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BUNDLES, STIEFELβWHITNEY CLASSES, & BRAID GROUPS PETER J. HAINE Abstract. We explain the basics of vector bundles and principal πΊ-bundles, as well as describe the relationship between vector bundles and principal Oπ -bundles. We then give an axiomatization of StiefelβWhitney classes and derive some basic consequences. Finally, we apply the theory of bundles and StiefelβWhitney to understand the space of unordered configurations of π distinct points in the plane. Contents 0. Overview 1. Preliminaries 2. Principal Bundles 3. Vector Bundles 4. Relating Principal Bundles & Vector Bundles 5. StiefelβWhitney Classes Classes & Universal Bundles 6. Configuration Spaces & Braid Groups References 1 2 3 6 9 13 15 19 0. Overview There are two main goals of this paper. The first is to provide a survey of the theory of principal πΊ-bundles, vector bundles, and StiefelβWhitney classes. The purpose of this is to highlight the main ideas of the theory in order to give the unfamiliar reader a general understanding of how the different pieces fit together, while also reminding the experienced reader about the central ideas of the theory. The second goal is to use these tools to study the space UCπ (C) of unordered configurations of π distinct points in the plane. We begin by outlining some preliminary definitions and notations, then in §§2 and 3 explain the basics of principal πΊ-bundles and vector bundles. In §4, we explain the relationship between principal bundles, and vector bundles, universal principal bundles and classifying spaces. In § 5 we give an axiomatization of StiefelβWhitney classes and derive some basic consequences from the axioms. In §6 we tie all of these subjects together to study braid groups, and the spaces UCπ (C), which are the classifying spaces of the braid groups. Prerequisites. We assume that the reader is familiar with basic algebraic topology: the fundamental group and covering spaces, basic homology and cohomology theory, and the definitions of the higher homotopy groups, i.e., the material from [7, Chs. 1β4] or [11, Chs. 1β20]. We also assume some familiarity with the basic terminology of category theory, namely, categories, functors, and natural transformations β we refer the unfamiliar reader to [16, Ch. 1] for an excellent introduction to the subject. Date: November 21, 2015. 1 2 PETER J. HAINE Acknowledgments. I would like to thank Nir Gadish for being a patient teacher and excellent mentor, Benson Farb for suggesting this project, and Peter May for making it possible for me to attend the REU program at the University of Chicago as well as making it such a pleasurable experience, and providing many helpful comments on this paper. I would also like to thank Randy Van Why for his helpful conversations about the material in this paper, and significant contributions to the ideas appearing in § 6. Finally, I would like to thank Janson Ng for many, many wonderful conversations while we were at UChicago and since. 1. Preliminaries In this section we outline some of our notational conventions and recall some preliminary definitions that we assume the reader has familiarity with and use extensively throughout this paper. In particular, we are interested in actions of topological groups on spaces. 1.1. Definition. Suppose that πΊ is a topological group. A right πΊ-action on a space π is a continuous map π βΆ π × πΊ π, written π(π₯, π) = π₯π, so that π₯1 = π₯, where 1 is the identity of πΊ, and (π₯π)πβ² = π₯(ππβ² ) for all π₯ β π and π, πβ² β πΊ. We call π a right πΊ-space. Left πΊ-spaces are defined analogously. We say that a πΊ-action is free if π₯π = π₯ if and only if π is the identity, i.e., the action has no fixed points. Suppose that π is a (right) πΊ-space. The orbit space π/πΊ is the quotient space of π under the equivalence relation βΌ on π given by saying that π₯ βΌ π₯π for each π₯ β π and all π β πΊ, i.e., the equivalence relation that associates all of the points in an orbit. 1.2. Example. For each positive integer π, the space Rπ is a left GLπ (R)-space, with the action given by left matrix multiplication. 1.3. Definition. Suppose that π and π are πΊ-spaces and that π βΆ π π is a continuous map. We say that π is πΊ-equivariant, or a πΊ-map if π commutes with the πΊ-actions on π and π, i.e., for all π β πΊ and π₯ β π we have π(π₯π) = π(π₯)π. 1.4. Definition. Suppose that π, πβ² βΆ π π are πΊ-equivariant maps. We say that π and πβ² are πΊ-homotopic if there exists a homotopy β βΆ π × [0, 1] π between π and πβ² so that β is πΊ-equivariant, with respect to the πΊ-action on π × πΌ given by (π₯, π‘)π β (π₯π, π‘). Such a homotopy is called a πΊ-homotopy. There are also a number of situations where the underlying point-set topology affects the theory of bundles. Here we recall some relevant definitions from point-set topology. 1.5. Definition. Let U be a collection of subsets of a space π. A collection V of subsets of π is called a refinement of U if every set in V is contained in a set in U. We say that V is an open refinement of U if all of the sets in V are open in π. 1.6. Example. A space π is compact if and only if every open cover U of π has a finite open refinement V which covers π. 1.7. Definition. A collection U of subsets of a space π is called locally finite if each point π₯ β π has a neighborhood that intersects only finitely many elements of U. 1.8. Definition. A space π is paracompact if every open cover U of π has a locally finite open refinement V which covers π. Milnor and Stasheff [12, Ch. 5 §8] also assume that paracompact spaces are Hausdorff, following the lead of Bourbaki [3, Pt. i Ch. 9 §1 Def. 1], which assumes that compact spaces are Hausdorff. We follow Munkres [13, §41] and do not assume the Hausdorff condition, as BUNDLES, STIEFELβWHITNEY CLASSES, & BRAID GROUPS 3 many students in the present generation learned topology from [13]. Detailed expositions on paracompactness can be found in [12, Ch. 5 §8; 13, §§39 and 41]. 1.9. Example. As Husemöller [9, §4.9] notes, a Hausdorff space π is paracompact if and only if every open cover of π is numerable. 1.10. Example. Metrizable spaces are paracompact. Various proofs of this fact can be found in [13, Tm. 41.4; 14; 17]. 1.11. Example. Manifolds are second-countable and regular, hence metrizable by the Urysohn metrization theorem. Thus manifolds are paracompact. 1.12. Remark. Milnor and Stasheff [12, §5.8] note that nearly all familiar spaces are paracompact and Hausdorff. 1.13. Definition. Suppose that π is a topological space and π βΆ π R is a continuous map. The support of π, denoted by supp(π), is the closure of the subspace πβ1 (R β {0}). 1.14. Definition. A partition of unity on a space π is a collection {ππΌ }πΌβπ΄ of continuous functions ππΌ βΆ π [0, 1] satisfying the following properties: (1.14.a) the family of supports {supp(ππΌ )}πΌβπ΄ is locally finite, (1.14.b) and for all π₯ β π we have βπΌβπ΄ ππΌ (π₯) = 1. 