLOGICALLY NON-TRIVIAL MAP. Contents 1. Introduction 1
ALGEBRAICALLY TRIVIAL, BUT TOPOLOGICALLY NON-TRIVIAL MAP HONG GYUN KIM Abstract. I studied the construction of an algebraically trivial, but topologically non-trivial map by Hopf map p : S 3 S 2 and a
More informationMath 231b Lecture 16. G. Quick
Math 231b Lecture 16 G. Quick 16. Lecture 16: Chern classes for complex vector bundles 16.1. Orientations. From now on we will shift our focus to complex vector bundles. Much of the theory for real vector
More informationBEN KNUDSEN. Conf k (f) Conf k (Y )
CONFIGURATION SPACES IN ALGEBRAIC TOPOLOGY: LECTURE 2 BEN KNUDSEN We begin our study of configuration spaces by observing a few of their basic properties. First, we note that, if f : X Y is an injective
More informationA Bridge between Algebra and Topology: Swan s Theorem
A Bridge between Algebra and Topology: Swan s Theorem Daniel Hudson Contents 1 Vector Bundles 1 2 Sections of Vector Bundles 3 3 Projective Modules 4 4 Swan s Theorem 5 Introduction Swan s Theorem is a
More informationTopological K-theory
Topological K-theory Robert Hines December 15, 2016 The idea of topological K-theory is that spaces can be distinguished by the vector bundles they support. Below we present the basic ideas and definitions
More informationB 1 = {B(x, r) x = (x 1, x 2 ) H, 0 < r < x 2 }. (a) Show that B = B 1 B 2 is a basis for a topology on X.
Math 6342/7350: Topology and Geometry Sample Preliminary Exam Questions 1. For each of the following topological spaces X i, determine whether X i and X i X i are homeomorphic. (a) X 1 = [0, 1] (b) X 2
More informationTopological K-theory, Lecture 3
Topological K-theory, Lecture 3 Matan Prasma March 2, 2015 1 Applications of the classification theorem continued Let us see how the classification theorem can further be used. Example 1. The bundle γ
More informationChern Classes and the Chern Character
Chern Classes and the Chern Character German Stefanich Chern Classes In this talk, all our topological spaces will be paracompact Hausdorff, and our vector bundles will be complex. Let Bun GLn(C) be the
More informationSome K-theory examples
Some K-theory examples The purpose of these notes is to compute K-groups of various spaces and outline some useful methods for Ma448: K-theory and Solitons, given by Dr Sergey Cherkis in 2008-09. Throughout
More informationCW complexes. Soren Hansen. This note is meant to give a short introduction to CW complexes.
CW complexes Soren Hansen This note is meant to give a short introduction to CW complexes. 1. Notation and conventions In the following a space is a topological space and a map f : X Y between topological
More informationp,q H (X), H (Y ) ), where the index p has the same meaning as the
There are two Eilenberg-Moore spectral sequences that we shall consider, one for homology and the other for cohomology. In contrast with the situation for the Serre spectral sequence, for the Eilenberg-Moore
More informationLecture on Equivariant Cohomology
Lecture on Equivariant Cohomology Sébastien Racanière February 20, 2004 I wrote these notes for a hours lecture at Imperial College during January and February. Of course, I tried to track down and remove
More informationL E C T U R E N O T E S O N H O M O T O P Y T H E O R Y A N D A P P L I C AT I O N S
L A U R E N T I U M A X I M U N I V E R S I T Y O F W I S C O N S I N - M A D I S O N L E C T U R E N O T E S O N H O M O T O P Y T H E O R Y A N D A P P L I C AT I O N S i Contents 1 Basics of Homotopy
More informationMath 6510 Homework 10
2.2 Problems 9 Problem. Compute the homology group of the following 2-complexes X: a) The quotient of S 2 obtained by identifying north and south poles to a point b) S 1 (S 1 S 1 ) c) The space obtained
More informationSolutions to Problem Set 1
Solutions to Problem Set 1 18.904 Spring 2011 Problem 1 Statement. Let n 1 be an integer. Let CP n denote the set of all lines in C n+1 passing through the origin. There is a natural map π : C n+1 \ {0}
More informationFREUDENTHAL SUSPENSION THEOREM
FREUDENTHAL SUSPENSION THEOREM TENGREN ZHANG Abstract. In this paper, I will prove the Freudenthal suspension theorem, and use that to explain what stable homotopy groups are. All the results stated in
More informationAn Outline of Homology Theory
An Outline of Homology Theory Stephen A. Mitchell June 1997, revised October 2001 Note: These notes contain few examples and even fewer proofs. They are intended only as an outline, to be supplemented
More information10 Excision and applications
22 CHAPTER 1. SINGULAR HOMOLOGY be a map of short exact sequences of chain complexes. If two of the three maps induced in homology by f, g, and h are isomorphisms, then so is the third. Here s an application.
