THE HOMOTOPY TYPE OF THE SPACE OF RATIONAL FUNCTIONS
|
|
- Geraldine Fields
- 6 years ago
- Views:
Transcription
1 THE HOMOTOPY TYPE OF THE SPACE OF RATIONAL FUNCTIONS by M.A. Guest, A. Kozlowski, M. Murayama and K. Yamaguchi In this note we determine some homotopy groups of the space of rational functions of degree d on the Riemann sphere, and describe a homogeneous space structure for the case of degree. As an application we show that C -structure on d 0 Hol d is not compatible with that on Ω S, where Hol d denotes the space of all based holomorphic maps of degree d from Riemann sphere to itself Mathematics subject classification: 55P15, 55Q5 1. Introduction. For each positive integer d, let Hol d denote the space of all holomorphic (equivalently, algebraic) maps of degree d from the Riemann sphere S = C to itself. This space is of interest both from a classical and a modern point of view (see [1], [5]). Let Hol d be the subspace of Hol d consisting of maps which preserve a basepoint of S. It is well known that Hol 1 is the group of fractional linear transformations PSL (C) and that Hol 1 may be identified with the affine transformation group of C. It is an elementary fact that Hol d and Hol d are connected spaces. The fundamental groups of these spaces are Z/d, Z respectively; these computations are due to Epshtein ([6]) and Jones (see [8]). The following more general result was obtained by Segal: Theorem 0 ([8]). Let Map d be the space of all continuous maps of degree d from S to itself and let Map d be the subspace consisting of maps f such that f( ) =1. Then the natural inclusion maps induce the following isomorphisms of homotopy groups: (1) If k<d, then π k (Hol d)=π k (Map d)=π k+ (S ). () If k<d, then π k (Hol d )=π k (Map d ). The stable homotopy type of Hol d was studied in [3]. In this note we shall extend the above results by determining some further homotopy groups of the space Hol d. Our results are as follows: 1
2 Theorem 1. (1) For k, π k (S 3 ) d=1 π k (Hol d )= π k (S 3 ) π k (S ) d= Z/ d 3,k = () If k 3 and d 3, then π k (Hol d )=π k (Hol d) π k (S 3 ). (3) In particular, if d>k 3, then π k (Hol d )=π k+ (S ) π k (S 3 ). Theorem. The space Hol may be identified with a homogeneous space of the form (SL (C) SL (C))/H, where H is isomorphic to C Z/4. In this semi-direct product, the action of Z/4 =<σ:σ 4 =1>is given by σ α = α 1 for α C. In particular, Hol is homotopy equivalent to (S 3 S 3 )/(S 1 Z/4). Theorem 3. (1) The universal cover of Hol is homotopy equivalent to S. () The universal cover of Hol may be identified with a homogeneous space of the form (SL (C) SL (C))/D, where D is isomorphic to C. In particular, it is homotopy equivalent to S 3 S. In Theorem 1, the case d = 1 follows from the fact that Hol 1 may be identified with PSL (C) and hence is homotopy equivalent to RP 3 ; the case d = is direct consequence of () of Theorem 3. In section, we shall consider the homogeneous structure of Hol based on the action of Hol 1 Hol 1 by pre- and post-composition, and give the proof of Theorem and () of Theorem 3. In section 3, we shall investigate the space Hol, and give the proof of (1) of Theorem 3. In section 4, we shall prove Theorem 1. In section 5 we shall give an application of these results to the C -operad structure on d 0 Hol d. In particular, we shall show that the C -structure on d 0 Hol d is not compatible with that on Ω S up to homotopy.. The Homogeneous Structure of Hol. From now on, we identify Hol d with the space of functions f = p 1 /p, where p 1,p are coprime polynomials such that max{deg(p 1 ), deg(p )} = d. The group Hol 1 acts on Hol d by pre- and post-compositions: for (A, B) Hol 1 Hol 1 and f Hol d we have (A, B) f (z) =A(f(B 1 (z)). The following proposition is well known, but we shall give a proof for the sake of completeness.
