DOCSLIB.ORG

TOPOLOGY of FIBER BUNDLES by HAINAN ZHANG B.S

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
TOPOLOGY of FIBER BUNDLES by HAINAN ZHANG B.S TOPOLOGY OF FIBER BUNDLES by HAINAN ZHANG B.S., Kansas State University, 2012 A THESIS submitted in partial fulfillment of the requirements for the degree MASTER OF SCIENCE Department of Mathematics College of Arts and Sciences KANSAS STATE UNIVERSITY Manhattan, Kansas 2014 Approved by: Major Professor David Auckly Copyright Hainan Zhang 2014 Abstract This report introduces the fiber bundles. It includes the definitions of fiber bundles such as vector bundles and principal bundles, with some interesting examples. Reduction of the structure groups, and covering homotopy theorem and some specific computation using obstruction classes, Cech cohomology, Stiefel-Whitney classes, and first Chern classes are included. Table of Contents Table of Contents iv List of Figuresv 1 Introduction to Fiber Bundles1 1.1 Trivial Bundles . .4 1.2 Vector Bundles . .4 1.3 Principal Bundles . .4 2 2 Example: The Bundle of Frames (π; SO3; S ; SO2) 6 3 Trivialization and Section 14 4 Reduction of the structure group 17 5 Covering Homotopy Theorem 21 5.1 Hopf-fibration: (S1 ! S3 ! S2)......................... 24 6 Obstruction Theory 25 6.1 Stiefel-Whitney Classes . 32 6.2 Chern Class . 33 7 Cechˇ Cohomology 35 Bibliography 40 iv List of Figures 6.1 Cell decomposition . 27 6.2 Graph one of 1st Stiefel Class under section sigma . 28 6.3 Graph two of 1st Stiefel Class under section sigma . 29 6.4 Graph of 1st Chern class . 31 v Chapter 1 Introduction to Fiber Bundles A fiber bundle is a space that is locally a product space, but globally may have a different topological structure. We can describe it as a continuous surjective map π : E ! B that in small neighborhood U behaves like the projection of U × F to U. Usually, we call the map π the projection of the bundle. The space E, B, F are known as total space, base space, and fiber space respectively. In general, there are two kinds of fiber bundles. The trivial bundle is B × F , the map π is the canonical projection from the product space to the first factor. In the non-trivial case, bundles will have totally different topological structures globally, such as the M¨obius strip and the Klein bottle, as well as non-trivial covering spaces. Definition 1. A fiber bundle is a structure (π; E; B; F ), where E, B, and F are topological spaces and π : E ! B is a continuous surjection satisfying a local triviality condition, which is called the bundle projection. The space E is the total space, B is called the base space of the bundle, and F is the fiber space. The locally triviality condition is that for any point x 2 B, there exists an open neighborhood 1 −1 Ux ⊂ B such that π (Ux) is homeomorphic to the product space Ux × F such that the following diagram commutes: −1 φ π (Ux) / Ux × F π proj Ux (1.1) where proj : Ux × F ! Ux is the natural projection. The set of all f(Uj; φj)g is called a local trivialization of the bundle. Thus for any p in B, the preimage π−1(p) is homeomorphic to F and is called the fiber over p. Every fiber bundle π : E ! B is an open map since the projection of products are open maps. Therefore B carries the quotient topology determined by the map π. Transition Functions If let U = fUαgα2I be a open cover of the base space, then one can construct many different local homeomorphisms by taking different open sets. Then, the nonempty interscetion of two local homeomorphisms will give two different systems for the same point in the over- lap. Transition functions give a invertible transformation of the fiber over the point in two different systems. Given a vector bundle E ! B, and a pair of neighborhood Uα and Uβ over which the bundle trivializes via two homeomorphisms, 2 −1 hα : Uα × F ! π (Uα) (1.2) −1 hβ : Uβ × F ! π (Uβ) (1.3) the composite function αβ is defined to be −1 hα ◦ hβ : Uαβ × F ! Uαβ × F (1.4) where Uαβ is Uα \ Uβ, which is well defined on the overlap by −1 hα ◦ hβ(x; v) := αβ(x; v) = (x; αβ(x)(v)) (1.5) where the map αβ : Uαβ ! Homeo(F) (1.6) sends x to (x) such that αβ(x): F ! F (1.7) is the automorphism of the fiber F