1.15. Definition. An open cover {ππΌ }πΌβπ΄ of a topological space π is called numerable if there exists a partition of unity {ππΌ }πΌβπ΄ so that supp(ππΌ ) β ππΌ for each πΌ β π΄. Finally, a bit of notation. 1.16. Notation. We often use the arrow β β to denote an inclusion or an embedding, the arrow β β to denote a surjection, and the arrow β βΌ β to denote an isomorphism. 2. Principal Bundles Bundles are important tools in topology and geometry. In this section we begin our study of bundles by studying principal πΊ-bundles, which are in many ways similar to covering spaces, but with the additional information of a topological group πΊ acting on the covering space. We only outline the basic theory; more complete accounts can be found in [9, Ch. 4 §§2β4; 20, Ch. 14 §1]. 2.1. Definition. Suppose that πΊ is a topological group. A principal πΊ-bundle π over a topological space π΅ consists of the following data: (2.1.a) a right πΊ-space πΈ(π), (2.1.b) and a continuous surjection π βΆ πΈ(π) π΅, subject to the following axioms. (2.1.1) The map π is constant-on-orbits: for each π₯ β πΈ and π β πΊ we have π(π₯π) = π(π₯). (2.1.2) There exists a numerableβ open cover {ππΌ }πΌβπ΄ of π΅ and a πΊ-equivariant homeomorphism ππΌ βΆ πβ1 (ππΌ ) βΌ ππΌ × πΊ β Following the conventions of May [11, Ch. 23 §8], we require that the cover be numerable. In light of Example 1.9, this is always true when the base space is paracompact and Hausdorff. Since most of the results are typically proven when the base space is paracompact and Hausdorff, there is nothing lost assuming this. 4 PETER J. HAINE making the following triangle commute πβ1 (ππΌ ) βΌ ππΌ ππΌ × πΊ prπ π πΌ ππΌ , where the πΊ-action on ππΌ ×πΊ is given by (π’, π)πβ² β (π’, ππβ² ). Such a map ππΌ is called a local trivialization of π over ππΌ . An open cover as in (2.1.2) is called a bundle atlas. 2.2. Example. For any space π΅ and any topological group πΊ, the space π΅ × πΊ is a right πΊspace with the action (π, π)πβ² β (π, ππβ² ). The projection prπ΅ βΆ π΅ × πΊ π΅ onto π΅, yields a principal πΊ-bundle over π΅ called the trivial πΊ-bundle. 2.3. Observation. It is an immediate consequence of Definition 2.1 that if π βΆ πΈ π΅ is a principal πΊ-bundle, then πΊ acts freely on πΈ, so π factors through the quotient map πβΆ πΈ πΈ/πΊ and induced a continuous bijection πΈ/πΊ π΅. Since π and π are open maps, they are quotient maps, the continuous bijection πΈ/πΊ π΅ is a homeomorphism. 2.4. Example. For each positive integer π there is a free Z/2 action on ππ given by the antipodal map sending π₯ βπ₯. Real projective π-space RPπ is the orbit space of ππ under this action, and the projection ππ βΆ ππ RPπ is a principal Z/2-bundle (where we equip Z/2 with the discrete topology). The elements of RPπ are unordered pairs {π₯, βπ₯}, where π₯ β ππ . 2.5. Example. Suppose that π β₯ 2 is an integer and consider the sphere π2πβ1 as a subspace of Cπ . There is a free π1 action on π2πβ1 given by coordinate-wise scalar multiplication. The orbit space under this action is the complex projective space CPπβ1 , and the orbit map CPπβ1 has the structure of a principal π1 -bundle. In the special case that π = 2, π βΆ π2πβ1 1 π3 π2 . since CP β π2 , this is the Hopf fibration π1 2.6. Definition. Suppose that π βΆ πΈ π΅ and πβ² βΆ πΈβ² π΅β² are principal πΊ-bundles. A morβ² phism of principal πΊ-bundles, or simply a bundle map π π, consists of a πΊ-equivariant map π Μ βΆ πΈβ² πΈ and a continuous map π βΆ π΅β² π΅ making the following square commute πΈβ² πΜ πΈ πβ² π΅β² We often denote a bundle map by (π,Μ π) βΆ πβ² π π π΅. π. 2.7. Example. For each positive integer π, the inclusion of ππ into the equator of ππ+1 is Z/2-equivariant, and induces an inclusion RPπ RPπ+1 given by taking Z/2-orbits. Since the inclusions respect the Z/2-action, the square ππ ππ RPπ ππ+1 ππ+1 RPπ+1 commutes, hence the two inclusions define a map of principal Z/2-bundles. BUNDLES, STIEFELβWHITNEY CLASSES, & BRAID GROUPS 5 We can also construct new bundles from old bundles, probably the most important construction being the pullback bundle. To explain this construction, we first must recall how to construct pullbacks of topological spaces. 2.8. Recollection. Suppose that we have continuous maps of topological spaces π βΆ π π and π βΆ π π. The pullback π ×π π is the terminal space equipped with maps so that the square πΜ π ×π π β πβ π π π π π π commutes. The pullback can be regarded as the following subset of the product π × π π ×π π = { (π₯, π¦) β π × π | π(π₯) = π(π¦) }, with the subspace topology. 2.9. Definition. Suppose that π βΆ πΈ π΅ is a principal πΊ-bundle and that π is a continuous map π΅β² π΅. We form the pullback π΅β² ×π΅ πΈ in Top. Regarding the pullback π΅β² ×π΅ πΈ as a subset of the product π΅β² × πΈ as above, we give a πΊ-action on π΅β² × πΈ by setting (πβ² , π)π β (πβ² , ππ) for all π β πΊ and (πβ² , π) β π΅β² × πΈ. This action is well-defined by the above construction of the pullback and because the map π is constant-on-orbits. If {ππΌ }πΌβπ΄ is a bundle atlas for π with trivializations {ππΌ }πΌβπ΄ , then setting ππΌβ² β πβ1 (ππΌ ) gives an open cover {ππΌβ² }πΌβπ΄ of π΅β² , and πβ πβΆ π΅β² ×π΅ πΈ π΅β² is trivial over each ππΌβ² . We call πβ π the bundle induced by π, or the pullback along π, and denote the total space of this bundle by πΈ(πβ π). The amazing fact about the pullback is the following homotopy theorem. 2.10. Theorem ([20, Tm. 14.3.3]). Suppose that π βΆ πΈ π΅ is a principal πΊ-bundle and that π0 , π1 βΆ π΅β² π΅ are homotopic maps. Then the induced bundles π0β π and π1β π are isomorphic. This theorem tells us that we can transform the rather rigid data of a principal bundle into purely homotopical data. Moreover, homotopies between base spaces can be lifted, giving homotopic bundle maps, with the appropriate notion of a homotopy of bundle maps. 2.11. Definition. Suppose that π βΆ πΈ π΅ and πβ² βΆ πΈβ² π΅β² are principal πΊ-bundles, and that Suppose that (π0Μ , π0 ), (π1Μ , π1 )βΆ πβ² π Μ Μ are bundle maps. A bundle homotopy from (π0 , π0 ) to (π1 , π1 ) is a pair of homotopies (β,Μ β), where βΜ is a πΊ-homotopy from π0Μ to π1Μ , and β is a homotopy from π0 to π1 , with the property that for each π‘ β [0, 1] the pair (β|Μ πΈβ² ×{π‘} , β|π΅β² ×{π‘} ) is a bundle map. 2.12. Theorem ([20, Tm. 14.3.4]). Suppose that π βΆ πΈ π΅ and πβ² βΆ πΈβ² π΅β² are principal β² Μ π is a bundle map, and π0 is homotopic to a map π1 βΆ π΅β² π΅ via πΊ-bundles, (π0 , π0 )βΆ π a homotopy β. Then there exists a πΊ-equivariant map π1Μ βΆ πΈβ² πΈ so that (π1Μ , π1 ) is a bundle map, as well as a πΊ-homotopy βΜ between π0Μ and π1Μ so that (β,Μ β) is a bundle homotopy. 