More informationCW-complexes. Stephen A. Mitchell. November 1997
CW-complexes Stephen A. Mitchell November 1997 A CW-complex is first of all a Hausdorff space X equipped with a collection of characteristic maps φ n α : D n X. Here n ranges over the nonnegative integers,
More informationTopological K-theory, Lecture 2
Topological K-theory, Lecture 2 Matan Prasma March 2, 2015 Again, we assume throughout that our base space B is connected. 1 Direct sums Recall from last time: Given vector bundles p 1 E 1 B and p 2 E
More information(1) Let π Ui : U i R k U i be the natural projection. Then π π 1 (U i ) = π i τ i. In other words, we have the following commutative diagram: U i R k
1. Vector Bundles Convention: All manifolds here are Hausdorff and paracompact. To make our life easier, we will assume that all topological spaces are homeomorphic to CW complexes unless stated otherwise.
More informationINTRO TO TENSOR PRODUCTS MATH 250B
INTRO TO TENSOR PRODUCTS MATH 250B ADAM TOPAZ 1. Definition of the Tensor Product Throughout this note, A will denote a commutative ring. Let M, N be two A-modules. For a third A-module Z, consider the
More information121B: ALGEBRAIC TOPOLOGY. Contents. 6. Poincaré Duality
121B: ALGEBRAIC TOPOLOGY Contents 6. Poincaré Duality 1 6.1. Manifolds 2 6.2. Orientation 3 6.3. Orientation sheaf 9 6.4. Cap product 11 6.5. Proof for good coverings 15 6.6. Direct limit 18 6.7. Proof
More informationTOPOLOGICAL K-THEORY
TOPOLOGICAL K-THEORY ZACHARY KIRSCHE Abstract. The goal of this paper is to introduce some of the basic ideas surrounding the theory of vector bundles and topological K-theory. To motivate this, we will
More information1. Classifying Spaces. Classifying Spaces
Classifying Spaces 1. Classifying Spaces. To make our lives much easier, all topological spaces from now on will be homeomorphic to CW complexes. Fact: All smooth manifolds are homeomorphic to CW complexes.