with f ! αβ(x)(f) for all f 2 F . Lemma 1. αβ ◦ βγ = αγ. −1 Proof. Note αβ is defined to be hα ◦ hβ : Uαβ × F ! Uαβ × F , therefore, αβ ◦ βγ = −1 −1 −1 hα ◦ hβ ◦ hβ ◦ hγ = hα ◦ hγ = αγ. Thus, by this fact, for all x 2 Uα \ Uβ \ Uγ and f 2 F we have αβ ◦ βγ(x; f) = αγ(x; f) ) αβ(x; βγ(x)(f)) = (x; αγ(x)(f)) ) (x; αβ(x)( βγ(x)(f))) = (x; αγ(x)(f)) ) (x; αβ ◦ βγ(x)(f)) = (x; αγ(x)(f)) ) αβ ◦ βγ = αγ 3 1.1 Trivial Bundles Let E = B × F and let π : E ! B be the projection onto the first factor. Then E is a fiber bundle over B. The bundle E is not just a locally product but globally one. Thus, a trivial bundle is homeomorphic to the Cartesian product. Lemma 2. Any fiber bundle over a contractible CW-complex is trivial. Proof. Let B be a contractible CW-complex. Since B is contractible, there is a point x 2 B and a homotopy between idB and the constant map f : B ! fxg. Then, we can define ∗ ∗ ∗ ∗ the pullback with idB(E) w f (E). Therefore, E = IdB and f = B × Ex imply that 1 E w B × Ex, which is a trivial bundle. 1.2 Vector Bundles Definition 2. A vector bundle is a fiber bundle in which every fiber is a vector space. For every point in the base space B, there is an open neighborhood U and a homeomorphism h : U × Rk ! π−1(U) such that for all x 2 U, 1. (π ◦ φ)(x; v) = x for all vectors v in Rk 2. v ! φ(x; v) is an isomorphism between the vector space Rk and π−1(x), and the vector bundle over the fields are defined similarly. 1.3 Principal Bundles A principal bundle P is a mathematical object which formalizes some essential features of the Cartesian product X ×G of a topological space with a structure group G equipped with: 1. An action of G on P , analogous to (x; g)h = (x; gh) for a product space 4 2. A natural projection onto X, this is just the projection onto the first factor, (x; g) ! x Definition 3. A principle G-bundle P , where G denotes any topological group, is a fiber bundle π : P ! X together with a continuous right action P ×G ! P such that G preserves the fibers of P and acts freely and transitively on them i.e. let Px be a fiber over any point x, then 8 x0 and x1 2 Px, 9 some g 2 G such that x0g = x1 The definition implies that each fiber of the bundle is homeomorphic to the group G itself. Unlike the product space, principal bundles P lack a preferred choice of identity section. Likewise, there is no general projection onto G generalizing the projection onto the second factor, X × G ! G which exists for the Cartesian product. 5 Chapter 2 Example: The Bundle of Frames 2 (π; SO3; S ; SO2) Every rotation maps an orthonormal basis of R3 to another orthonormal basis. Like any lin- ear transformation of a finite-dimensional vector space, a rotation can always be represented by a matrix. The orthonormality condition can be expressed as RT R = I, where R is any ro- tation matrix in R3 and I is the 3×3 identity matrix. Matrices for which this property holds are called orthogonal matrices. The group of the orthogonal matrices is called orthogonal T group, and SO3 := fA 2 Mat3×3jA A = I; det(A) = 1g is the subgroup of the orthogo- nal group with determinant +1. It is called the special orthogonal group, and the det(A) = 1 2 Similiarly, SO2 is the orientation preserving orthogonal group in R . In order to carry out a rotation using the point (x, y) to be rotated is written as a vector, then multiplied by a matrix from the rotating angle θ: 2 3 2 3 2 3 x0 cos θ − sin θ x 6 7 6 7 6 7 4 5 = 4 5 4 5 (2.1) y0 sin θ cos θ y 6 where (x0, y0) are the coordinates of the point after rotation, and the formulae for x0 and y0 can be seen to be x0 = x cos θ − y sin θ y0 = x sin θ + y cos θ: Note that unlike higher dimension, the group of vector rotations in R2 is commutative. 2 Before showing the bundle structure of (π; SO3; S ; SO2), let us consider the map πe : 2 T SO3=SO2 ! S defined by πe([A]) = A · e3 where A 2 SO3 and e3 = [0; 0; 1] . Here, the equivalence class of [A] is defined via the equivalence 2 3 B 0 6 7 A ∼ A · 4 5 0 1 where B 2 SO2. We are trying to show that this map is a bijective continuous closed map, therefore, a homeomorphism. 2 3 B 0 6 7 Proof. We first show that this map is well-defined. Let [A1] = [A2], so A1 = A2 · 4 5.