6 PETER J. HAINE 3. Vector Bundles In this section we continue our study of bundles by studying vector bundles, which are in many ways similar to covering spaces, but with the additional data of a vector space structure on each fiber. Vector bundles appear throughout topology and differential geometry, among many other areas, and are the basis of topological πΎ-theory, a generalized cohomology theory widely used in geometric and low-dimensional topology. We only outline the basic theory; more complete accounts can be found in [8, Ch. 1; 9, Ch. 3 §§1β4; 12, Ch. 2; 20, Ch. 14 §2]. The definition of a vector bundle is very similar to that of a principal πΊ-bundle. 3.1. Definition. A real vector bundle π over a topological space π΅, called the base space, consists of the following data: (3.1.a) a topological space πΈ, called the total space, equipped with a continuous surjection πβΆ πΈ π΅, (3.1.b) and for each π β π΅, the structure of a finite-dimensional real vector space on the fiber πβ1 (π). subject to the following axiom. (3.1.1) There exists a numerableβ‘ open cover {ππΌ }πΌβπ΄ of π΅, a natural number ππΌ for each πΌ β π΄, and a homeomorphism ππΌ βΆ πβ1 (ππΌ ) ππΌ × RππΌ so that the triangle πβ1 (ππΌ ) βΌ ππΌ ππΌ × RππΌ prπ π ππΌ πΌ commutes, and, moreover, for any π β ππΌ the restriction of ππΌ to the fiber πβ1 (π) is a linear isomorphism of real vector spaces. Such a map ππΌ is called a local trivialization of π over ππΌ . An open cover as in (3.1.1) is called a bundle atlas. We say that a real vector bundle π is an π-plane bundle if each fiber πβ1 (π) for π β π΅ has the structure of an π-dimensional real vector space, and a line bundle in the special case that π = 1. 3.2. Example. For any space π΅ and any natural number π, the projection prπ΅ βΆ π΅ × Rπ π΅ has the structure of an π-plane bundle. The vector space structure on each fiber is given by πΌ(π, π’) + π½(π, π£) β (π, πΌπ£ + π½π’), for πΌ, π½ β R and π’, π£ β Rπ . This bundle is called the trivial π-plane bundle. The following example is one of the most important examples of vector bundles, and is integral to the axiomatization of StiefelβWhitney classes in §5. 3.3. Example. Define a line bundle βπ over RPπ in the following manner. The total space πΈ(βπ ) is the subset of RPπ ×Rπ+1 consisting of pairs ({±π₯}, π’), where π’ lies on the line in Rπ+1 spanned by π₯, and the projection βπ βΆ πΈ(βπ ) RPπ is given by sending ({±π₯}, π’) {±π₯}. π+1 The fiber over {±π₯} is simply the line in R spanned by π₯. We endow each fiber with its usual 1-dimensional real vector space structure. We call βπ is called the canonical line bundle over RPπ . In particular, since RP1 β π1 , the total space of β1 is an open Möbius band. A generalization of this example is given by the Stiefel and Grassmanian manifolds. β‘Again, following May [11, Ch. 23 §1] we assume that the cover is numerable. BUNDLES, STIEFELβWHITNEY CLASSES, & BRAID GROUPS 7 3.4. Definition. An orthonormal π-frame in Rπ is a π-tuple of linearly independent orthonormal vectors in Rπ , i.e., an orthonormal basis for a π-dimensional subspace of Rπ . The set of all orthonormal π-frames in Rπ is a subset of the π-fold product Rπ × β― × Rπ , called the Stiefel manifold, denoted St π (Rπ ). There is a natural action of the real orthogonal group Oπ on the Stiefel manifold St π (Rπ ) given by multiplying each vector in the an orthonormal π-frame by an orthogonal matrix. The Grassmann manifold, or simply Grassmannian, Grπ (Rπ ) is the orbit space St π (Rπ )/ Oπ . The elements of Grπ (Rπ ) are associated with π-dimensional subspaces of Rπ . Moreover, the projection St π (Rπ ) Grπ (Rπ ) is a principal Oπ -bundle. 3.5. Warning. The notation ππ (Rπ ) is very commonly used instead of St π (Rπ ) for the Stiefel manifold, but we prefer the notation St π (Rπ ) as we find it more evocative. Likewise, the notation πΊπ (Rπ ) is commonly used for the Grassmanian Grπ (Rπ ), but, again, we prefer the notation Grπ (Rπ ) as we find it more evocative, and clearer in the context of this paper as we always use πΊ to denote a (topological) group. 3.6. Example. For each positive integer π, there is a homeomorphism Gr1 (Rπ ) β RPπ . It is a standard result [12, Lem. 5.1] that the Grassmannian manifolds are compact topological manifolds. Moreover, Grassmanians have a beautifully combinatorial cell structure; for a description see [12, Ch. 6]. We can start to see the connection between principal bundles and vector bundles by constructing a vector bundle from the principal Oπ -bundle St π (Rπ ) Grπ (Rπ ). 3.7. Example. Construct a canonical π-plane bundle πΎπ,π over Grπ (Rπ ) as follows. Regarding elements of Grπ (Rπ ) as π-dimensional subspaces of Rπ , the total space is πΈ(πΎπ,π ) β {(π, π£) | π£ β π β Grπ (Rπ )}, topologized as a subspace of Grπ (Rπ ) × Rπ . The projection πΎπ,π βΆ πΈ(πΎπ,π ) Grπ (Rπ ) is deπ fined by πΎπ,π (π, π£) = π, and the fiber over a π-plane π β Grπ (R ) has the obvious vector space structure. Since the proof that the vector bundle πΎπ,π is locally trivial is not the case of primary interest, we omit it β a proof can be found in [12, Lem. 5.2]. The canonical π-plane bundles πΎπ,π are integral to the relationship between vector bundles and principal bundles. To begin to explain this relationship, we need to discuss morphisms of vector bundles. Just like with principal πΊ-bundles, morphisms are given by commutative squares, however, with vector bundles we have two reasonable choices for the morphisms. 3.8. Definition. Suppose that π βΆ πΈ π΅ and πβ² βΆ πΈβ² π΅β² are vector bundles. A bundle β² morphism π π consists of a pair (π,Μ π) of maps making the following square commute πΈβ² πΜ πβ² π΅β² πΈ π π π΅, where π Μ is linear on each fiber. We call a bundle morphism (π,Μ π) a bundle map if π Μ is a linear isomorphism on each fiber. Milnor and Stasheff show that any π-plane bundle admits a bundle map to the canonical bundle πΎπ,π , at least when π is large enough. 8 PETER J. HAINE 3.9. Lemma ([12, Lem. 5.3]). For any π-plane bundle π over a base space π΅, there exists a bundle map π πΎπ,π , for sufficiently large π. There are two categories at play here: the category of vector bundles and bundle morphisms, and the category of vector bundles and bundle maps. For the rest of this section we present a few methods to construct new vector bundles from old β we will see that both the category of vector bundles and bundle morphisms and the category of vector bundles and bundle maps are important to consider for these constructions. The main construction that we are interested in is a direct sum operation for vector bundles which is called the Whitney sum. We only present the constructions that are relevant to giving an axiomatization of StiefelβWhitney classes (which is the goal of §5), but almost all of the structures on vector spaces, such as images, (co)kernels, tensor products, and exterior powers, lift to suitable structures of vector bundles, defined fiber-wise. Probably the most elementary construction is restriction to a subspace. 3.10. Definition. Suppose that π βΆ πΈ π΅ is a vector bundle and that π΅β² is a subspace β² β1 β² of π΅. Defining πΈ β π (π΅ ), we see that the restriction of π to πΈβ² gives a vector bundle π|π΅β² βΆ πΈβ² π΅β² , with the property that the fiber of π|π΅β² at π β π΅β² is πβ1 (π). We call π|π΅β² the restriction of π to π΅β² . 