3 Proposition.1. The group Hol 1 Hol 1 acts transitively on Hol. Proof. Let f = p/q Hol. It suffices to show that A(f(B(z))) = z for some A, B Hol 1. Since Hol 1 acts transitively on S, there is a function A Hol 1 such that A(f( )) =. Hence, without loss of generality, we may suppose that f( ) =, i.e. that deg(p) => deg(q). Claim: If f( ) =, then there is some (A, B) Hol 1 Hol 1 such that A(f(B(z))) = z or (z + z 1 )/. We shall prove this by considering separately the cases deg(q) =0,deg(q) =1. (i) If deq(q) = 0, we may suppose that f(z) =p(z)=a(z+b) +cfor some a 0,b,c C. If we put A(z) =a 1 (z c), B(z) =z bthen A(f(B(z))) = z, as required. (ii) If deg(q) = 1, we may suppose that q(z) =z+a,andp(z)=b{(z+a) +c(z+a)+d } with b 0,d 0. Putting A(z) =(z bc)/bd, B(z) =dz a we see that A(f(B(z))) = (z + z 1 )/. This completes the proof of the claim. Let g(z) = (z + z 1 )/. If A(z) = (z +1)/(z 1), B(z) = (z 1)/(z + 1) then A(g(B(z))) = z. This completes the proof of Proposition.1. Remark: It follows from the claim that Hol consists of two Hol 1 Hol 1 orbits. It is well known that the map ( ) a b SL (C) Hol 1, az + b c d cz + d, is a double covering and induces an isomorphism PSL (C) = Hol 1 of Lie groups. Thus the group G = SL (C) SL (C) acts (transitively) on Hol. Lemma.. Let H denote the isotropy subgroup of SL (C) SL (C) at z Hol. Then H = {( ( ) ( )) ( ( ) ( )) } α 0 α 0 0 iα 0 α ± 0 α, 0 α 1, ± iα, 0 α 1 : α C. 0 Proof. This follows by direct calculation. Next we determine the group structure of H. Lemma.3. Let K = C Z/4 be the group defined by the action of Z/4 =<σ:σ 4 =1> on C by σ α = α 1 for α C. Then H and K are isomorphic Lie groups. Proof. We put σ 1 = ( ) 0 i, σ i 0 = ( )
4 Define ϕ : K H by (( ) ( ) (α, σ m α 0 ) 0 α σ1 m α 0, 0 α 1 σ m It is easy to check that ϕ is an isomorphism. Proof of Theorem. The first part of Theorem follows from Proposition.1, Lemma. and Lemma.3. The inclusion map of the maximal compact subgroup SU() = S 3 of SL (C) induces the homotopy equivalence (S 3 S 3 )/(S 1 Z/4) Hol. Next we consider the universal cover of Hol. Let D be the subgroup {(( ) ( )) } α 0 α 0 0 α, 0 α 1 : α C of SL (C) SL (C), and let D c be its maximal compact subgroup {(( ) ( )) } α 0 α 0 0 α, 0 α 1 : α =1,α C. Then D is a normal subgroup of H = C Z/4 isomorphic to C. Similarly D c is is isomorphic to S 1. Clearly the projection E = (SL (C) SL (C))/D (SL (C) SL (C))/H = Hol is a regular covering. To show that E is simply connected, we shall consider the fibre bundle SL (C) E SL (C)/D whose projection map is induced by the projection onto the second factor. Lemma.4. (1) E is fibre homotopy equivalent to the fibre bundle SU() Y S, where Y = (SU() SU())/D c. () Y can be identified with the unit sphere bundle S(η η ), where η denotes the Hopf complex line bundle over S = CP 1. ). Proof. (1) The inclusion map SU() SU() SL (C) SL (C) induces the desired fibre homotopy equivalence Y E. () The second factor S = {( ) } α 0 0 α 1 : α =1,α C of D c is the standard embedding of S 1 into SU(), and S 1 SU() SU()/S S is the Hopf bundle. We may use the identifications SU() = Sp(1) = S 3 H, and extend the action of D c naturally to SU() H. By considering the transition functions of the vector bundle (SU() H)/D c, it is not difficult to see that Y is equivalent to S(η η ). Proof of () of Theorem 3. It follows from the homotopy exact sequence that Y (and hence E) is simply connected. Hence E is the universal covering of Hol. Since π (BU()) = Z, a -dimensional complex vector bundle ξ over S is determined by its first Chern class c 1 (ξ). As c 1 (η η ) = 0, it follows that η η is trivial. Hence E Y S 3 S. 4
5 3. The Space Hol. In this section we shall use the actions of Hol 1 (and Hol 1) on Hol d by post-composition: A f(z) =A(f(z)) for (A, f) Hol 1 Hol d. First, we have two easy lemmas: Lemma 3.1. Let d 1. The group Hol 1 acts freely on Hol d by post-composition. Similarly, Hol 1 acts freely on Hol d by post-composition. Proof. This follows immediately from the fact that any map of non-zero degree is surjective. Lemma 3.. Let d 1. Then the natural inclusion map j d : Hol d Hol d induces a homeomorphism j d : Hol 1 \ Hol d Hol 1 \ Hol d. Proof. Since Hol 1 acts transitively on S, the induced map j d : Hol 1 \ Hol d Hol 1 \ Hol d is surjective. Since (Hol 1 f) Hol d = Hol 1 f for any f Hol d, j d is injective. If we identify these spaces by j d, it is easy to see that the topologies coincide. Proposition 3.3 ([4]). There is a fibration S 1 Hol RP. Remark: Cohen and Shimamoto ([4]) deduce this from results of Donaldson ([5]) and Atiyah and Hitchin ([1]) on monopoles. We shall give a direct and elementary proof. Proof. By Lemma 3.1 we have a principal bundle Hol 1 Hol Hol 1 \ Hol. By Theorem and Lemma 3., Hol 1 \ Hol Hol 1 \ Hol (S 3 /S 1 )/{±1} S /{±1} = RP. Since Hol 1 S 1, we have the required fibration S 1 Hol RP. Proof of (1) of Theorem 3. Consider the above fibration S 1 Hol π RP. Let p : S RP and q : X Hol be the universal coverings. Since X is simply connected, there is a lift θ : X S such that p θ = π q. It follows by diagram chasing that θ : π k (X) π k (S ) is an isomorphism for all k. Hence θ is a homotopy equivalence. 4. The Proof of Theorem 1. Let ι n π n (S n ) be the oriented generator and η π 3 (S ) be the class of the Hopf map. We put η n =Σ n η π n+1 (S n )forn>. The following three results are well known and we omit the proofs. 5