Load more

Recommended publications
  • Persistent Obstruction Theory for a Model Category of Measures with Applications to Data Merging
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, SERIES B Volume 8, Pages 1–38 (February 2, 2021) https://doi.org/10.1090/btran/56 PERSISTENT OBSTRUCTION THEORY FOR A MODEL CATEGORY OF MEASURES WITH APPLICATIONS TO DATA MERGING ABRAHAM D. SMITH, PAUL BENDICH, AND JOHN HARER Abstract. Collections of measures on compact metric spaces form a model category (“data complexes”), whose morphisms are marginalization integrals. The fibrant objects in this category represent collections of measures in which there is a measure on a product space that marginalizes to any measures on pairs of its factors. The homotopy and homology for this category allow measurement of obstructions to finding measures on larger and larger product spaces. The obstruction theory is compatible with a fibrant filtration built from the Wasserstein distance on measures. Despite the abstract tools, this is motivated by a widespread problem in data science. Data complexes provide a mathematical foundation for semi- automated data-alignment tools that are common in commercial database software. Practically speaking, the theory shows that database JOIN oper- ations are subject to genuine topological obstructions. Those obstructions can be detected by an obstruction cocycle and can be resolved by moving through a filtration. Thus, any collection of databases has a persistence level, which measures the difficulty of JOINing those databases. Because of its general formulation, this persistent obstruction theory also encompasses multi-modal data fusion problems, some forms of Bayesian inference, and probability cou- plings. 1. Introduction We begin this paper with an abstraction of a problem familiar to any large enterprise. Imagine that the branch offices within the enterprise have access to many data sources.
    [Show full text]
  • Obstructions, Extensions and Reductions. Some Applications Of
    Obstructions, Extensions and Reductions. Some applications of Cohomology ∗ Luis J. Boya Departamento de F´ısica Te´orica, Universidad de Zaragoza. E-50009 Zaragoza, Spain [email protected] October 22, 2018 Abstract After introducing some cohomology classes as obstructions to ori- entation and spin structures etc., we explain some applications of co- homology to physical problems, in especial to reduced holonomy in M- and F -theories. 1 Orientation For a topological space X, the important objects are the homology groups, H∗(X, A), with coefficients A generally in Z, the integers. A bundle ξ : E(M, F ) is an extension E with fiber F (acted upon by a group G) over an space M, noted ξ : F → E → M and it is itself a Cech˘ cohomology element, arXiv:math-ph/0310010v1 8 Oct 2003 ξ ∈ Hˆ 1(M,G). The important objects here the characteristic cohomology classes c(ξ) ∈ H∗(M, A). Let M be a manifold of dimension n. Consider a frame e in a patch U ⊂ M, i.e. n independent vector fields at any point in U. Two frames e, e′ in U define a unique element g of the general linear group GL(n, R) by e′ = g · e, as GL acts freely in {e}. An orientation in M is a global class of frames, two frames e (in U) and e′ (in U ′) being in the same class if det g > 0 where e′ = g · e in the overlap of two patches. A manifold is orientable if it is ∗To be published in the Proceedings of: SYMMETRIES AND GRAVITY IN FIELD THEORY.