More informationCELLULAR HOMOLOGY AND THE CELLULAR BOUNDARY FORMULA. Contents 1. Introduction 1
CELLULAR HOMOLOGY AND THE CELLULAR BOUNDARY FORMULA PAOLO DEGIORGI Abstract. This paper will first go through some core concepts and results in homology, then introduce the concepts of CW complex, subcomplex
More informationINVERSE LIMITS AND PROFINITE GROUPS
INVERSE LIMITS AND PROFINITE GROUPS BRIAN OSSERMAN We discuss the inverse limit construction, and consider the special case of inverse limits of finite groups, which should best be considered as topological
More informationAlgebraic Topology II Notes Week 12
Algebraic Topology II Notes Week 12 1 Cohomology Theory (Continued) 1.1 More Applications of Poincaré Duality Proposition 1.1. Any homotopy equivalence CP 2n f CP 2n preserves orientation (n 1). In other
More informationThe Hopf invariant one problem
The Hopf invariant one problem Ishan Banerjee September 21, 2016 Abstract This paper will discuss the Adams-Atiyah solution to the Hopf invariant problem. We will first define and prove some identities
More informationOn the Diffeomorphism Group of S 1 S 2. Allen Hatcher
On the Diffeomorphism Group of S 1 S 2 Allen Hatcher This is a revision, written in December 2003, of a paper of the same title that appeared in the Proceedings of the AMS 83 (1981), 427-430. The main
More informationLecture 6: Classifying spaces
Lecture 6: Classifying spaces A vector bundle E M is a family of vector spaces parametrized by a smooth manifold M. We ask: Is there a universal such family? In other words, is there a vector bundle E
More informationTopology Hmwk 6 All problems are from Allen Hatcher Algebraic Topology (online) ch 2
Topology Hmwk 6 All problems are from Allen Hatcher Algebraic Topology (online) ch 2 Andrew Ma August 25, 214 2.1.4 Proof. Please refer to the attached picture. We have the following chain complex δ 3
More informationA Lower Bound for Immersions of Real Grassmannians. Marshall Lochbaum
A Lower Bound for Immersions of eal Grassmannians By Marshall Lochbaum Senior Honors Thesis Mathematics University of North Carolina at Chapel Hill November 11, 2014 Approved: Justin Sawon, Thesis Advisor
More informationMATH540: Algebraic Topology PROBLEM SET 3 STUDENT SOLUTIONS
Key Problems 1. Compute π 1 of the Mobius strip. Solution (Spencer Gerhardt): MATH540: Algebraic Topology PROBLEM SET 3 STUDENT SOLUTIONS In other words, M = I I/(s, 0) (1 s, 1). Let x 0 = ( 1 2, 0). Now
More informationLecture 8: More characteristic classes and the Thom isomorphism
Lecture 8: More characteristic classes and the Thom isomorphism We begin this lecture by carrying out a few of the exercises in Lecture 1. We take advantage of the fact that the Chern classes are stable
More informationCharacteristic Classes, Chern Classes and Applications to Intersection Theory
Characteristic Classes, Chern Classes and Applications to Intersection Theory HUANG, Yifeng Aug. 19, 2014 Contents 1 Introduction 2 2 Cohomology 2 2.1 Preliminaries................................... 2
More informationTHE POINCARE-HOPF THEOREM
THE POINCARE-HOPF THEOREM ALEX WRIGHT AND KAEL DIXON Abstract. Mapping degree, intersection number, and the index of a zero of a vector field are defined. The Poincare-Hopf theorem, which states that under
More informationBott Periodicity and Clifford Algebras
Bott Periodicity and Clifford Algebras Kyler Siegel November 27, 2012 Contents 1 Introduction 1 2 Clifford Algebras 3 3 Vector Fields on Spheres 6 4 Division Algebras 7 1 Introduction Recall for that a
More informationAn Algebraic View of the Relation between Largest Common Subtrees and Smallest Common Supertrees
An Algebraic View of the Relation between Largest Common Subtrees and Smallest Common Supertrees Francesc Rosselló 1, Gabriel Valiente 2 1 Department of Mathematics and Computer Science, Research Institute
More informationEQUIVARIANT COHOMOLOGY. p : E B such that there exist a countable open covering {U i } i I of B and homeomorphisms
EQUIVARIANT COHOMOLOGY MARTINA LANINI AND TINA KANSTRUP 1. Quick intro Let G be a topological group (i.e. a group which is also a topological space and whose operations are continuous maps) and let X be
More informationAlgebraic Topology I Homework Spring 2014
Algebraic Topology I Homework Spring 2014 Homework solutions will be available http://faculty.tcu.edu/gfriedman/algtop/algtop-hw-solns.pdf Due 5/1 A Do Hatcher 2.2.4 B Do Hatcher 2.2.9b (Find a cell structure)
More informationDefinitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations
Page 1 Definitions Tuesday, May 8, 2018 12:23 AM Notations " " means "equals, by definition" the set of all real numbers the set of integers Denote a function from a set to a set by Denote the image of
More informationCohomology and Vector Bundles
Cohomology and Vector Bundles Corrin Clarkson REU 2008 September 28, 2008 Abstract Vector bundles are a generalization of the cross product of a topological space with a vector space. Characteristic classes
More informationSMSTC Geometry & Topology 1 Assignment 1 Matt Booth
SMSTC Geometry & Topology 1 Assignment 1 Matt Booth Question 1 i) Let be the space with one point. Suppose X is contractible. Then by definition we have maps f : X and g : X such that gf id X and fg id.