6 Lemma 4.1 ([9]). (1) π n (S n )=Z{ι n }. () π 3 (S )=Z{η },π n+1 (S n )=Z/{η n }for n>. (3) π n+ (S n )=Z/{ηn}for n>1.hereweputηn=η n η n+1. Lemma 4. ([7],[9]). (1) [ι,ι ]=η. () If k>,[ι,α]=0for any α π k (S ). Here [, ] denotes the Whitehead product. Let Map(S n,x) denote the space of all continuous maps from S n to X, and let Map (S n,x) be the subspace consisting of based maps. For a map f we denote by Map f (S n,x)ormap f(s n,x) the path-component containing f. Lemma 4.3 ([10]). Let f Map (S n,x) and let Map f (S n,x) Map f (S n,x) ev X be the evaluation fibration. If we use the identification π k (Map f (S n,x)) = π k+n (X), then the boundary operator : π k+n (X) π k 1 (X) of the homotopy exact sequence associated with the evaluation fibration is given (up to sign) by the Whitehead product: (α) =[α, f]. Proof of Theorem 1. (1) It suffices to consider the case d 3. Let I : Hol d Map d and J : Hol d Map d be the inclusion maps. By Theorem 0, I and J are homotopy equivalences up to dimension d. Consider the following commutative diagram: Hol ev d Hol d S I J = Map d Map d ev S in which the horizontal sequences are evaluation fibre sequences. The result follows from the induced diagram of homotopy groups, by using Lemmas 4.1, 4., and 4.3 and the Five Lemma. This completes the proof of (1). () Suppose that d 3andk 3 are integers. Consider the commutative diagram of principal bundles Hol 1 Hol p d d Hol 1 \ Hol d j d j d q d Hol 1 Hol d Hol1 \ Hol d where j d is a natural inclusion map. In the induced homotopy exact sequences, since Hol 1 S 1, (p d ) : π k (Hol d) π k (Hol 1 \ Hol d) is an isomorphism for k 3. Hence (j d ) (p d ) 1 ( j d ) 1 : π k (Hol 1 \ Hol d ) π k (Hol d ) gives a splitting of (q d ).Sowehave π k (Hol d )=π k (Hol d) π k (Hol 1 ). 6
7 Because Hol 1 RP 3, π k (Hol 1 )=π k (S 3 ) and this completes the proof of (). (3) It follows from Theorem 0 that π k (Hol d)=π k+ (S )fork<dand the assertion easily follows from (). Remark: The above method allows one to deduce the result of Epshtein ([6]) that π 1 (Hol d )= Z/dfrom the result of Jones (see [8]) that π 1 (Hol d)=z. 5. The C -operad structure on d 0 Hol d. Consider d 0 Hol d, the disjoint union of the based rational functions of degree d. It is known that this is a C -operad space ([]). Let µ d : F (C,d) (S 1 ) d Hol d be the structure map, where we identify Hol 1 up to homotopy with S 1. Let i d : Hol d Ω d S and i : Hol Ω S be the inclusion maps. It is known that i is a C -map up to homotopy ([]). It follows from the May-Milgram model of Ω Σ X that we can identify Ω S 3 with J(S 1 ), where J(X) denotes the space J(X) = d 1 (F(C,d) Σd X d )/ and is a well known equivalence relation. On the other hand, there is a well known equivalence Ω S 3 Ω 0S. The following observation of Professor F.R. Cohen shows that the C -structure on Ω 0S is incompatible with the one on d 0 Hol d. (This contradicts the statement of [3] that diagram (3.4) of that paper is homotopy commutative.) Let J d (X) bethed-th term of the May-Milgram filtration on J(X). 7
8 Proposition 5.1. There is no homotopy equivalence θ :Map (S,S )=Ω S Ω S 3 such that the following diagram is homotopy commutative: F (C, ) Σ (S 1 ) µ Hol i Ω S j θ J (S 1 ) J (S 1 ) Ω S 3 Here j denotes the inclusion map. Proof. Suppose that the above diagram is homotopy commutative. Since F (C, ) S 1, there is a non-zero element e 1 e 1 H 3 (F (C, ) Σ (S 1 ), Z/) = Z/ which represents a generator. Using the above diagram, if we put α =(µ ) (e 1 e 1), we have 0 Q 1 (e 1 )=(θ i ) (α) H 3 (Ω S 3, Z/). Since Q 1 (e 1 ) H 3 (Ω S 3, Z/) is primitive, and θ is an equivalence, the element (i ) α = θ 1 (Q 1 (e 1 )) H 3 (Hol, Z/) = Z/ is also primitive (the image of a primitive element is primitive). By [8], (i ) is injective, so α is primitive. Thus the generator of H 3 (Hol, Z/) = Z/ is indecomposable. On the other hand, because the universal cover of Hol is homotopy equivalent to S and π 1 (Hol )=Z, there is a fibration S Hol BZ S 1. Consider the mod Serre spectral sequence of this fibration. This collapses at the E level, and the generator of H 3 (Hol, Z/) = Z/ is decomposable. This is a contradiction. Remark: The above result implies that the C -structure of Hol d andthatofω S 3 are not compatible (up to homotopy) at least when d =. This was also pointed out in [4]. Acknowlegements: The first, second and fourth authors are indebted to the Mathematics Department of Tokyo Institute of Technology for its hospitality. The application to C - operad structures is due to Professor F.R. Cohen. A gap in an earlier version of the proof of Theorem 3 was filled thanks to Professor A. Kono, who suggested the present version. The authors are grateful to Professors Cohen and Kono for their kind assistance. 8