3.11. Definition. Suppose that π βΆ πΈ π΅ is a vector bundle and that π is a continuous map π΅β² π΅. We form the pullback π΅β² ×π΅ πΈ in Top. Regarding the pullback π΅β² ×π΅ πΈ as a subset of the product π΅β² × πΈ, we give a vector space structure on each fiber (πβ π)β1 (π) by defining πΌ(πβ² , π₯) + π½(πβ² , π¦) β (πβ² , πΌπ₯ + π½π¦), where πΌ, π½ β R and π₯, π¦ β πΈ are elements in the fiber of π over π(πβ² ), which is a vector space because π is a vector bundle. Then the map π Μ βΆ π΅β² ×π΅ πΈ πΈ given by the pullback sends the vector space (πβ π)β1 (π) isomorphically to πβ1 (π(π)). If {ππΌ }πΌβπ΄ is a bundle atlas for π with trivializations {ππΌ }πΌβπ΄ , then setting ππΌβ² β πβ1 (ππΌ ) gives an open cover {ππΌβ² }πΌβπ΄ of π΅, and defining a homeomorphsim ππΌβ² βΆ ππΌβ² × RππΌ βΌ (πβ π)β1 (ππΌβ² ) by ππΌβ² (π, π₯) β (π, ππΌβ1 (π(π), π₯)) gives a trivialization of the cover. It is clear that gives πβ πβΆ π΅β² ×π΅ πΈ π΅β² the structure of a vector bundle. We call πβ π the bundle induced by π, or the pullback along π, and denote the total space of this induced bundle by πΈ(πβ π). 3.12. Example. Suppose that π βΆ πΈ π΅ is a vector bundle and that π΅β² is a subspace of π΅. β² Let π denote the inclusion π΅ π΅. Then π|π΅β² β πβ π. In analogy with principal πΊ-bundles, homotopic maps between base spaces induce isomorphic vector bundles. 3.13. Theorem. Suppose that π βΆ πΈ π΅ is a vector and that π0 , π1 βΆ π΅β² β maps. Then the induced bundles π0 π and π1β π are isomorphic. π΅ are homotopic Again, theorem tells us that we can transform the rather rigid data of a vector bundle into purely homotopical data. Moreover, homotopies between base spaces can be lifted, giving homotopic bundle maps, which are defined in the same way they were for principal bundles. 3.14. Definition. Suppose that π βΆ πΈ π΅ and πβ² βΆ πΈβ² π΅β² are vector bundles, and that Suppose that (π0Μ , π0 ), (π1Μ , π1 )βΆ πβ² π BUNDLES, STIEFELβWHITNEY CLASSES, & BRAID GROUPS 9 are bundle maps. A bundle homotopy from (π0Μ , π0 ) to (π1Μ , π1 ) is a pair of homotopies (β,Μ β), where βΜ is a homotopy from π0Μ to π1Μ , β is a homotopy from π0 to π1 , so that for each π‘ β [0, 1] the pair (β|Μ πΈβ² ×{π‘} , β|π΅β² ×{π‘} ) is a bundle map. 3.15. Theorem. Suppose that π βΆ πΈ π΅ and πβ² βΆ πΈβ² (π0Μ , π0 )βΆ πβ² π π΅β² are vector bundles, is a bundle map, and π0 is homotopic to a map π1 βΆ π΅β² π΅ via a homotopy β. Then there πΈ so that (π1Μ , π1 ) is a bundle map, along with a homotopy βΜ between exists a map π1Μ βΆ πΈβ² π0Μ and π1Μ so that (β,Μ β) is a bundle homotopy. We can also use pullback bundles to define a direct sum operation on vector bundles, which on fibers is just the direct sum of the vector spaces. The most illuminating way to do this is to first define the product of vector bundles, and then define the direct sum as a pullback of the product. 3.16. Definition. Given vector bundles π βΆ πΈ π΅ and πβ² βΆ πΈβ² π΅β² , we can form the β² β² product bundle π × π whose base space is π΅ × π΅ and total space is πΈ × πΈβ² . The projection πΈ × πΈβ² π΅ × π΅β² is simply the product map π × πβ² . The vector space structure on the fiber β² β1 (π × π ) (π, πβ² ) is given as the product of the vector space structures on the individual fibers, and the bundle atlas on π × πβ² is the obvious one. Notice that since the product of vector spaces agrees with the direct sum, the fiber of π × πβ² over (π, πβ² ) is isomorphic to the direct sum of the fiber of π over π with the fiber of πβ² over πβ² . In this sense, the product vector bundle can be seen as an external direct sum. The product π × πβ² is clearly the categorical product in the category of vector bundles and bundle morphisms. However, the product projections are not isomorphisms on each fiber, so the product projections are bundle morphisms, but not bundle maps. Thus, in order to be able to do standard constructions such as taking products, it often makes sense to work in the category of vector bundles and bundle morphisms, rather than bundle maps. 3.17. Definition. Suppose that π and πβ² are vector bundles a space π΅. Let Ξ βΆ π΅ π΅×π΅ denote the diagonal embedding. The induced bundle Ξβ (π × πβ² ) is called the Whitney sum of π and πβ² , denoted by π β πβ² . In a similar manner to forming that product, or external direct sum, of two vector bundles π βΆ πΈ π΅ and πβ² βΆ πΈβ² π΅β² , we can form an external tensor product π β πβ² of vector bundles, where the fiber of π β πβ² over (π, πβ² ) is the tensor product of the fiber of π over π with the fiber of πβ² over πβ² . Having done this, the tensor product π β πβ² of vector bundles over the same base space is defined as the pullback Ξβ (π β πβ² ) as in Definition 3.17, though there are point-set topological considerations that one must consider. The basic idea is the fiber-wise operations pass to operations on bundles, when suitably topologized. We refer the interested reader to [9, Ch. 3 §8; 12, Ch. 3; 20, Ch. 14 §§2 and 5] for more thorough treatments. 4. Relating Principal Bundles & Vector Bundles We are now ready to describe the amazing relationships between vector bundles and principal bundles, and how this theory lets us translate the much more strict data of bundle maps into homotopical data. This is the result of a very general phenomenon discussed in [11, Ch. 23 §8]; we only provide a brief summary of this phenomenon here. One might hope that given a topological group πΊ, there is some sort of universal principal πΊ-bundle ππΊ with the property that any principal πΊ-bundle π admits a bundle map π ππΊ , 10 PETER J. HAINE which is unique in an appropriate sense. Assuming mild additional hypotheses, this is a reasonable thing to ask for. 4.1. Definition. A principal πΊ-bundle ππΊ βΆ πΈπΊ π΅πΊ is called universal if any principal πΊ-bundle π admits a πΊ-bundle map π ππΊ which is unique up to bundle homotopy. The space π΅πΊ is called the classifying space of πΊ. Amazingly, universal bundles exist, and, in principle, we should provide a construction π΅πΊ, but to avoid unnecessary technical details, of a universal principal πΊ-bundle ππΊ βΆ πΈπΊ we only summarize a common construction using simplicial spaces. Regarding a topological group πΊ as a one-object topological groupoid, we form the nerve ππΊ of πΊ, which is a simplicial space. The geometric realization |ππΊ| of ππΊ is a model for the classifying space π΅πΊ. In this setting, the realization of the simplicial space whose π-simplices are given by πΊπ+1 is a model for πΈπΊ. An early source for this construction is [18]. From the definition, it is immediate that universal πΊ-bundles are unique up to bundle homotopy. This is just one way of characterizing the universal principal πΊ-bundle, though there are many others. One extremely convenient characterization is given by the total space being contractible. 