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The Classifying Space of the G-Cobordism Category in Dimension Two

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The Classifying Space of the G-Cobordism Category in Dimension Two The classifying space of the G-cobordism category in dimension two Carlos Segovia Gonz´alez Instituto de Matem´aticasUNAM Oaxaca, M´exico 29 January 2019 The classifying space I Graeme Segal, Classifying spaces and spectral sequences (1968) I A. Grothendieck, Th´eorie de la descente, etc., S´eminaire Bourbaki,195 (1959-1960) I S. Eilenberg and S. MacLane, Relations between homology and homotopy groups of spaces I and II (1945-1950) I P.S. Aleksandrov-E.Chec,ˇ Nerve of a covering (1920s) I B. Riemann, Moduli space (1857) Definition A simplicial set X is a contravariant functor X : ∆ ! Set from the simplicial category to the category of sets. The geometric realization, denoted by jX j, is the topological space defined as ` jX j := n≥0(∆n × Xn)= ∼, where (s; X (f )a) ∼ (∆f (s); a) for all s 2 ∆n, a 2 Xm, and f :[n] ! [m] in ∆. Definition For a small category C we define the simplicial set N(C), called the nerve, denoted by N(C)n := Fun([n]; C), which consists of all the functors from the category [n] to C. The classifying space of C is the geometric realization of N(C). Denote this by BC := jN(C)j. Properties I The functor B : Cat ! Top sends a category to a topological space, a functor to a continuous map and a natural transformation to a homotopy. 0 0 I B conmutes with products B(C × C ) =∼ BC × BC . I A equivalence of categories gives a homotopy equivalence. I A category with initial or final object is contractil. I A filtered category is contractil, where J is filtered if it is non empty with the properties: 0 I a) for objects j and j in J there exists a third object k and two morphisms f : j ! k and f 0 : j 0 ! k in J; and I b) for every pair of morphisms u; v : i ! j in J, there exists an object k and a morphism w : j ! k such that w ◦ u = w ◦ v. Example n+1 Pn I B[n] = ∆n = (x0; ··· ; xn) 2 R : i=0 xi = 1; xi ≥ 1 , I (Milnor's construction) EG = G ∗ G ∗ · · · ∗ G with the fiber bundle G ! EG ! BG 1 1 1 1 1 1 I Z ! R ! S , Z2 ! S ! RP , S ! S ! CP , then 1 1 1 1 1 BZ = S , BZ2 = RP , BS = CP , BZn = Ln , 1 1 1 1 1 BΣn = Fn(R ), BQ8 = L4 ×Z2 S , BD2n = Ln ×Z2 S . The cobordism category Definition Let Σ and Σ0 be two closed oriented manifolds of dimension d and consider an oriented manifold M of dimension d + 1 with boundary diffeomorphic to Σ t −Σ0, We represent this by (Σ; M; Σ0) and we say that (Σ; M; Σ0) and (Σ; M; Σ0) are equivalent if there exists a diffeomorphism φ : M −! M0 which commutes the diagram M > a 0 Σ Φ Σ ~ M0 : The class of these elements is called a cobordism of dimension (d + 1) from Σ to Σ0. Principal G-bundle Definition For G a finite group, a principal G bundle over a topological space X , consists of a fiber bundle π : E ! X where G acts free a transitive over each fiber. Definition 1 For each g 2 G, we construct a principal G bundle Pg −! S obtained as the gluing of the product space [0; 2π] × G with the identification of (0; h) with (1; gh). Every principal G bundle over 1 S is isomorphic to this construction and two bundle Pg and Pg 0 are isomorphic y and only if g is conjugate to g 0. This construction is called G-circle, denoted by Pg . Example I Take G = Z4 = f0; 1; 2; 3g and consider the G-circles 1 associated to 1; 2 2 Z4. In that case P1 = S and 1 1 P2 = S t S with projections of degree four and two, respectively. G-cobordism Definition Consider principal G bundles π : P ! Σ, π0 : P0 ! Σ0 and ξ : Q ! M, for triples (P; Q; P0) and (Σ; M; Σ0) as before, and such that we have the commutative diagram P / Q o P0 π ξ π0 Σ / M o Σ0 : We say that (π; ξ; π0) and (π; ξ0; π0) are equivalent if they satify: 1. the triples (Σ; M; Σ0) and (Σ; M0; Σ0) are equivalent by means of a diffeomorphism φ : M ! M0, 2. the triples (P; Q; P0) and (P; Q0; P0) are equivalent by means of a G-equivariant diffeomorphism : Q −! Q0, and 3. the diffeomorphisms satisfy φ ◦ ξ = ξ0 ◦ The class of these elements are called a G-cobordism. Examples 1. G-cylinder: this is the G-cobordism from Pg to Ph (g; h 2 G) with base space a cylinder. Since the cylinder is homotopic to the circle, hence Pg and Ph are isomorphic, and as a consequence h is conjugated to g by h = kgk−1. The identification by diffeomorphisms is given by the Dehn twist. 