9 References [1] M.F. Atiyah and N.J. Hitchin, The Geometry and Dynamics of Magnetic Monopoles, Princeton Univ. Press, [] C.P. Boyer and B.M. Mann, Monopoles, non-linear σ models and two-fold loop spaces, Commun. Math. Phys. 115 (1988), [3] F.R. Cohen, R.L. Cohen, B.M. Mann and R.J. Milgram, The topology of rational functions and divisors of surfaces, Acta Math. 166 (1991), [4] R.L. Cohen and D.H. Shimamoto, Rational functions, labelled configurations and Hilbert schemes, J. Lond. Math. Soc. 43 (1991), [5] S.K. Donaldson, Nahm s equations and the classification of monopoles, Commun. Math. Phys. 96 (1984), [6] S.I. Epshtein, Fundamental groups of spaces of coprime polynomials, Functional Analysis and its Applications 7 (1973), [7] P.J. Hilton and J.H.C. Whitehead, Note on Whitehead product, Annals of Math. 58 (1953), [8] G.B. Segal, The topology of spaces of rational functions, Acta Math. 143 (1979), [9] H. Toda, Composition Methods in Homotopy Groups of Spheres, Vol. 49, Princeton Univ. Press, 196. [10]G.W. Whitehead, On products in homotopy groups, Annals of Math. 47 (1946), Department of Mathematics, University of Rochester, Rochester, New York 1467, USA Department of Mathematics, Toyama International University, Kamishinkawa, Toyama, Japan Department of Mathematics, Tokyo Institute of Technology, Oh-Okayama, Meguro, Tokyo 15, Japan Department of Mathematics, The University of Electro-Communications, Chofu-gaoka, Chofu, Tokyo 18, Japan 9
THE HOMOTOPY TYPES OF SU(5)-GAUGE GROUPS. Osaka Journal of Mathematics. 52(1) P.15-P.29
Title THE HOMOTOPY TYPES OF SU(5)-GAUGE GROUPS Author(s) Theriault, Stephen Citation Osaka Journal of Mathematics. 52(1) P.15-P.29 Issue Date 2015-01 Text Version publisher URL https://doi.org/10.18910/57660
More informationCohomology of the classifying spaces of gauge groups over 3-manifolds in low dimensions
Cohomology of the classifying spaces of gauge groups over 3-manifolds in low dimensions by Shizuo Kaji Department of Mathematics Kyoto University Kyoto 606-8502, JAPAN e-mail: kaji@math.kyoto-u.ac.jp Abstract
More informationTitle multiplicity (New topics of transfo. Citation 数理解析研究所講究録 (2015), 1968:
Title Note on the space of polynomials wi multiplicity (New topics of transfo Author(s) 山口, 耕平 Citation 数理解析研究所講究録 (2015), 1968: 126-129 Issue Date 2015-11 URL http://hdl.handle.net/2433/224267 Right Type
More informationMATH 8253 ALGEBRAIC GEOMETRY WEEK 12
MATH 8253 ALGEBRAIC GEOMETRY WEEK 2 CİHAN BAHRAN 3.2.. Let Y be a Noetherian scheme. Show that any Y -scheme X of finite type is Noetherian. Moreover, if Y is of finite dimension, then so is X. Write f
More informationComparing the homotopy types of the components of Map(S 4 ;BSU(2))
Journal of Pure and Applied Algebra 161 (2001) 235 243 www.elsevier.com/locate/jpaa Comparing the homotopy types of the components of Map(S 4 ;BSU(2)) Shuichi Tsukuda 1 Department of Mathematical Sciences,
More informationA note on Samelson products in the exceptional Lie groups
A note on Samelson products in the exceptional Lie groups Hiroaki Hamanaka and Akira Kono October 23, 2008 1 Introduction Samelson products have been studied extensively for the classical groups ([5],
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 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY
LECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY WEIMIN CHEN, UMASS, SPRING 07 1. Blowing up and symplectic cutting In complex geometry the blowing-up operation amounts to replace a point in
More informationThe Global Defect Index
Communications in Mathematical Physics Manuscript-Nr. (will be inserted by hand later) The Global Defect Index Stefan Bechtluft-Sachs, Marco Hien Naturwissenschaftliche Fakultät I, Universität Regensburg,
More informationb 0 + b 1 z b d z d
I. Introduction Definition 1. For z C, a rational function of degree d is any with a d, b d not both equal to 0. R(z) = P (z) Q(z) = a 0 + a 1 z +... + a d z d b 0 + b 1 z +... + b d z d It is exactly
More informationHOMOTOPY THEORY OF THE SUSPENSIONS OF THE PROJECTIVE PLANE
HOMOTOPY THEORY OF THE SUSPENSIONS OF THE PROJECTIVE PLANE J. WU Abstract. The homotopy theory of the suspensions of the real projective plane is largely investigated. The homotopy groups are computed
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 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 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 informationREAL INSTANTONS, DIRAC OPERATORS AND QUATERNIONIC CLASSIFYING SPACES PAUL NORBURY AND MARC SANDERS Abstract. Let M(k; SO(n)) be the moduli space of ba
REAL INSTANTONS, DIRAC OPERATORS AND QUATERNIONIC CLASSIFYING SPACES PAUL NORBURY AND MARC SANDERS Abstract. Let M(k; SO(n)) be the moduli space of based gauge equivalence classes of SO(n) instantons on
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 informationON THE HOMOTOPY TYPE OF INFINITE STUNTED PROJECTIVE SPACES FREDERICK R. COHEN* AND RAN LEVI
ON THE HOMOTOPY TYPE OF INFINITE STUNTED PROJECTIVE SPACES FREDERICK R. COHEN* AND RAN LEVI 1. introduction Consider the space X n = RP /RP n 1 together with the boundary map in the Barratt-Puppe sequence
More informationSETS OF DEGREES OF MAPS BETWEEN SU(2)-BUNDLES OVER THE 5-SPHERE
SETS OF DEGREES OF MAPS BETWEEN SU(2)-BUNDLES OVER THE 5-SPHERE JEAN-FRANÇOIS LAFONT AND CHRISTOFOROS NEOFYTIDIS ABSTRACT. We compute the sets of degrees of maps between SU(2)-bundles over S 5, i.e. between