4.2. Theorem. A principal πΊ-bundle π βΆ πΈ π΅ is universal if and only if πΈ is contractible. There is an extremely interesting formal consequence of the existence of universal principal πΊ-bundles; informally, isomorphism classes of principal πΊ-bundles are classified by homotopy classes of maps of the base space into π΅πΊ. To state this formally, we must first introduce a bit of notation and terminology. 4.3. Notation. Write βTop for the homotopy category of topological spaces, whose objects are topological spaces, and a morphism π π is a homotopy class of maps π π. 4.4. Definition. For any topological space π΅, let ππΊ (π΅) be the set of isomorphism classes of numerable principal πΊ-bundles over π΅. Given a map of topological spaces π βΆ π΅β² π΅, sending the isomorphism class of a bundle π βΆ πΈ π΅ to the isomorphism class of it pullback πβ πβΆ πΈ(πβ π) π΅β² defines a set-map ππΊ (π)βΆ ππΊ (π΅) ππΊ (π΅β² ). ππΊ (π) is homotopy invariant. Moreover, it is In light of Theorem 3.13, the assignment π obviously functorial, hence these data assemble into a well defined functor ππΊ βΆ βTopop Set. 4.5. Notation. Suppose that πΆ is a category and π, πβ² β πΆ. We write πΆ(π, πβ² ) for the collection of morphisms π πβ² in πΆ. We write πΆ(β, πβ² )βΆ πΆop Set for the contravariant functor β² represented by π . The assignment on morphisms is given as follows: given a morphism πβΆ π₯ π¦ in πΆ, the morphism πΆ(π, πβ² )βΆ πΆ(π¦, πβ² ) πΆ(π₯, πβ² ) is given by pre-composition β² by π, i.e., the set-map that sends a morphism π βΆ π¦ πβ² to the composite πβ² β πβΆ π₯ πβ² . Given categories πΆ and π· and functors πΉ, πΊ βΆ πΆ π·, we denote a natural transformation πΌ from πΉ to πΊ by πΌ βΆ πΉ βΉ πΊ. Now we are ready to formally state the classification theorem. In order to avoid unnecessary technical detail for readers unfamiliar with formal properties of pullbacks, we only give a sketch of the proof, which provides sufficient detail for the reader familiar with these formalities to check the remaining details. 4.6. Theorem. Suppose that πΊ is a topological group. The contravariant functor ππΊ is representable, with representing object the classifying space π΅πΊ, i.e., there is a natural isomorphism ππΊ β βTop(β, π΅πΊ). BUNDLES, STIEFELβWHITNEY CLASSES, & BRAID GROUPS 11 For the reader with little experience in category theory, morally what Theorem 4.6 says is that the data of an isomorphism class of a numerable principal πΊ-bundle over space π΅ is βthe sameβ as the data of a homotopy class of maps π΅ π΅πΊ. This is an amazing result as it translates the data of a homeomorphism into seemingly weaker homotopical data. This theorem also explains the name classifying space β π΅πΊ classifies principal πΊ-bundles. Moreover, from basic category theory this implies that π΅πΊ is unique up to unique homotopy equivalence. Before we provide a sketch of the proof of Theorem 4.6, let us take a slight diversion in order to explain how Theorem 4.6 fits into the theory. 4.7. Definition. Suppose that π is a right πΊ-space and that π is a left πΊ-space. The balanced product π ×πΊ π is the quotient of π × π under the equivalence relation (π₯π, π¦) βΌ (π₯, ππ¦), for (π₯, π¦) β π × π and π β πΊ. Alternatively, if we convert π into a right πΊ-space by setting π¦π β πβ1 π¦, then π × π is the orbit space of the right πΊ-space π × π under the diagonal action. Given right πΊ-spaces π and πβ² , left πΊ-spaces π and πβ² , and a pair of πΊ-equivariant maps πβΆ π πβ² and π βΆ π πβ² , taking orbit spaces induces a map π ×πΊ πβΆ π ×πΊ π π β² ×πΊ π β² , called the balanced product of the maps π and π. 4.8. Observations. Here are some basic properties of the balanced product. (4.8.a) For any right πΊ-space π, we have π ×πΊ β β π/πΊ, where β is the one-point space. (4.8.b) Regarding πΊ as a left πΊ-space, for any right πΊ-space π, the right action of πΊ on itself by multiplication makes π ×πΊ πΊ into a right πΊ-space, and the action map π×πΊ π induces a πΊ-equivariant homeomorphism π ×πΊ πΊ βΌ π. 4.9. Construction. Suppose that π βΆ πΈ π΅ is a principal πΊ-bundle, and that πΉ is a free left πΊ-space. Taking the quotient of the projection prπΈ βΆ πΈ × πΉ πΈ gives a map π β prπΈ /πΊβΆ πΈ ×πΊ πΉ πΈ/π΅ β π΅ Then the fiber of π over every point of π΅ is πΉ. Moreover, a local trivialization π βΆ prβ1 πΈ (π) π×πΊ of π yields a local trivialization of π via the map πβ1 (π) = prβ1 πΈ (π) ×πΊ πΉ βΌ π×πΊ idπΉ (π × πΊ) ×πΊ πΉ β π × πΉ . We call π βΆ πΈ ×πΊ πΉ πΊ the associated fiber bundle with structure group πΊ and fiber πΉ. An extremely important point of this theory, whose technical details we suppress in order to maintain concise, is that with the correct formal definition of a fiber bundle with structure group πΊ and fiber πΉ, every fiber bundle πΈβ² π΅ is equivalent to one of the form πΈ/πΊ β π΅ for some principal πΊ-bundle πΈ π΅. The conclusion is that for any πΈ ×πΊ πΉ fiber πΉ with a free πΊ-action, the set ππΊ (π΅) is (naturally) isomorphic to the set of equivalence classes of bundles over π΅ with structure group πΊ and fiber πΉ. The key insight into the theory that we are interested lies in the following very specific example of the general phenomenon. 4.10. Example ([11, Ch. 23 §8]). Consider the case of Construction 4.9 when πΊ is the orthogonal group Oπ , and the fiber πΉ is the space Rπ with the usual left Oπ -action given by left matrix multiplication. Fiber bundles over a base space π΅ with structure group Oπ and fiber 12 PETER J. HAINE Rπ are real π-plane bundles. Hence Construction 4.9 gives an explicit way of connecting the theory of vector bundles with theory of principal bundles. Now we return to our original path and present a sketch of the proof of Theorem 4.6. Sketch of the proof of Theorem 4.6. Let ππΊ βΆ πΈπΊ π΅πΊ be the universal principal πΊ-bundle. Since homotopic maps π΅ π΅πΊ have isomorphic induced bundles by Theorem 2.10, the map ππ΅ βΆ βTop(π΅, π΅πΊ) ππΊ (π΅) defined by sending the homotopy class of a map π to the isomorphism class of the induced bundle πβ ππΊ is well-defined. Moreover, it is immediate from the definitions and basic properties of pullbacks that the components ππ΅ assemble into a natural transformation π βΆ βTop(β, π΅πΊ) βΉ ππΊ . Suppose that π βΆ πΈ π΅ is a numerable principal πΊ-bundle. By the universal property of the universal bundle ππΊ βΆ πΈπΊ π΅πΊ, there exists a πΊ-bundle map (π,Μ π) βΆ π ππΊ which is unique up to bundle homotopy. Define a map ππ΅ βΆ ππΊ (π΅) βTop(π΅, π΅πΊ) by sending π to the homotopy class of π βΆ π΅ π΅πΊ. It is an immediate consequence of the definitions that the components ππ΅ assemble into a natural transformation π βΆ ππΊ βΉ βTop(β, π΅πΊ). Again, it is an immediate consequence of the definitions that the composites ππ΅ ππ΅ and ππ΅ ππ΅ are the identity. β‘ The rest of the section is devoted to an extremely important class of examples, which are a special case of the general theory. In § 3 we saw that the projection St π (Rπ ) Grπ (Rπ ) provided a principal Oπ -bundle, and constructed canonical π-plane bundles πΎπ,π . Under the general framework of Construction 4.9, the associated fiber bundles of the principal Oπ bundles St π (Rπ ) Grπ (Rπ ) are the canonical bundles πΎπ,π . Another important aspect of this theory that we saw was that any π-plane bundle admits a bundle map to πΎπ,π for sufficiently large π. What this suggests is that π β, the π-plane bundle πΎπ,π becomes universal. To state this precisely, let us first recall how to take colimits of a sequence of topological spaces. 