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

Home , Classifying space

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 |X |, is the topological space defined as ` |X | := n≥0(∆n × Xn)/ ∼, where (s, X (f )a) ∼ (∆f (s), a) for all s ∈ ∆n, a ∈ 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 := |N(C)|. 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) ∈ R : i=0 xi = 1, xi ≥ 1 , I (Milnor’s construction) EG = G ∗ G ∗ · · · ∗ G with the ﬁber bundle G → EG → BG 1 ∞ ∞ 1 ∞ ∞ I Z → R → S , Z2 → S → RP , S → S → CP , then 1 ∞ 1 ∞ ∞ BZ = S , BZ2 = RP , BS = CP , BZn = Ln , ∞ ∞ ∞ ∞ ∞ 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 ∈ 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 = {0, 1, 2, 3} and consider the G-circles 1 associated to 1, 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 ψ : Q −→ Q0, and 3. the diffeomorphisms satisfy φ ◦ ξ = ξ0 ◦ ψ 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 ∈ 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 ∈ 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 ∈ 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 ∞ 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 .

pgˆ Pgˆ / g

Σ Φ(Σ) Pˆ / h , h phˆ 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.

I The number r1(G) is given by the quotient of the commuting pairs {(k, g):[k, g] = 1} under the identiﬁcation

k 1 0 k g 0 1 k = y = . k + g 1 1 g −k −1 0 g

Therefore, the number r1(G) is given by two identiﬁcations,

(k, g) ∼ (k, mk + g) y (k, g) ∼ (g, −k) ,

for m ∈ Z. Properties of r1(G)

I For G abelian r(G) = r1(G). I It is multiplicative, so for a pair of groups (not necessarily abelian) G1 and G2 with relative prime orders (|G1|, |G2|) = 1. We have r1(G1 × G2) = r1(G1)r1(G2)

I For G = Zn, we obtain X r(Zn) = 1 , d|n

usually denoted by τ(n). I For a prime number p and positive integers l and n, we obtain

l(2n−3) n n (p − 1)(p − 1) n r(Zpl ) = 1 + Ap(l, n) + 2n−3 , 1 p p − 1 2 p where for n > 2, " # pl(n−1) − 1 pl(2n−3) − 1 pl(n−1) pn−1 − 1 A (l, n) = + − . p pn−1 − 1 p2n−3 − 1 pn−1 − 1 pn−1 − p

l−1 with initial values Ap(l, 2) = lp and Ap(l, 1) = l.

I For the dihedral D2n and the dicyclic Dicn, the numbers r1(D2n) and r1(Dicn) have the values, c(D2n) if n = 2k + 1 , r1(D2n) = c(D2n) + k if n = 2k .

and r1(Dicn) = c(Dicn), where c(D2n) = n + τ(n) y c(Dicn) = n + τ(2n). ORDERS DESCRIPTION OF THE GROUPS SUBGROUPS ABE-SUBGROUPS r1(G) 2 4 Cyclic(Z4), Z2 3,5 3,5 3,5 6 Z6, symmetric(Σ3) 4,6 4,5 4,5 8 Z8, octic(D8), quaternion(Q8) 4,10+2,6+1 4,9,5 4,9,5 3 Z4 × Z2, Z2 8,16 8,16 8,15 2 9 Z9, Z3 3,6 3,6 3,7 10 Z10, dihedral(D10) 4,8 4,7 4,7 12 Z12, tetrahedral(A4), D12 6,10,16 6,9,13 6,9,13 2 Dicyclic(Dic3), Z2 × Z3 8,10 7,10 7,10 14 Z14, D14 4,10 4,9 4,9 15 Z15 4 4 4 16 Z16, Dic4, D16, Q8 × Z2 5,11,19,19 5,8,16,14 5,8,16 2 2 4 Z8 × Z2, Z4, Z4 × Z2 , Z2 11,15,27,67 11,15,27,67 11,16,25,51 Modular group of order 16 11 10 10 Quasihedral of order 16 15 12 12 D8 × Z2 35 30 28 (Z4 × Z2) o Z2 23 22 21 Z4 o Z4 or G4,4 15 14 14 Q8 o Z2 23 18 18 18 Z18, Z3 × Z6, D18 6,12,16 6,12,12 6,14,12 (Z3 × Z3) o Z2,Σ 3 × Z3 28,14 15,12 16+,13 20 Z20, Z10 × Z2, D20 6,10,22 6,10,19 6,10,19 Dic5, metacyclic 10,14 9,12 9+2,12 21 Z21, Z7 o Z3 4,10 4,9 4,9 22 Z22, D22 4,14 4,13 4,13 24 Z24, Z2 × Z12, Z2 × Z2 × Z6 8,16,32 8,16,32 8,16,30 D8 × Z3, Q8 × Z3 20,12 18,10 18,10 Sl(2, 3), A4 × Z2 15,26 13,24 13,23 Σ4, D24, Dic6 30,34,18 21,24,12 21,24,12 Z2 × Z2 × Σ3, Z2 × (Z3 o Z4) 54,22 43,19 40,19 Z4 × Σ3, Z3 o Z8 26,10 21,9 21,9 (Z6 × Z2) o Z2 30 22 22 2 25 Z25, Z5 3,8 3,8 3,11 26 Z26, D26 4,16 4,15 4,15 3 27 Z27, Z9 × Z3, Z3 4,10,28 4,10,28 4,12,40 (Z3 × Z3) o Z3, Z9 o Z3 19,10 18,9 22,10 28 Z28, Z14 × Z2, D28, Dic7 6,10,28,12 6,10,25,11 6,10,25,11 30 Z30, D30, D10 × Z3, D6 × Z5 8,28,16,12 8,19,14,10 8,19,14,10 Languages, codes and coverings n The sequence r(Z2) = 2, 5, 15, 51, 187, 715, 2795, 11051, 43947, ··· n (2n+1)(2n−1+1) has the form r(Z2) = 3 , found in the web

A007581: (1) The density of a language with four letters. n (2) The number of non-equivalent states for a Hanoi graph H4 . (3) The number of isomorphism classes of regular covering graphs of a graph L with Betti number n = β(L) and voltage group Z2 × Z2. ⊗k (4) The dimension of the centralizer algebra EndH1 V10 where H1 is a group of orden 96. (5) The dimension of the universal embedding of the symplectic dual polar space. (6) The rank of the fundamental group of the n Z2-cobordismo category. Bibliography

I U. Tillmann, The classifying space of the 1+1 dimensional cobordism category, J. f¨urdie reine und angewandte Mathematik 479 (1996), 67-75.

I C. Segovia, The classifying space of the G-cobordism category in dimension two, in a few days in the Arxivs.

I B. Cisneros and C. Segovia, An approximation of the number of subgroups of a groups, https://arxiv.org/pdf/1805.04633.pdf (2018).