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 informationRiemann sphere and rational maps
Chapter 3 Riemann sphere and rational maps 3.1 Riemann sphere It is sometimes convenient, and fruitful, to work with holomorphic (or in general continuous) functions on a compact space. However, we wish
More informationz, w = z 1 w 1 + z 2 w 2 z, w 2 z 2 w 2. d([z], [w]) = 2 φ : P(C 2 ) \ [1 : 0] C ; [z 1 : z 2 ] z 1 z 2 ψ : P(C 2 ) \ [0 : 1] C ; [z 1 : z 2 ] z 2 z 1
3 3 THE RIEMANN SPHERE 31 Models for the Riemann Sphere One dimensional projective complex space P(C ) is the set of all one-dimensional subspaces of C If z = (z 1, z ) C \ 0 then we will denote by [z]
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 informationQUALIFYING EXAM, Fall Algebraic Topology and Differential Geometry
QUALIFYING EXAM, Fall 2017 Algebraic Topology and Differential Geometry 1. Algebraic Topology Problem 1.1. State the Theorem which determines the homology groups Hq (S n \ S k ), where 1 k n 1. Let X S
More informationPart II. Riemann Surfaces. Year
Part II Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 96 Paper 2, Section II 23F State the uniformisation theorem. List without proof the Riemann surfaces which are uniformised
More informationLECTURE 6: J-HOLOMORPHIC CURVES AND APPLICATIONS
LECTURE 6: J-HOLOMORPHIC CURVES AND APPLICATIONS WEIMIN CHEN, UMASS, SPRING 07 1. Basic elements of J-holomorphic curve theory Let (M, ω) be a symplectic manifold of dimension 2n, and let J J (M, ω) be
More informationnx ~p Us x Uns2"'-1,» i
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 96, Number 4, April 1986 LOOP SPACES OF FINITE COMPLEXES AT LARGE PRIMES C. A. MCGIBBON AND C. W. WILKERSON1 ABSTRACT. Let X be a finite, simply
More informationMODULI OF ALGEBRAIC SL 3 -VECTOR BUNDLES OVER ADJOINT REPRESENTATION
Masuda, K. Osaka J. Math. 38 (200), 50 506 MODULI OF ALGEBRAIC SL 3 -VECTOR BUNDLES OVER ADJOINT REPRESENTATION KAYO MASUDA (Received June 2, 999). Introduction and result Let be a reductive complex algebraic
More informationFundamental group. Chapter The loop space Ω(X, x 0 ) and the fundamental group
Chapter 6 Fundamental group 6. The loop space Ω(X, x 0 ) and the fundamental group π (X, x 0 ) Let X be a topological space with a basepoint x 0 X. The space of paths in X emanating from x 0 is the space
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 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 informationSYMPLECTIC GEOMETRY: LECTURE 5
SYMPLECTIC GEOMETRY: LECTURE 5 LIAT KESSLER Let (M, ω) be a connected compact symplectic manifold, T a torus, T M M a Hamiltonian action of T on M, and Φ: M t the assoaciated moment map. Theorem 0.1 (The
More informationFINITE CHEVALLEY GROUPS AND LOOP GROUPS
FINITE CHEVALLEY GROUPS AND LOOP GROUPS MASAKI KAMEKO Abstract. Let p, l be distinct primes and let q be a power of p. Let G be a connected compact Lie group. We show that there exists an integer b such
More information1 Structures 2. 2 Framework of Riemann surfaces Basic configuration Holomorphic functions... 3
Compact course notes Riemann surfaces Fall 2011 Professor: S. Lvovski transcribed by: J. Lazovskis Independent University of Moscow December 23, 2011 Contents 1 Structures 2 2 Framework of Riemann surfaces
More informationA GLIMPSE OF ALGEBRAIC K-THEORY: Eric M. Friedlander
A GLIMPSE OF ALGEBRAIC K-THEORY: Eric M. Friedlander During the first three days of September, 1997, I had the privilege of giving a series of five lectures at the beginning of the School on Algebraic
More informationTRANSITIVE HOLONOMY GROUP AND RIGIDITY IN NONNEGATIVE CURVATURE. Luis Guijarro and Gerard Walschap
TRANSITIVE HOLONOMY GROUP AND RIGIDITY IN NONNEGATIVE CURVATURE Luis Guijarro and Gerard Walschap Abstract. In this note, we examine the relationship between the twisting of a vector bundle ξ over a manifold
More informationHomotopy and geometric perspectives on string topology
Homotopy and geometric perspectives on string topology Ralph L. Cohen Stanford University August 30, 2005 In these lecture notes I will try to summarize some recent advances in the new area of study known
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 informationA Problem of Hsiang-Palais-Terng on Isoparametric Submanifolds
arxiv:math/0312251v1 [math.dg] 12 Dec 2003 A Problem of Hsiang-Palais-Terng on Isoparametric Submanifolds Haibao Duan Institute of Mathematics, Chinese Academy of Sciences, Beijing 100080, dhb@math.ac.cn
More informationRIMS-1897 On multiframings of 3-manifolds By Tatsuro SHIMIZU December 2018 RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES KYOTO UNIVERSITY, Kyoto, Japan
RIMS-1897 On multiframings of 3-manifolds By Tatsuro SHIMIZU December 2018 RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES KYOTO UNIVERSITY, Kyoto, Japan On multiframings of 3 manifolds Tatsuro Shimizu 1
More informationPacific Journal of Mathematics
Pacific Journal of Mathematics GROUP ACTIONS ON POLYNOMIAL AND POWER SERIES RINGS Peter Symonds Volume 195 No. 1 September 2000 PACIFIC JOURNAL OF MATHEMATICS Vol. 195, No. 1, 2000 GROUP ACTIONS ON POLYNOMIAL
More informationLemma 1.3. The element [X, X] is nonzero.