More informationQuiz-1 Algebraic Topology. 1. Show that for odd n, the antipodal map and the identity map from S n to S n are homotopic.
Quiz-1 Algebraic Topology 1. Show that for odd n, the antipodal map and the identity map from S n to S n are homotopic. 2. Let X be an Euclidean Neighbourhood Retract space and A a closed subspace of X
More information1 Whitehead s theorem.
1 Whitehead s theorem. Statement: If f : X Y is a map of CW complexes inducing isomorphisms on all homotopy groups, then f is a homotopy equivalence. Moreover, if f is the inclusion of a subcomplex X in
More informationLECTURE 2: THE THICK SUBCATEGORY THEOREM
LECTURE 2: THE THICK SUBCATEGORY THEOREM 1. Introduction Suppose we wanted to prove that all p-local finite spectra of type n were evil. In general, this might be extremely hard to show. The thick subcategory
More informationMath 440 Problem Set 2
Math 440 Problem Set 2 Problem 4, p. 52. Let X R 3 be the union of n lines through the origin. Compute π 1 (R 3 X). Solution: R 3 X deformation retracts to S 2 with 2n points removed. Choose one of them.
More informationThe Real Grassmannian Gr(2, 4)
The Real Grassmannian Gr(2, 4) We discuss the topology of the real Grassmannian Gr(2, 4) of 2-planes in R 4 and its double cover Gr + (2, 4) by the Grassmannian of oriented 2-planes They are compact four-manifolds
More informationwhere Σ is a finite discrete Gal(K sep /K)-set unramified along U and F s is a finite Gal(k(s) sep /k(s))-subset
Classification of quasi-finite étale separated schemes As we saw in lecture, Zariski s Main Theorem provides a very visual picture of quasi-finite étale separated schemes X over a henselian local ring
More informationELEMENTARY SUBALGEBRAS OF RESTRICTED LIE ALGEBRAS
ELEMENTARY SUBALGEBRAS OF RESTRICTED LIE ALGEBRAS J. WARNER SUMMARY OF A PAPER BY J. CARLSON, E. FRIEDLANDER, AND J. PEVTSOVA, AND FURTHER OBSERVATIONS 1. The Nullcone and Restricted Nullcone We will need
More informationGEOMETRY FINAL CLAY SHONKWILER
GEOMETRY FINAL CLAY SHONKWILER 1 Let X be the space obtained by adding to a 2-dimensional sphere of radius one, a line on the z-axis going from north pole to south pole. Compute the fundamental group and
More informationNotes 10: Consequences of Eli Cartan s theorem.
Notes 10: Consequences of Eli Cartan s theorem. Version 0.00 with misprints, The are a few obvious, but important consequences of the theorem of Eli Cartan on the maximal tori. The first one is the observation
More informationA SHORT PROOF OF ROST NILPOTENCE VIA REFINED CORRESPONDENCES
A SHORT PROOF OF ROST NILPOTENCE VIA REFINED CORRESPONDENCES PATRICK BROSNAN Abstract. I generalize the standard notion of the composition g f of correspondences f : X Y and g : Y Z to the case that X
More informationDedekind Domains. Mathematics 601
Dedekind Domains Mathematics 601 In this note we prove several facts about Dedekind domains that we will use in the course of proving the Riemann-Roch theorem. The main theorem shows that if K/F is a finite
More informationEILENBERG-ZILBER VIA ACYCLIC MODELS, AND PRODUCTS IN HOMOLOGY AND COHOMOLOGY
EILENBERG-ZILBER VIA ACYCLIC MODELS, AND PRODUCTS IN HOMOLOGY AND COHOMOLOGY CHRIS KOTTKE 1. The Eilenberg-Zilber Theorem 1.1. Tensor products of chain complexes. Let C and D be chain complexes. We define
More informationHomework 3: Relative homology and excision
Homework 3: Relative homology and excision 0. Pre-requisites. The main theorem you ll have to assume is the excision theorem, but only for Problem 6. Recall what this says: Let A V X, where the interior
More informationCutting and pasting. 2 in R. 3 which are not even topologically
Cutting and pasting We begin by quoting the following description appearing on page 55 of C. T. C. Wall s 1960 1961 Differential Topology notes, which available are online at http://www.maths.ed.ac.uk/~aar/surgery/wall.pdf.