    [Show full text]
  • PIECEWISE LINEAR TOPOLOGY Contents 1. Introduction 2 2. Basic
    PIECEWISE LINEAR TOPOLOGY J. L. BRYANT Contents 1. Introduction 2 2. Basic Definitions and Terminology. 2 3. Regular Neighborhoods 9 4. General Position 16 5. Embeddings, Engulfing 19 6. Handle Theory 24 7. Isotopies, Unknotting 30 8. Approximations, Controlled Isotopies 31 9. Triangulations of Manifolds 33 References 35 1 2 J. L. BRYANT 1. Introduction The piecewise linear category offers a rich structural setting in which to study many of the problems that arise in geometric topology. The first systematic ac- counts of the subject may be found in [2] and [63]. Whitehead’s important paper [63] contains the foundation of the geometric and algebraic theory of simplicial com- plexes that we use today. More recent sources, such as [30], [50], and [66], together with [17] and [37], provide a fairly complete development of PL theory up through the early 1970’s. This chapter will present an overview of the subject, drawing heavily upon these sources as well as others with the goal of unifying various topics found there as well as in other parts of the literature. We shall try to give enough in the way of proofs to provide the reader with a flavor of some of the techniques of the subject, while deferring the more intricate details to the literature. Our discussion will generally avoid problems associated with embedding and isotopy in codimension 2. The reader is referred to [12] for a survey of results in this very important area. 2. Basic Definitions and Terminology. Simplexes. A simplex of dimension p (a p-simplex) σ is the convex closure of a n set of (p+1) geometrically independent points {v0, .
    [Show full text]
  • Math 527 - Homotopy Theory Obstruction Theory
    Math 527 - Homotopy Theory Obstruction theory Martin Frankland April 5, 2013 In our discussion of obstruction theory via the skeletal filtration, we left several claims as exercises. The goal of these notes is to fill in two of those gaps. 1 Setup Let us recall the setup, adopting a notation similar to that of May x 18.5. Let (X; A) be a relative CW complex with n-skeleton Xn, and let Y be a simple space. Given two maps fn; gn : Xn ! Y which agree on Xn−1, we defined a difference cochain n d(fn; gn) 2 C (X; A; πn(Y )) whose value on each n-cell was defined using the following \double cone construction". Definition 1.1. Let H; H0 : Dn ! Y be two maps that agree on the boundary @Dn ∼= Sn−1. The difference construction of H and H0 is the map 0 n ∼ n n H [ H : S = D [Sn−1 D ! Y: Here, the two terms Dn are viewed as the upper and lower hemispheres of Sn respectively. 1 2 The two claims In this section, we state two claims and reduce their proof to the case of spheres and discs. Proposition 2.1. Given two maps fn; gn : Xn ! Y which agree on Xn−1, we have fn ' gn rel Xn−1 if and only if d(fn; gn) = 0 holds. n n−1 Proof. For each n-cell eα of X n A, consider its attaching map 'α : S ! Xn−1 and charac- n n−1 teristic map Φα :(D ;S ) ! (Xn;Xn−1).