Math 210C. The remarkable SU(2) Let G be a non-commutative connected compact Lie group, and assume that its rank (i.e., dimension of maximal tori) is 1; equivalently, G is a compact connected Lie group
More informationOn the homotopy invariance of string topology
On the homotopy invariance of string topology Ralph L. Cohen John Klein Dennis Sullivan August 25, 2005 Abstract Let M n be a closed, oriented, n-manifold, and LM its free loop space. In [3] a commutative
More informationLinear connections on Lie groups
Linear connections on Lie groups The affine space of linear connections on a compact Lie group G contains a distinguished line segment with endpoints the connections L and R which make left (resp. right)
More informationAN EXTENTION OF MILNOR'S INEQUALITY
Kasagawa, R. Osaka J. Math. 36 (1999), 27-31 AN EXTENTION OF MILNOR'S INEQUALITY RYOJI KASAGAWA (Received June 16, 1997) 1. Introduction Milnor showed the following theorem. Theorem 1.1 (Milnor [4]). Let
More informationNon-separable AF-algebras
Non-separable AF-algebras Takeshi Katsura Department of Mathematics, Hokkaido University, Kita 1, Nishi 8, Kita-Ku, Sapporo, 6-81, JAPAN katsura@math.sci.hokudai.ac.jp Summary. We give two pathological
More informationQUATERNIONIC LINE BUNDLES OVER QUATERNIONIC PROJECTIVE SPACES
Math. J. Okayama Univ. 48 (2006), 87 101 QUATERNIONIC LINE BUNDLES OVER QUATERNIONIC PROJECTIVE SPACES Daciberg L. GONÇALVES and Mauro SPREAFICO 1. Introduction Let ξ be an F-line bundle over a space X,
More informationSOME EXERCISES. By popular demand, I m putting up some fun problems to solve. These are meant to give intuition for messing around with spectra.
SOME EXERCISES By popular demand, I m putting up some fun problems to solve. These are meant to give intuition for messing around with spectra. 1. The algebraic thick subcategory theorem In Lecture 2,
More informationNote that the first map is in fact the zero map, as can be checked locally. It follows that we get an isomorphism
11. The Serre construction Suppose we are given a globally generated rank two vector bundle E on P n. Then the general global section σ of E vanishes in codimension two on a smooth subvariety Y. If E is
More information1. Algebraic vector bundles. Affine Varieties
0. Brief overview Cycles and bundles are intrinsic invariants of algebraic varieties Close connections going back to Grothendieck Work with quasi-projective varieties over a field k Affine Varieties 1.
More informationLarge automorphism groups of 16-dimensional planes are Lie groups
Journal of Lie Theory Volume 8 (1998) 83 93 C 1998 Heldermann Verlag Large automorphism groups of 16-dimensional planes are Lie groups Barbara Priwitzer, Helmut Salzmann Communicated by Karl H. Hofmann
More informationA DISCONNECTED DEFORMATION SPACE OF RATIONAL MAPS. for Tan Lei
A DISCONNECTED DEFORMATION SPACE OF RATIONAL MAPS ERIKO HIRONAKA AND SARAH KOCH for Tan Lei Abstract. The deformation space of a branched cover f : (S 2, A) (S 2, B) is a complex submanifold of a certain
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 informationSAMELSON PRODUCTS IN Sp(2) 1. Introduction We calculated certain generalized Samelson products of Sp(2) and give two applications.
SAMELSON PRODUCTS IN Sp(2) HIROAKI HAMANAKA, SHIZUO KAJI, AND AKIRA KONO 1. Introduction We calculated certain generalized Samelson products of Sp(2) and give two applications. One is the classification
More informationLIMITING CASES OF BOARDMAN S FIVE HALVES THEOREM
Proceedings of the Edinburgh Mathematical Society Submitted Paper Paper 14 June 2011 LIMITING CASES OF BOARDMAN S FIVE HALVES THEOREM MICHAEL C. CRABB AND PEDRO L. Q. PERGHER Institute of Mathematics,
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 informationMultiplicative properties of Atiyah duality
Multiplicative properties of Atiyah duality Ralph L. Cohen Stanford University January 8, 2004 Abstract Let M n be a closed, connected n-manifold. Let M denote the Thom spectrum of its stable normal bundle.