More informationNOTES IN COMMUTATIVE ALGEBRA: PART 2
NOTES IN COMMUTATIVE ALGEBRA: PART 2 KELLER VANDEBOGERT 1. Completion of a Ring/Module Here we shall consider two seemingly different constructions for the completion of a module and show that indeed they
More information4. Images of Varieties Given a morphism f : X Y of quasi-projective varieties, a basic question might be to ask what is the image of a closed subset
4. Images of Varieties Given a morphism f : X Y of quasi-projective varieties, a basic question might be to ask what is the image of a closed subset Z X. Replacing X by Z we might as well assume that Z
More informationHomework 3 MTH 869 Algebraic Topology
Homework 3 MTH 869 Algebraic Topology Joshua Ruiter February 12, 2018 Proposition 0.1 (Exercise 1.1.10). Let (X, x 0 ) and (Y, y 0 ) be pointed, path-connected spaces. Let f : I X y 0 } and g : I x 0 }
More informationA BRIEF GUIDE TO ORDINARY K-THEORY
A BRIEF GUIDE TO ORDINARY K-THEORY SPENCER STIRLING Abstract. In this paper we describe some basic notions behind ordinary K- theory. Following closely [Hat02b] we ll first study unreduced K-theory. Without
More informationHomology of a Cell Complex
M:01 Fall 06 J. Simon Homology of a Cell Complex A finite cell complex X is constructed one cell at a time, working up in dimension. Each time a cell is added, we can analyze the effect on homology, by
More informationManifolds and Poincaré duality
226 CHAPTER 11 Manifolds and Poincaré duality 1. Manifolds The homology H (M) of a manifold M often exhibits an interesting symmetry. Here are some examples. M = S 1 S 1 S 1 : M = S 2 S 3 : H 0 = Z, H
More informationMath 752 Week s 1 1
Math 752 Week 13 1 Homotopy Groups Definition 1. For n 0 and X a topological space with x 0 X, define π n (X) = {f : (I n, I n ) (X, x 0 )}/ where is the usual homotopy of maps. Then we have the following
More informationThe Hurewicz Theorem
The Hurewicz Theorem April 5, 011 1 Introduction The fundamental group and homology groups both give extremely useful information, particularly about path-connected spaces. Both can be considered as functors,
More informationSmith theory. Andrew Putman. Abstract
Smith theory Andrew Putman Abstract We discuss theorems of P. Smith and Floyd connecting the cohomology of a simplicial complex equipped with an action of a finite p-group to the cohomology of its fixed
More informationFINITE CONNECTED H-SPACES ARE CONTRACTIBLE
FINITE CONNECTED H-SPACES ARE CONTRACTIBLE ISAAC FRIEND Abstract. The non-hausdorff suspension of the one-sphere S 1 of complex numbers fails to model the group s continuous multiplication. Moreover, finite
More informationFINITE SPECTRA CARY MALKIEWICH
FINITE SPECTRA CARY MALKIEWICH These notes were written in 2014-2015 to help me understand how the different notions of finiteness for spectra are related. I am usually surprised that the basics are not
More informationOn Eilenberg-MacLanes Spaces (Term paper for Math 272a)
On Eilenberg-MacLanes Spaces (Term paper for Math 272a) Xi Yin Physics Department Harvard University Abstract This paper discusses basic properties of Eilenberg-MacLane spaces K(G, n), their cohomology
More informationThe Proj Construction
The Proj Construction Daniel Murfet May 16, 2006 Contents 1 Basic Properties 1 2 Functorial Properties 2 3 Products 6 4 Linear Morphisms 9 5 Projective Morphisms 9 6 Dimensions of Schemes 11 7 Points of
More informationALGEBRAIC PROPERTIES OF BIER SPHERES
LE MATEMATICHE Vol. LXVII (2012 Fasc. I, pp. 91 101 doi: 10.4418/2012.67.1.9 ALGEBRAIC PROPERTIES OF BIER SPHERES INGA HEUDTLASS - LUKAS KATTHÄN We give a classification of flag Bier spheres, as well as
More informationDerived Algebraic Geometry III: Commutative Algebra
Derived Algebraic Geometry III: Commutative Algebra May 1, 2009 Contents 1 -Operads 4 1.1 Basic Definitions........................................... 5 1.2 Fibrations of -Operads.......................................