    [Show full text]
  • Arxiv:2101.06841V2 [Math.AT] 1 Apr 2021
    C2-EQUIVARIANT TOPOLOGICAL MODULAR FORMS DEXTER CHUA Abstract. We compute the homotopy groups of the C2 fixed points of equi- variant topological modular forms at the prime 2 using the descent spectral sequence. We then show that as a TMF-module, it is isomorphic to the tensor product of TMF with an explicit finite cell complex. Contents 1. Introduction 1 2. Equivariant elliptic cohomology 5 3. The E2 page of the DSS 9 4. Differentials in the DSS 13 5. Identification of the last factor 25 6. Further questions 29 Appendix A. Connective C2-equivariant tmf 30 Appendix B. Sage script 35 References 38 1. Introduction Topological K-theory is one of the first examples of generalized cohomology theories. It admits a natural equivariant analogue — for a G-space X, the group 0 KOG(X) is the Grothendieck group of G-equivariant vector bundles over X. In 0 particular, KOG(∗) = Rep(G) is the representation ring of G. As in the case of non-equivariant K-theory, this extends to a G-equivariant cohomology theory KOG, and is represented by a genuine G-spectrum. We shall call this G-spectrum KO, omitting the subscript, as we prefer to think of this as a global equivariant spectrum — one defined for all compact Lie groups. The G-fixed points of this, written KOBG, is a spectrum analogue of the representation ring, BG 0 BG −n with π0KO = KOG(∗) = Rep(G) (more generally, πnKO = KOG (∗)). These fixed point spectra are readily computable as KO-modules. For example, KOBC2 = KO _ KO; KOBC3 = KO _ KU: arXiv:2101.06841v2 [math.AT] 1 Apr 2021 This corresponds to the fact that C2 has two real characters, while C3 has a real character plus a complex conjugate pair.
    [Show full text]
  • ON TOPOLOGICAL and PIECEWISE LINEAR VECTOR FIELDS-T
    ON TOPOLOGICAL AND PIECEWISE LINEAR VECTOR FIELDS-t RONALD J. STERN (Received31 October1973; revised 15 November 1974) INTRODUCTION THE existence of a non-zero vector field on a differentiable manifold M yields geometric and algebraic information about M. For example, (I) A non-zero vector field exists on M if and only if the tangent bundle of M splits off a trivial bundle. (2) The kth Stiefel-Whitney class W,(M) of M is the primary obstruction to obtaining (n -k + I) linearly independent non-zero vector fields on M [37; 0391. In particular, a non-zero vector field exists on a compact manifold M if and only if X(M), the Euler characteristic of M, is zero. (3) Every non-zero vector field on M is integrable. Now suppose that M is a topological (TOP) or piecewise linear (IX) manifold. What is the appropriate definition of a TOP or PL vector field on M? If M were differentiable, then a non-zero vector field is just a non-zero cross-section of the tangent bundle T(M) of M. In I%2 Milnor[29] define! the TOP tangent microbundle T(M) of a TOP manifold M to be the microbundle M - M x M G M, where A(x) = (x, x) and ~(x, y) = x. If M is a PL manifold, the PL tangent microbundle T(M) is defined similarly if one works in the category of PL maps of polyhedra. If M is differentiable, then T(M) is CAT (CAT = TOP or PL) microbundle equivalent to T(M).
    [Show full text]
  • Symmetric Obstruction Theories and Hilbert Schemes of Points On
    Symmetric Obstruction Theories and Hilbert Schemes of Points on Threefolds K. Behrend and B. Fantechi March 12, 2018 Abstract Recall that in an earlier paper by one of the authors Donaldson- Thomas type invariants were expressed as certain weighted Euler char- acteristics of the moduli space. The Euler characteristic is weighted by a certain canonical Z-valued constructible function on the moduli space. This constructible function associates to any point of the moduli space a certain invariant of the singularity of the space at the point. In the present paper, we evaluate this invariant for the case of a sin- gularity which is an isolated point of a C∗-action and which admits a symmetric obstruction theory compatible with the C∗-action. The an- swer is ( 1)d, where d is the dimension of the Zariski tangent space. − We use this result to prove that for any threefold, proper or not, the weighted Euler characteristic of the Hilbert scheme of n points on the threefold is, up to sign, equal to the usual Euler characteristic. For the case of a projective Calabi-Yau threefold, we deduce that the Donaldson- Thomas invariant of the Hilbert scheme of n points is, up to sign, equal to the Euler characteristic. This proves a conjecture of Maulik-Nekrasov- Okounkov-Pandharipande. arXiv:math/0512556v1 [math.AG] 24 Dec 2005 1 Contents Introduction 3 Symmetric obstruction theories . 3 Weighted Euler characteristics and Gm-actions . .. .. 4 Anexample........................... 5 Lagrangian intersections . 5 Hilbertschemes......................... 6 Conventions........................... 6 Acknowledgments. .. .. .. .. .. .. 7 1 Symmetric Obstruction Theories 8 1.1 Non-degenerate symmetric bilinear forms .