More informationCobordant differentiable manifolds
Variétés différentiables cobordant, Colloque Int. du C. N. R. S., v. LII, Géométrie différentielle, Strasbourg (1953), pp. 143-149. Cobordant differentiable manifolds By R. THOM (Strasbourg) Translated
More informationIntroduction to Braid Groups Joshua Lieber VIGRE REU 2011 University of Chicago
Introduction to Braid Groups Joshua Lieber VIGRE REU 2011 University of Chicago ABSTRACT. This paper is an introduction to the braid groups intended to familiarize the reader with the basic definitions
More informationApplications of Homotopy
Chapter 9 Applications of Homotopy In Section 8.2 we showed that the fundamental group can be used to show that two spaces are not homeomorphic. In this chapter we exhibit other uses of the fundamental
More informationLECTURE 15-16: PROPER ACTIONS AND ORBIT SPACES
LECTURE 15-16: PROPER ACTIONS AND ORBIT SPACES 1. Proper actions Suppose G acts on M smoothly, and m M. Then the orbit of G through m is G m = {g m g G}. If m, m lies in the same orbit, i.e. m = g m for
More informationOn stable homotopy equivalences
On stable homotopy equivalences by R. R. Bruner, F. R. Cohen 1, and C. A. McGibbon A fundamental construction in the study of stable homotopy is the free infinite loop space generated by a space X. This
More informationA FUCHSIAN GROUP PROOF OF THE HYPERELLIPTICITY OF RIEMANN SURFACES OF GENUS 2
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 28, 2003, 69 74 A FUCHSIAN GROUP PROOF OF THE HYPERELLIPTICITY OF RIEMANN SURFACES OF GENUS 2 Yolanda Fuertes and Gabino González-Diez Universidad
More informationHiggs Bundles and Character Varieties
Higgs Bundles and Character Varieties David Baraglia The University of Adelaide Adelaide, Australia 29 May 2014 GEAR Junior Retreat, University of Michigan David Baraglia (ADL) Higgs Bundles and Character
More informationTHE BORSUK-ULAM THEOREM FOR GENERAL SPACES
THE BORSUK-ULAM THEOREM FOR GENERAL SPACES PEDRO L. Q. PERGHER, DENISE DE MATTOS, AND EDIVALDO L. DOS SANTOS Abstract. Let X, Y be topological spaces and T : X X a free involution. In this context, a question
More informationMathematical Research Letters 3, (1996) EQUIVARIANT AFFINE LINE BUNDLES AND LINEARIZATION. Hanspeter Kraft and Frank Kutzschebauch
Mathematical Research Letters 3, 619 627 (1996) EQUIVARIANT AFFINE LINE BUNDLES AND LINEARIZATION Hanspeter Kraft and Frank Kutzschebauch Abstract. We show that every algebraic action of a linearly reductive
More information0.1 Complex Analogues 1
0.1 Complex Analogues 1 Abstract In complex geometry Kodaira s theorem tells us that on a Kähler manifold sufficiently high powers of positive line bundles admit global holomorphic sections. Donaldson
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 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 informationHYPERKÄHLER MANIFOLDS
HYPERKÄHLER MANIFOLDS PAVEL SAFRONOV, TALK AT 2011 TALBOT WORKSHOP 1.1. Basic definitions. 1. Hyperkähler manifolds Definition. A hyperkähler manifold is a C Riemannian manifold together with three covariantly
More informationChromatic homotopy theory at height 1 and the image of J
Chromatic homotopy theory at height 1 and the image of J Vitaly Lorman Johns Hopkins University April 23, 2013 Key players at height 1 Formal group law: Let F m (x, y) be the p-typification of the multiplicative
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 informationCourse 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra
Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra D. R. Wilkins Contents 3 Topics in Commutative Algebra 2 3.1 Rings and Fields......................... 2 3.2 Ideals...............................
More informationStable bundles on CP 3 and special holonomies
Stable bundles on CP 3 and special holonomies Misha Verbitsky Géométrie des variétés complexes IV CIRM, Luminy, Oct 26, 2010 1 Hyperkähler manifolds DEFINITION: A hyperkähler structure on a manifold M
More information1 Hermitian symmetric spaces: examples and basic properties
Contents 1 Hermitian symmetric spaces: examples and basic properties 1 1.1 Almost complex manifolds............................................ 1 1.2 Hermitian manifolds................................................