4.11. Recollection. Suppose that we have a sequence of spaces ordered by inclusion, as displayed below (4.11.1) π1 π2 π3 β―. The colimit of the sequence (4.11.1), denoted by colimπβ₯1 ππ is the space whose underlying set is the union βπβ₯1 ππ , topologized in the following manner: a subset π β colimπβ₯1 ππ is open (resp., closed) if and only if π β© ππ is open (resp., closed) for each π β₯ 1. 4.12. Example. A CW-complex π is the colimit colimπβ₯0 ππ , where ππ is the π-skeleton of π. 4.13. Example. Infinite real projective space RPβ is the colimit colimπβ₯1 RPπ . Similarly, infinite complex projective space CPβ is the colimit colimπβ₯1 CPπ . 4.14. Notation. Write Rβ for vector space whose vectors are infinite sequences (π₯1 , π₯2 , β¦), where only finitely many π₯π are nonzero, with addition and scalar multiplication defined component-wise. As a topological space, Rβ is the colimit colimπβ₯1 Rπ . BUNDLES, STIEFELβWHITNEY CLASSES, & BRAID GROUPS 13 4.15. Definition. The infinite Stiefel manifold St π (Rβ ) is the set of orthonormal π-frames in Rβ topologized as the colimit colimπβ₯0 St π (Rπ+π ). Similarly, the infinite Grassmannian Grπ (Rβ ) is the orbit space St π (Rβ )/ Oπ , where the Oπ -action on St π (Rβ ) is given by multiplying each π-frame by an orthogonal matrix. Alternatively, Grπ (Rβ ) is the set of π-dimensional linear subspaces of Rβ , topologized as the colimit colimπβ₯0 Grπ (Rπ+π ). Like at each finite stage, the projection St π (Rβ ) Grπ (Rβ ) is a principal Oπ -bundle, and by taking the colimit, this bundle becomes the universal principal Oπ -bundle, so Grπ (Rβ ) is the classifying space of Oπ . One particularly easy way to see this is to show that St π (Rβ ) is contractible, a fun exercise which we leave to the reader. In light of Construction 4.9 and Example 4.10, the associated fiber bundle to the universal Grπ (Rβ ) will also have a universal property. Because principal Oπ -bundle St π (Rβ ) the description is enlightening, we provide a description of the fiber bundle associated to the principal Oπ -bundle St π (Rβ ) Grπ (Rβ ); it is the colimit of the canonical π-plane bundles πΎπ,π . 4.16. Construction. In analogy with Example 3.7, construct a canonical π-plane bundle πΎπ over Grπ (Rβ ) as follows. Regarding elements of Grπ (Rβ ) as π-dimensional subspaces of Rβ , the total space is πΈ(πΎπ ) β { (π, π£) | π£ β π β Grπ (Rβ ) }, topologized as a subspace of Grπ (Rβ ) × Rβ . The projection πΎπ βΆ πΈ(πΎπ ) Grπ (Rβ ) is deβ fined by πΎπ (π, π£) = π, and the fiber over a π-plane π β Grπ (R ) has the obvious vector space structure. Applying the general theory from this section to this very specific example, we get the universal property of the canonical π-plane bundle from the universal property of the universal Oπ -bundle St π (Rβ ) Grπ (Rβ ). 4.17. Proposition. Every π-plane bundle π admits a bundle map π to bundle homotopy. πΎπ which is unique up 5. StiefelβWhitney Classes Classes & Universal Bundles In this section we give an axiomatic definition of invariants of vector bundles called StiefelβWhitney classes. StiefelβWhitney classes are cohomology classes of the base space, and provide an incredible amount of information about vector bundles. Most of material in this section can be found in [12, §4]. 5.1. Notation. Suppose that π is a space and π΄ is a commutative ring. We write π»β (π; π΄) for the cohomology ring of π with coefficients in π΄ and with multiplication given by the cup product. Throughout this section, all cohomology is with coefficients in Z/2. With this notation in hand we are ready to describe a set of axioms that completely characterize StiefelβWhiteny classes. 5.2. Axioms. StiefelβWhitney classes are cohomology classes π€π (π) β π»π (π΅; Z/2) for each natural number π assigned to each vector bundle π βΆ πΈ π΅ satisfying the following axioms. 0 (5.2.a) The class π€0 (π) is the identity element of π» (π΅; Z/2), and if π is a π-plane bundle, then π€π (π) = 0 for π > π. (5.2.b) Naturality: If (π,Μ π) βΆ πβ² π is a bundle map, then for each π β N we have π€π (πβ² ) = πβ π€π (π), where πβ βΆ π»π (π΅; Z/2) π»π (π΅β² ; Z/2) is the induced map on cohomology. 14 PETER J. HAINE (5.2.c) Whitney sum formula: If π and πβ² are vector bundles on the same base space π΅, then π π€π (π β πβ² ) = β π€π (π) β£ π€πβπ (πβ² ), π=0 where β£ denotes the cup product on the cohomology ring π»β (π΅; Z/2). (5.2.d) The line bundle β1 on RP1 β π1 has the property that π€1 (β1 ) β 0. It is not obvious that there exists cohomology classes π€π (π) satisfying these axioms, or that, assuming existence, these axioms uniquely characterize cohomology classes satisfying the axioms. Since we are more interested in the properties of StiefelβWhitney classes rather than the proofs of their existence and uniqueness, we refer the reader to [12, Ch. 7] for the proof of uniqueness of StiefelβWhitney classes, which uses the cohomology of Grassmann manifolds (see Definition 3.4) and projective space, and [12, Ch. 8] for a proof of the existence of StiefelβWhitney classes, which makes use of Steenrod squares. An alternative proof utilizing the LerayβHirsch theorem can be found in [8, Ch. 3 §1]. Yet another proof of these facts can be found in [9, Ch. 17 §§1β6]. Assuming that StiefelβWhitney classes exist and are unique, we can derive some immediate consequences from the axioms. 5.2.1. Corollary. If π β πβ² , then π€π (π) = π€π (πβ² ) for all natural numbers π. Proof. This follows immediately from axiom (5.2.b) because π»π (π΅; Z/2) β π»π (π΅β² ; Z/2) for all π β N. β‘ 5.2.2. Corollary. If π is a trivial vector bundle, then π€π (π) = 0 for π > 0. Proof. Since π is trivial, there exists a bundle map from π to a vector bundle over the onepoint space β. Since π»π (β; Z/2) = 0 for π > 0, axiom (5.2.b) implies that π€π (π) = 0 for π > 0. β‘ 5.2.3. Corollary. If π is the trivial vector bundle over a base space π΅, and π is any other vector bundle over π΅, then π€π (π β π) = π€π (π) for all π β N. Proof. This follows immediately from the previous corollary and the Whitney sum formula (5.2.c). β‘ There is another cohomology ring which is not generally isomorphic to π»β (π΅; Z/2) that we can associate to any space. In this ring we can define a total StiefelβWhitney class, which is a compact way of expressing of the StiefelβWhitney classes π€π all at once. 5.3. Definition. For a topological space π΅, write π»ββ (π΅; Z/2) β βπβ₯0 π»π (π΅; Z/2). Regarding the elements of the abelian group π»ββ (π΅; Z/2) as formal power series βπβ₯0 ππ π‘π , where ππ β π»π (π΅; Z/2), we can make π»ββ (π΅; Z/2) into a ring by giving it the following product structure: the product on π»ββ (π΅; Z/2), suggestively denoted by β£, is given by β β β π ( β ππ π‘π ) β£ ( β ππ π‘π ) β β π‘π β ππ β£ ππβπ . π=0 π=0 π=0 π=0 The total StiefelβWhitney class of a π-plane bundle π over π΅ is defined to be the element π€(π) β βπβ₯0 π€π (π). 