    [Show full text]
  • Almost Complex Structures and Obstruction Theory
    ALMOST COMPLEX STRUCTURES AND OBSTRUCTION THEORY MICHAEL ALBANESE Abstract. These are notes for a lecture I gave in John Morgan's Homotopy Theory course at Stony Brook in Fall 2018. Let X be a CW complex and Y a simply connected space. Last time we discussed the obstruction to extending a map f : X(n) ! Y to a map X(n+1) ! Y ; recall that X(k) denotes the k-skeleton of X. n+1 There is an obstruction o(f) 2 C (X; πn(Y )) which vanishes if and only if f can be extended to (n+1) n+1 X . Moreover, o(f) is a cocycle and [o(f)] 2 H (X; πn(Y )) vanishes if and only if fjX(n−1) can be extended to X(n+1); that is, f may need to be redefined on the n-cells. Obstructions to lifting a map p Given a fibration F ! E −! B and a map f : X ! B, when can f be lifted to a map g : X ! E? If X = B and f = idB, then we are asking when p has a section. For convenience, we will only consider the case where F and B are simply connected, from which it follows that E is simply connected. For a more general statement, see Theorem 7.37 of [2]. Suppose g has been defined on X(n). Let en+1 be an n-cell and α : Sn ! X(n) its attaching map, then p ◦ g ◦ α : Sn ! B is equal to f ◦ α and is nullhomotopic (as f extends over the (n + 1)-cell).
    [Show full text]
  • MATH 227A – LECTURE NOTES 1. Obstruction Theory a Fundamental Question in Topology Is How to Compute the Homotopy Classes of M
    MATH 227A { LECTURE NOTES INCOMPLETE AND UPDATING! 1. Obstruction Theory A fundamental question in topology is how to compute the homotopy classes of maps between two spaces. Many problems in geometry and algebra can be reduced to this problem, but it is monsterously hard. More generally, we can ask when we can extend a map defined on a subspace and then how many extensions exist. Definition 1.1. If f : A ! X is continuous, then let M(f) = X q A × [0; 1]=f(a) ∼ (a; 0): This is the mapping cylinder of f. The mapping cylinder has several nice properties which we will spend some time generalizing. (1) The natural inclusion i: X,! M(f) is a deformation retraction: there is a continous map r : M(f) ! X such that r ◦ i = IdX and i ◦ r 'X IdM(f). Consider the following diagram: A A × I f◦πA X M(f) i r Id X: Since A ! A × I is a homotopy equivalence relative to the copy of A, we deduce the same is true for i. (2) The map j : A ! M(f) given by a 7! (a; 1) is a closed embedding and there is an open set U such that A ⊂ U ⊂ M(f) and U deformation retracts back to A: take A × (1=2; 1]. We will often refer to a pair A ⊂ X with these properties as \good". (3) The composite r ◦ j = f. One way to package this is that we have factored any map into a composite of a homotopy equivalence r with a closed inclusion j.