More informationSegre classes of tautological bundles on Hilbert schemes of surfaces
Segre classes of tautological bundles on Hilbert schemes of surfaces Claire Voisin Abstract We first give an alternative proof, based on a simple geometric argument, of a result of Marian, Oprea and Pandharipande
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 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 informationTHE FIRST ADAMS-NOVIKOV DIFFERENTIAL FOR THE SPECTRUM T (m)
THE FIRST ADAMS-NOVIKOV DIFFERENTIAL FOR THE SPECTRUM T (m) DOUGLAS C. RAVENEL Abstract. There are p-local spectra T (m) with BP (T (m)) = BP [t,..., t m]. In this paper we determine the first nontrivial
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 informationMorse theory and stable pairs
Richard A. SCGAS 2010 Joint with Introduction Georgios Daskalopoulos (Brown University) Jonathan Weitsman (Northeastern University) Graeme Wilkin (University of Colorado) Outline Introduction 1 Introduction
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 information5.3 The Upper Half Plane
Remark. Combining Schwarz Lemma with the map g α, we can obtain some inequalities of analytic maps f : D D. For example, if z D and w = f(z) D, then the composition h := g w f g z satisfies the condition
More informationBUNDLES, COHOMOLOGY AND TRUNCATED SYMMETRIC POLYNOMIALS
BUNDLES, COHOMOLOGY AND TRUNCATED SYMMETRIC POLYNOMIALS ALEJANDRO ADEM AND ZINOVY REICHSTEIN Abstract. The cohomology of the classifying space BU(n) of the unitary group can be identified with the the
More informationNONNEGATIVE CURVATURE AND COBORDISM TYPE. 1. Introduction
NONNEGATIVE CURVATURE AND COBORDISM TYPE ANAND DESSAI AND WILDERICH TUSCHMANN Abstract. We show that in each dimension n = 4k, k 2, there exist infinite sequences of closed simply connected Riemannian
More informationThe Yang-Mills equations over Klein surfaces
The Yang-Mills equations over Klein surfaces Chiu-Chu Melissa Liu & Florent Schaffhauser Columbia University (New York City) & Universidad de Los Andes (Bogotá) Seoul ICM 2014 Outline 1 Moduli of real
More informationEquivalence of the Combinatorial Definition (Lecture 11)
Equivalence of the Combinatorial Definition (Lecture 11) September 26, 2014 Our goal in this lecture is to complete the proof of our first main theorem by proving the following: Theorem 1. The map of simplicial
More informationFUNDAMENTAL GROUPS AND THE VAN KAMPEN S THEOREM. Contents
FUNDAMENTAL GROUPS AND THE VAN KAMPEN S THEOREM SAMUEL BLOOM Abstract. In this paper, we define the fundamental group of a topological space and explore its structure, and we proceed to prove Van-Kampen
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 informationMath. Res. Lett. 13 (2006), no. 1, c International Press 2006 ENERGY IDENTITY FOR ANTI-SELF-DUAL INSTANTONS ON C Σ.
Math. Res. Lett. 3 (2006), no., 6 66 c International Press 2006 ENERY IDENTITY FOR ANTI-SELF-DUAL INSTANTONS ON C Σ Katrin Wehrheim Abstract. We establish an energy identity for anti-self-dual connections
More informationMultiplicity of singularities is not a bi-lipschitz invariant
Multiplicity of singularities is not a bi-lipschitz invariant Misha Verbitsky Joint work with L. Birbrair, A. Fernandes, J. E. Sampaio Geometry and Dynamics Seminar Tel-Aviv University, 12.12.2018 1 Zariski
More informationA UNIVERSAL PROPERTY FOR Sp(2) AT THE PRIME 3
Homology, Homotopy and Applications, vol. 11(1), 2009, pp.1 15 A UNIVERSAL PROPERTY FOR Sp(2) AT THE PRIME 3 JELENA GRBIĆ and STEPHEN THERIAULT (communicated by Donald M. Davis) Abstract We study a universal
More informationArtin s map in stable homology. Ulrike Tillmann
Artin s map in stable homology Ulrike Tillmann Abstract: Using a recent theorem of Galatius [G] we identify the map on stable homology induced by Artin s injection of the braid group β n into the automorphism
More informationarxiv: v1 [math.co] 25 Jun 2014
THE NON-PURE VERSION OF THE SIMPLEX AND THE BOUNDARY OF THE SIMPLEX NICOLÁS A. CAPITELLI arxiv:1406.6434v1 [math.co] 25 Jun 2014 Abstract. We introduce the non-pure versions of simplicial balls and spheres
More informationRational Hopf G-spaces with two nontrivial homotopy group systems
F U N D A M E N T A MATHEMATICAE 146 (1995) Rational Hopf -spaces with two nontrivial homotopy group systems by Ryszard D o m a n (Poznań) Abstract. Let be a finite group. We prove that every rational
More informationEXERCISES IN MODULAR FORMS I (MATH 726) (2) Prove that a lattice L is integral if and only if its Gram matrix has integer coefficients.
EXERCISES IN MODULAR FORMS I (MATH 726) EYAL GOREN, MCGILL UNIVERSITY, FALL 2007 (1) We define a (full) lattice L in R n to be a discrete subgroup of R n that contains a basis for R n. Prove that L is
More informationINSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD
INSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD () Instanton (definition) (2) ADHM construction (3) Compactification. Instantons.. Notation. Throughout this talk, we will use the following notation:
More informationON A DISTANCE DEFINED BY THE LENGTH SPECTRUM ON TEICHMÜLLER SPACE
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 28, 2003, 315 326 ON A DISTANCE DEFINED BY THE LENGTH SPECTRUM ON TEICHMÜLLER SPACE Hiroshige Shiga Tokyo Institute of Technology, Department of
More informationk=0 /D : S + S /D = K 1 2 (3.5) consistently with the relation (1.75) and the Riemann-Roch-Hirzebruch-Atiyah-Singer index formula
20 VASILY PESTUN 3. Lecture: Grothendieck-Riemann-Roch-Hirzebruch-Atiyah-Singer Index theorems 3.. Index for a holomorphic vector bundle. For a holomorphic vector bundle E over a complex variety of dim
More information1 Spaces and operations Continuity and metric spaces Topological spaces Compactness... 3
Compact course notes Topology I Fall 2011 Professor: A. Penskoi transcribed by: J. Lazovskis Independent University of Moscow December 23, 2011 Contents 1 Spaces and operations 2 1.1 Continuity and metric
More informationTutorial on Differential Galois Theory III
Tutorial on Differential Galois Theory III T. Dyckerhoff Department of Mathematics University of Pennsylvania 02/14/08 / Oberflockenbach Outline Today s plan Monodromy and singularities Riemann-Hilbert
More information