5.4. Remarks. First notice that the product β£ on π»ββ (π΅; Z/2) is commutative since we are working modulo 2. Second, the Whitney sum formula Axiom 5.2.c can be compactly expressed by saying that π€(π β πβ² ) = π€(π) β£ π€(πβ² ) in π»ββ (π΅; Z/2). BUNDLES, STIEFELβWHITNEY CLASSES, & BRAID GROUPS 15 Now comes one of the major theorems that ties all of the theory together β the Stiefelβ Whitney classes of the canonical π-plane bundles πΎπ generate the Z/2-cohomology ring of Grπ (Rβ ). 5.5. Theorem ([12, Thm. 7.1]). The cohomology ring π»β (Grπ (Rβ ); Z/2) is given by the polynomial algebra Z/2[π€1 (πΎπ ), β¦ , π€π (πΎπ )]. 6. Configuration Spaces & Braid Groups In this section we explore configuration spaces, which have been studied by many, for example, Arnolβd [1, 2] and Segal [19], and have ties to classical homotopy theory. Recently, these spaces have been studied by ChurchβFarb [5] and ChurchβEllenbergβFarb [4], as well as others, and have led to many new discoveries and the theory of representation stability. Configuration spaces are intimately related to braid groups, as the configuration spaces are the classifying spaces of the braid groups. We present the very basics of these subjects; the interested reader should see [10, 15] for more detailed expository accounts. We conclude by reviewing computations of Fuchs [6] on the cohomology of braid groups, and explain how this relates to vector bundles over configuration spaces and a problem posed by Farb. The material from this section is mostly joint work with Randy Van Why, and I would like to once again thank him for his helpful discussions regarding this material. 6.1. Definition. Suppose that π is a topological space and π is a positive integer. The ordered configuration space OCπ (π) is the set OCπ (π) β { (π₯1 , β¦ , π₯π ) β ππ | π₯π β π₯π for π β π } of π-tuples of distinct points in π, topologized as a subspace of the π-fold product ππ . 6.2. Definition. There is a natural action of the symmetric group Ξ£π on OCπ (π) given by permuting the order of the coordinates. The unordered configuration space UCπ (π) is the orbit space OCπ (π)/Ξ£π under the symmetric group action. Elements of UCπ (π) are unordered sets of π distinct points of π. 6.3. Example. If π is a manifold, then the π-fold product ππ is a manifold. The space OCπ (π) is open in ππ , hence is also a manifold. Since the symmetric group Ξ£π acts freely on OCπ (π), the orbit space UCπ (π) is also a manifold. In light of Example 1.11 we see that UCπ (π) is paracompact. There are many definitions for the braid group, a number of which can be found in [10, §§1.1β1.3]. Since we are concerned with their relation to configuration spaces, we simply define them to be the fundamental groups of configurations in the plane. 6.4. Definition. Suppose that π is a positive integer. The braid group π΅π is the fundamental group of the unordered configuration space UCπ (C). The pure braid group ππ is the fundamental group of OCπ (C). 6.5. Remark. As Arnolβd [1] notes, the space UCπ (C) is homeomorphic to the space Polyπ (C) of monic square-free complex polynomials. The elements of Polyπ (C) are sets of roots of monic square-free polynomials, which are precisely sets of π unordered points in C, moreover, the topology given on Polyπ (C) agrees with the topology on UCπ (C). What is perhaps more important is the interpretation of what these groups are. A loop in OCπ (C) based at the point (1, β¦ , π) is an π-tuple of loops (π½1 (π‘), β¦ , π½π (π‘)) so that for all 16 PETER J. HAINE π‘ β [0, 1] and π β π we have π½π (π‘) β π½π (π‘). A loop in UCπ (C) based at the point {1, β¦ , π} is a set of paths {π½1 (π‘), β¦ , π½π (π‘)} so that {π½1 (0), β¦ , π½π (0)} = {1, β¦ , π} = {π½1 (1), β¦ , π½π (1)}, and for all π‘ β [0, 1] and π β π we have π½π (π‘) β π½π (π‘). That is, a loop in UCπ (C) is a collection of paths that collectively start and end at {1, β¦ , π}, but the individual paths do not necessarily start and end at the same point. 6.6. Observation. Notice that the orbit map OCπ (C) UCπ (C) satisfies the hypothesis of [7, Prop. 1.40(a)], hence is a normal covering map. Thus OCπ (C) is aspherical if and only if UCπ (C) is. Also notice that there is a natural fibration π βΆ OCπ+1 (C) OCπ (C) given by forgetting the last coordinate. Each fiber of π is homeomorphic to C β {1, β¦ , π}. Since the spaces C β {1, β¦ , π} are aspherical, the long exact sequence in homotopy shows that all of the spaces OCπ (C) are aspherical. 6.7. Observation. Since the configuration space OCπ (C) is path-connected and locally pathconnected, by basic covering space theory [7, Prop. 1.40(c)] the pure braid group is a subgroup of the braid group, and the symmetric group Ξ£π is isomorphic to π΅π /ππ . Let ππ denote the quotient map π΅π Ξ£π , and let ππ βΆ Ξ£π Oπ denote the regular representation sending a permutation to its permutation matrix. Applying the classifying space functor, the composite map ππ ππ yields a classifying map π΅(ππ ππ )βΆ UCπ (C) β π΅π΅π π΅Oπ β Grπ (Rβ ), which we simply denote by ππ from now on. Fuchs [6, Remark 1.2] notes that the pullback bundle ππβ πΎπ of the canonical π-plane bundle over Grπ (Rβ ) can be described as follows. For simplicity we write ππ for ππβ πΎπ and πΈπ for the total space πΈ(ππβ πΎπ ). The total space πΈπ is the orbit space of OCπ (C) × Rπ under the diagonal action of the symmetric group Ξ£π . The elements of πΈπ are unordered sets {(π§1 , π1 ), β¦ , (π§π , ππ )} where the π§π are pairwise distinct complex numbers, and the ππ are real numbers. The projection πΈπ UCπ (C) is given by forgetting the real numbers ππ , formally, {(π§1 , π1 ), β¦ , (π§π , ππ )} {π§1 , β¦ , π§π }. Even though Fuchs is able to describe the bundle, he never gives an explicit geometric or algebraic description of the classifying map ππ , and no explicit description appears to be in the literature. Because the space UCπ (C) is so fundamental, Farb posed the following problem. 6.8. Problem (Farb). Give an explicit algebraic or geometric description of the classifying Grπ (Rβ ). In other words, describe a universal way of constructing a map ππ βΆ UCπ (C) specific π-dimensional subspace of Rβ from a set of π distinct points in C (or, equivalently from a square-free monic degree π polynomial). One might naïvely propose a solution to Problem 6.8 by simply sending a set {π§1 , β¦ , π§π } to the free real vector space on {π§1 , β¦ , π§π }, considered as a subspace of Rβ . Unfortunately, this approach is too naïve as it does not specify a specific π-dimensional subspace of Rβ . The rest of this section is devoted to answering Problem 6.8. The main tools we use come from Fuchsβ [6] computation of the cohomology of UCπ (C), so before proceeding any further we briefly review Fuchs method of computation. BUNDLES, STIEFELβWHITNEY CLASSES, & BRAID GROUPS 17 The main tools of Fuchsβ calculation are a cell structure on the one-point compactification UCβπ (C) and an application of PoincaréβLefschetz duality. The cell structure on UCβπ (C) is combinatorial in nature. Specifically, it uses compositions, which are like partitions, where order matters. 