    [Show full text]
  • Differential Topology Forty-Six Years Later
    Differential Topology Forty-six Years Later John Milnor n the 1965 Hedrick Lectures,1 I described its uniqueness up to a PL-isomorphism that is the state of differential topology, a field that topologically isotopic to the identity. In particular, was then young but growing very rapidly. they proved the following. During the intervening years, many problems Theorem 1. If a topological manifold Mn without in differential and geometric topology that boundary satisfies Ihad seemed totally impossible have been solved, 3 n 4 n often using drastically new tools. The following is H (M ; Z/2) = H (M ; Z/2) = 0 with n ≥ 5 , a brief survey, describing some of the highlights then it possesses a PL-manifold structure that is of these many developments. unique up to PL-isomorphism. Major Developments (For manifolds with boundary one needs n > 5.) The corresponding theorem for all manifolds of The first big breakthrough, by Kirby and Sieben- dimension n ≤ 3 had been proved much earlier mann [1969, 1969a, 1977], was an obstruction by Moise [1952]. However, we will see that the theory for the problem of triangulating a given corresponding statement in dimension 4 is false. topological manifold as a PL (= piecewise-linear) An analogous obstruction theory for the prob- manifold. (This was a sharpening of earlier work lem of passing from a PL-structure to a smooth by Casson and Sullivan and by Lashof and Rothen- structure had previously been introduced by berg. See [Ranicki, 1996].) If B and B are the Top PL Munkres [1960, 1964a, 1964b] and Hirsch [1963].
    [Show full text]
  • ALMOST COMPLEX STRUCTURES and OBSTRUCTION THEORY Let
    ALMOST COMPLEX STRUCTURES AND OBSTRUCTION THEORY MICHAEL ALBANESE Abstract. These are notes for a lecture I gave in John Morgan's Homotopy Theory course at Stony Brook in Fall 2018. Let X be a CW complex and Y a simply connected space. Last time we discussed the obstruction to extending a map f : X(n) ! Y to a map X(n+1) ! Y ; recall that X(k) denotes the k-skeleton n+1 of X. There is an obstruction o(f) 2 C (X; πn(Y )) which vanishes if and only if f can be (n+1) n+1 extended to X . Moreover, o(f) is a cocycle and [o(f)] 2 H (X; πn(Y )) vanishes if and only (n+1) if fjX(n−1) can be extended to X ; that is, f may need to be redefined on the n-cells. Obstructions to lifting a map p Given a fibration F ! E −! B and a map f : X ! B, when can f be lifted to a map g : X ! E? If X = B and f = idB, then we are asking when p has a section. For convenience, we will only consider the case where F and B are simply connected, from which it follows that E is simply connected. For a more general statement, see Theorem 7.37 of [2]. Suppose g has been defined on X(n). Let en+1 be an n-cell and α : Sn ! X(n) its attaching map, then p ◦ g ◦ α : Sn ! B is equal to f ◦ α and is nullhomotopic (as f extends over the (n + 1)-cell).
    [Show full text]
  • Topological Modular Forms with Level Structure
    Topological modular forms with level structure Michael Hill,∗ Tyler Lawsony February 3, 2015 Abstract The cohomology theory known as Tmf, for \topological modular forms," is a universal object mapping out to elliptic cohomology theories, and its coefficient ring is closely connected to the classical ring of modular forms. We extend this to a functorial family of objects corresponding to elliptic curves with level structure and modular forms on them. Along the way, we produce a natural way to restrict to the cusps, providing multiplicative maps from Tmf with level structure to forms of K-theory. In particular, this allows us to construct a connective spectrum tmf0(3) consistent with properties suggested by Mahowald and Rezk. This is accomplished using the machinery of logarithmic structures. We construct a presheaf of locally even-periodic elliptic cohomology theo- ries, equipped with highly structured multiplication, on the log-´etalesite of the moduli of elliptic curves. Evaluating this presheaf on modular curves produces Tmf with level structure. 1 Introduction The subject of topological modular forms traces its origin back to the Witten genus. The Witten genus is a function taking String manifolds and producing elements of the power series ring C[[q]], in a manner preserving multiplication and bordism classes (making it a genus of String manifolds). It can therefore be described in terms of a ring homomorphism from the bordism ring MOh8i∗ to this ring of power series. Moreover, Witten explained that this should factor through a map taking values in a particular subring MF∗: the ring of modular forms. An algebraic perspective on modular forms is that they are universal func- tions on elliptic curves.
    [Show full text]