2. G-pair of pants: consider the G-cobordism with entry the disjoint union Pg t Ph and exit Pgh, and with base space the pair of pants. We take the principal G-bundles over the pair of pants, which are a G-retract of deformation of principal G-bundles over S1 _ S1. 3. G-disc: there exists only one G-cobordism with base space the disc and where every representative is a trivial bundle. 4. Take a handlebody with n-handles and with a one boundary circle. Every principal G-cobordism over this handlebody depends on elements gi ; ki 2 G, for 1 ≤ i ≤ n, where the Qn boundary circle has the G-circle associated to the i=1[ki ; gi ]. The G-cobordism category Definition The G-cobordism category CobG in dimension two is the category with objects disjoint union of G-circles Pg1 t Pg2 t · · · t Pgn (with gi 2 G for 1 ≤ i ≤ n). The morphisms of CobG are G-cobordisms of dimension two. The composition of morphisms is given by means of the gluing of boundaries. G I the category Cob has a monoidal structure given by the disjoint union t : CobG × CobG ! CobG with unit the empty G-bundle P0; G I the category Cob has a symmetric structure given by the mapping cylinder associated to the transposition diffeomorphism. Note When the group G is trivial the category is denoted by Cob. The classifying space of Cob U. Tillmann, The classifying space of the 1+1 dimensional cobordism category (1996) Resultados I Let Cob0 be the full subcategory of Cob with only one object given by the empty 1-manifold. The classifying space 1 B Cob0 ' T the infinite dimensional torus. I Let Cob>0 be the subcategory of Cob with the same objects of Cob except the empty 1-manifold and where every connected component of every morphism has non empty initial boundary and non empty final boundary. We have the functor Φ : Cob>0 ! N 1 given by Φ(Σ) := 2 [m − n − χ(Σ)] with Σ: n ! m. I Let Cob1 be the full subcategory of Cob>0 with only one object given by one circle. The inclusion Cob1 ,! Cob>0 has left adjoint given by Φ. There is a natural transformation pn n / 1 Σ Φ(Σ) m / 1 pm ∼ 1 1 Since Cob1 = N and BN ' S , we obtain B Cob>0 ' S . I The functor Φ admits and extension Φ : Cob ! Z and the composition Φ N = Cob1 ,! Cob ! Z is a homotopy equivalence. 1 I There is a homotopy equivalence B Cob ' F × S , with F an infinite loop space. Connected components and fundamental group G I The connected components of the classifying space of Cob , G ∼ has the bijection π0(Cob ) = G=[G; G]. G I The classifying space B Cob is the product of G=[G; G] with the connected component of the trivial G bundle Pe . G I The fundamentel group of the classifying space of Cob , has G ∼ r(G) an isomorphism π1(Cob ) = Z with r(G) some positive integer. −1 I For a small category C, the category of fractions C[C ] is a good representative of the fundamental groupoid. Generalization of Tillman's work I Let M be the monoid defined by all the G-cobordisms with base space a two dimensional handlebody with one boundary circle with trivial monodromy. The composition is by gluing in a G-pair of pants Pe t Pe ! Pe . I The monoid M is abelian, free torsion and finitely generated. Therefore, ∼ r(G) M = N : I The composition of inclusion functor and the localization −1 M ,! CobG ! CobG [CobG ] is a homotopy equivalence over the image. I There is a homotopy equivalence r(G) G G G 1 1 G B Cob ' [G;G] × F1 × S × · · · ×S with F1 an infinite loop space. G G I The inclusion functor Cob1 ,! Cob>0 has left adjoint G G Φ : Cob>0 ! Cob1 . pg^ Pg^ / g Σ Φ(Σ) P^ / h ; h ph^ Σ > h g 1 1 g 1 Φ(Σ) h > 2 g g 2 2 h = g h h g 3 3 > g 3 h g 4 4 g 4 > The invariant r(G) I The generators of M are given by the closed G-cobordism where there is not a cut with trivial monodromy. I The positive number r(G) writes as a sum r1(G) + r2(G) + r3(G) + ··· , where ri (G) denotes the number of different generators of M of genus i.
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