6.9. Definition. Suppose that π and π are positive integers. A composition of π of length π is a π-tuple (π1 , β¦ , ππ ) of positive integers so that π1 + β― + ππ = π. We write πΆ(π) for the set of all compositions of π. 6.10. Construction ([6, Remark 3.2]). The cell structure on UCβπ (C) has a single 0-cell, the point β. For each composition π = (π1 , β¦ , ππ ) of π, let πΏπ denote the subset of UCπ (C) consisting of elements {π§1 , β¦ , π§π } so that the points π§1 , β¦ , π§π lie on π distinct vertical lines in C, where ππ of the points π§1 , β¦ , π§π lie on the πth line, counting from left to right. Notice that the sets πΏπ , for π β πΆ(π) are disjoint and partition UCπ (C). The (π + π)-cells are indexed by compositions of π of length π, and the image of the interior of the cell corresponding to a composition π is the set πΏπ . The main results of Fuchsβ computation are the following. 6.11. Theorem ([6, Thm. 4.8]). For each positive integer π and nonnegative integer π, the rank of the group π»π (UCπ (C); Z/2) is equal to the number of ways that π can be partitioned into π β π (nonnegative) powers of 2. Hence each π»π (UCπ (C); Z/2) has a set of distinguished generators indexed by the partitions of π into π β π powers of 2. 6.12. Theorem ([6, Thm. 5.2]). For each natural number π, the StiefelβWhitney class π€π (ππ ) is the sum of the distinguished generators of π»π (UCπ (C); Z/2). To make a convenient bundle atlas for ππ we expand the sets πΏπ that appear in Construction 6.10 to open sets by replacing the vertical lines with disjoint vertical strips in C. 6.13. Definition. A vertical strip π in C is an open set of the form π = { π§ β C | π < Re(π§) < π } β (π, π) × R, for real numbers π < π. With this definition we are ready to construct a bundle atlas for ππ . 6.14. Construction. Construct an open cover for UCπ (C) in the following manner. For each composition π = (π1 , β¦ , ππ ) of π, let ππ denote the subset of UCπ (C) of elements {π§1 , β¦ , π§π } with the property that there exist π disjoint open vertical strips π1 , β¦ , ππ in C, so that Re(π£π ) < Re(π£π+1 ), for all π£π β ππ and π£π+1 β ππ+1 , ππ elements of {π§1 , β¦ , π§π } lie in ππ , and all of the elements of {π§1 , β¦ , π§π } lying in ππ have distinct imaginary part, for each π. To see that the sets ππ are open, notice that for any element {π§1 , β¦ , π§π } of ππ , for sufficiently small π > 0, the set π΅π (π§1 ) βͺ β― βͺ π΅π (π§π ), where π΅π (π§π ) denotes the open ball of radius π in C centered at π§π , is an open neighborhood of {π§1 , β¦ , π§π } contained in ππ . Moreover, for each π β πΆ(π) we have πΏπ β ππ , so the sets ππ cover UCπ (C). Now extend the open cover {ππ }πβπΆ(π) of UCπ (C) to a bundle atlas for ππ as follows. For each composition π = (π1 , β¦ , ππ ) of π, there is a linear order β€π on a set {π§1 , β¦ , π§π } β ππ given in the following manner. (6.14.a) First choose a π vertical strips π1 , β¦ , ππ as above, so that ππ of the points π§1 , β¦ , π§π lie in ππ . 18 PETER J. HAINE (6.14.b) Within each vertical strip ππ order the elements of {π§1 , β¦ , π§π } lying in ππ by saying that π§π β€π π§β if and only if Im(π§π ) β€ Im(π§β ), and equality holds if and only if π = β. Notice that this defines a linear order on the elements of {π§1 , β¦ , π§π } in each ππ since all of the elements of {π§1 , β¦ , π§π } in ππ have distinct imaginary part. (6.14.c) Extend the linear orders on the elements of {π§1 , β¦ , π§π } in each ππ to a linear order on {π§1 , β¦ , π§π } by saying that if π§π β ππ and π§πβ² β ππβ² with π < πβ² , then π§π <π π§πβ² . (6.14.d) Finally, notice that this ordering on {π§1 , β¦ , π§π } is independent of the choice of vertical strips π1 , β¦ , ππ satisfying the above hypotheses. Now define a homeomorphism ππ βΆ ππ × Rπ βΌ ππβ1 (ππ ) sending ({π§1 , β¦ , π§π }, (π1 , β¦ , ππ )) to the set of pairs (π§π , ππ ), where ππ is paired with the π§π which is the πth (smallest) element of {π§1 , β¦ , π§π } in the linear order β€π . Moreover, it is clear that ππ is a continuous. The inverse bijection is given in the following manner. For each π β πΆ(π) there is a map ππ βΆ ππβ1 (ππ ) Rπ given by sending a set of pairs {(π§1 , π1 ), β¦ , (π§π , ππ )} to the vector ππ ({(π§1 , π1 ), β¦ , (π§π , ππ )}) in Rπ whose πth coordinate is the real number ππ so that π§π is the πth smallest element of the set {π§1 , β¦ , π§π } under the order relation β€π . Moreover, ππ is linear on each fiber of ππ . The inverse of ππ is given by the assignment ({π§1 , β¦ , π§π }, ππ ({(π§1 , π1 ), β¦ , (π§π , ππ )})), {(π§1 , π1 ), β¦ , (π§π , ππ )} which is clearly continuous, so ππ is a homeomorphism. Now we can use an adapted version of the proof of [12, Lem. 5.3] to construct the classifying map ππ βΆ UCπ (C) Grπ (Rβ ), giving an answer to Problem 6.8. 6.15. Construction. Since C is a manifold, in light of Examples 1.9, 1.11 and 6.3, we can choose a partition of unity {ππ }πβπΆ(π) so that supp(ππ ) β ππ , for each π β πΆ(π). Define a map ππ βΆ πΈπ Rπ for each π β πΆ(π) by setting ππ (π¦) β { ππ ππ (π¦)ππ (π¦), 0, π¦ β ππβ1 (ππ ) π¦ β ππβ1 (ππ ) . Notice that each ππ is continuous because ππ ππ vanishes outside of ππβ1 (ππ ). Now we combine all of these maps ππ together to define a map πΈπ Rβ . First notice that there is a linear order on πΆ(π) given by ordering compositions of length π lexicographically, and extending these orders to πΆ(π) by saying that compositions of length π are less than compositions of length π + 1, for each π. Because we are interested in constructing a specific plane in Rβ , we use the order relation on πΆ(π) to order the factors in the direct sum β¨πβπΆ(π) Rπ , and define a map π βΆ πΈπ β¨πβπΆ(π) Rπ by π(π¦) β (ππ (π¦))πβπΆ(π) , i.e., π is the map induced by the maps ππ by the universal property of the product. Now extend π to a map πΜ βΆ πΈπ Rβ by post-composing the map π with the standard inclusion β¨ Rπ Rβ . πβπΆ(π) The classifying map ππ βΆ UCπ (C) Grπ (Rβ ) is given by sending the element {π§1 , β¦ , π§π } of UCπ (C) to the image of the fiber ππβ1 ({π§1 , β¦ , π§π }) under π,Μ which is an π-dimensional subspace of Rβ . BUNDLES, STIEFELβWHITNEY CLASSES, & BRAID GROUPS 19 References 1. V. I. Arnolβd, On some topological invariants of algebraic functions, Trudy Moskov. Mat. ObΕ‘Δ. 21 (1970), 27β46. 2. , The cohomology ring of the colored braid group, Matematicheskie Zametik 5 (Feb. 1969), no. 2, 227β 231. 3. 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