More informationMath 121 Homework 4: Notes on Selected Problems
Math 121 Homework 4: Notes on Selected Problems 11.2.9. If W is a subspace of the vector space V stable under the linear transformation (i.e., (W ) W ), show that induces linear transformations W on W
More informationHomotopy and homology groups of the n-dimensional Hawaiian earring
F U N D A M E N T A MATHEMATICAE 165 (2000) Homotopy and homology groups of the n-dimensional Hawaiian earring by Katsuya E d a (Tokyo) and Kazuhiro K a w a m u r a (Tsukuba) Abstract. For the n-dimensional
More informationNONSINGULAR CURVES BRIAN OSSERMAN
NONSINGULAR CURVES BRIAN OSSERMAN The primary goal of this note is to prove that every abstract nonsingular curve can be realized as an open subset of a (unique) nonsingular projective curve. Note that
More informationTHE FUNDAMENTAL GROUP AND CW COMPLEXES
THE FUNDAMENTAL GROUP AND CW COMPLEXES JAE HYUNG SIM Abstract. This paper is a quick introduction to some basic concepts in Algebraic Topology. We start by defining homotopy and delving into the Fundamental
More informationConcentrated Schemes
Concentrated Schemes Daniel Murfet July 9, 2006 The original reference for quasi-compact and quasi-separated morphisms is EGA IV.1. Definition 1. A morphism of schemes f : X Y is quasi-compact if there
More informationTHE TATE MODULE. Seminar: Elliptic curves and the Weil conjecture. Yassin Mousa. Z p
THE TATE MODULE Seminar: Elliptic curves and the Weil conjecture Yassin Mousa Abstract This paper refers to the 10th talk in the seminar Elliptic curves and the Weil conjecture supervised by Prof. Dr.
More informationCitation Osaka Journal of Mathematics. 43(2)
TitleIrreducible representations of the Author(s) Kosuda, Masashi Citation Osaka Journal of Mathematics. 43(2) Issue 2006-06 Date Text Version publisher URL http://hdl.handle.net/094/0396 DOI Rights Osaka
More information0.1 Universal Coefficient Theorem for Homology
0.1 Universal Coefficient Theorem for Homology 0.1.1 Tensor Products Let A, B be abelian groups. Define the abelian group A B = a b a A, b B / (0.1.1) where is generated by the relations (a + a ) b = a
More informationAlgebraic Topology Final
Instituto Superior Técnico Departamento de Matemática Secção de Álgebra e Análise Algebraic Topology Final Solutions 1. Let M be a simply connected manifold with the property that any map f : M M has a
More informationMATH 215B HOMEWORK 5 SOLUTIONS
MATH 25B HOMEWORK 5 SOLUTIONS. ( marks) Show that the quotient map S S S 2 collapsing the subspace S S to a point is not nullhomotopic by showing that it induces an isomorphism on H 2. On the other hand,
More informationON A THEOREM OF CAMPANA AND PĂUN
ON A THEOREM OF CAMPANA AND PĂUN CHRISTIAN SCHNELL Abstract. Let X be a smooth projective variety over the complex numbers, and X a reduced divisor with normal crossings. We present a slightly simplified
More informationCharacteristic classes in the Chow ring
arxiv:alg-geom/9412008v1 10 Dec 1994 Characteristic classes in the Chow ring Dan Edidin and William Graham Department of Mathematics University of Chicago Chicago IL 60637 Let G be a reductive algebraic
More informationChern classes à la Grothendieck
Chern classes à la Grothendieck Theo Raedschelders October 16, 2014 Abstract In this note we introduce Chern classes based on Grothendieck s 1958 paper [4]. His approach is completely formal and he deduces
More information32 Proof of the orientation theorem
88 CHAPTER 3. COHOMOLOGY AND DUALITY 32 Proof of the orientation theorem We are studying the way in which local homological information gives rise to global information, especially on an n-manifold M.
More informationEXCERPT FROM ON SOME ACTIONS OF STABLY ELEMENTARY MATRICES ON ALTERNATING MATRICES
EXCERPT FROM ON SOME ACTIONS OF STABLY ELEMENTARY MATRICES ON ALTERNATING MATRICES RAVI A.RAO AND RICHARD G. SWAN Abstract. This is an excerpt from a paper still in preparation. We show that there are
More informationLecture 6. s S} is a ring.
Lecture 6 1 Localization Definition 1.1. Let A be a ring. A set S A is called multiplicative if x, y S implies xy S. We will assume that 1 S and 0 / S. (If 1 / S, then one can use Ŝ = {1} S instead of
More informationEXOTIC SMOOTH STRUCTURES ON TOPOLOGICAL FIBRE BUNDLES B
EXOTIC SMOOTH STRUCTURES ON TOPOLOGICAL FIBRE BUNDLES B SEBASTIAN GOETTE, KIYOSHI IGUSA, AND BRUCE WILLIAMS Abstract. When two smooth manifold bundles over the same base are fiberwise tangentially homeomorphic,
More informationAlgebraic Geometry Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
More informationExercises for Algebraic Topology
Sheet 1, September 13, 2017 Definition. Let A be an abelian group and let M be a set. The A-linearization of M is the set A[M] = {f : M A f 1 (A \ {0}) is finite}. We view A[M] as an abelian group via
More informationFiberwise two-sided multiplications on homogeneous C*-algebras
Fiberwise two-sided multiplications on homogeneous C*-algebras Ilja Gogić Department of Mathematics University of Zagreb XX Geometrical Seminar Vrnjačka Banja, Serbia May 20 23, 2018 joint work with Richard
More informationLecture 4: Stabilization
Lecture 4: Stabilization There are many stabilization processes in topology, and often matters simplify in a stable limit. As a first example, consider the sequence of inclusions (4.1) S 0 S 1 S 2 S 3
More informationMATH730 NOTES WEEK 8
MATH730 NOTES WEEK 8 1. Van Kampen s Theorem The main idea of this section is to compute fundamental groups by decomposing a space X into smaller pieces X = U V where the fundamental groups of U, V, and
More informationFILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS.
FILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS. Let A be a ring, for simplicity assumed commutative. A filtering, or filtration, of an A module M means a descending sequence of submodules M = M 0
More informationLecture 2. x if x X B n f(x) = α(x) if x S n 1 D n
Lecture 2 1.10 Cell attachments Let X be a topological space and α : S n 1 X be a map. Consider the space X D n with the disjoint union topology. Consider further the set X B n and a function f : X D n
More informationLECTURE 3: RELATIVE SINGULAR HOMOLOGY
LECTURE 3: RELATIVE SINGULAR HOMOLOGY In this lecture we want to cover some basic concepts from homological algebra. These prove to be very helpful in our discussion of singular homology. The following
More informationCHAPTER 2. Ordered vector spaces. 2.1 Ordered rings and fields
CHAPTER 2 Ordered vector spaces In this chapter we review some results on ordered algebraic structures, specifically, ordered vector spaces. We prove that in the case the field in question is the field
More information