String Topology of Classifying Spaces

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1 KTH Royal Institute of Technology Manifolds, K -theory, and Related Topics Dubrovnik, June 2014 Joint work with Richard Hepworth Preprint: arxiv:

2 Z/2 coefficients

3 Part 1: HCFTs

4 String Topology String topology studies algebraic structures on H (LX). (LX = map(s 1, X)) Theorem (Chas and Sullivan 1999) Suppose M is a closed d-manifold. Then H +d (LM) is a BV algebra. Theorem (Godin 2007) Suppose M is a closed d-manifold. Then H (LM) is the value on S 1 in a degree d Homological Conformal Field Theory (HCFT).

5 Definition of HCFT Rough Definition An HCFT Φ of degree d is an assignment ( 1-manifold X ) ( graded vector space Φ (X) ) ( ) cobordism ( H (BDiff(Σ)) Φ (X) Φ Σ: X Y +dχ(σ,x) (Y ) ) compatible with disjoint unions and composition of cobordisms. Here { } orientation-preserving self-diffeos of Σ Diff(Σ) =. fixing X and Y pointwise

6 Two perspectives on HCFTs ( ) cobordism Σ: X Y ( H (BDiff(Σ)) Φ (X) Φ +shift (Y ) ) 1. An HCFT Φ is an algebraic structure on Φ (S 1 ). A closed 1-manifold X is isomorphic to p S 1 for some p, and then Φ (X) Φ (S 1 ) p. Each z H (BDiff(Σ)) for Σ: p S 1 q S 1 gives an operation Φ (S 1 ) p Φ (S 1 ) q. ( ) ( ) Eg. the generators of H 0 BDiff and H 0 BDiff make Φ (S 1 ) into an algebra and a coalgebra. This algebraic structure sheds light on Φ (S 1 ), for example H (LM).

7 Two perspectives on HCFTs (continued) ( ) cobordism Σ: X Y ( H (BDiff(Σ)) Φ (X) Φ +shift (Y ) ) 2. An HCFT is a representation of H (BDiff(Σ)) s. BDiff(Σ) s are very interesting spaces: they classify bundles with fibre Σ, and under mild conditions on Σ, Diff(Σ) mapping class group of Σ BDiff(Σ) moduli space of Riemann surfaces modelled on Σ. HCFTs give a tool for studying the homology of these spaces.

8 String Topology of BG Theorem (Godin 2007) Suppose M is a closed manifold. Then H (LM) is the value on S 1 in an HCFT of degree dim(m). Theorem (Chataur and Menichi 2007) Suppose G is a compact Lie group. Then H (LBG) is the value on S 1 in an HCFT of degree dim(g). As stated, the results seem exactly analogous...

9 Comparison... but in fact they differ significantly in details. Godin C M type of cobordisms open closed closed only

10 Closed and open closed cobordisms Closed: incoming and outgoing boundaries consist of circles ( closed strings ) Open closed: also allow intervals ( open strings ) as incoming and outgoing boundaries Φ (S 1 ) Φ (S 1 ) and Φ (I)

11 Comparison (continued) Godin C M type of cobordisms open closed closed only unit for Φ (S 1 ) counit for Φ (S 1 ) unit for Φ (I ) n/a counit for Φ (I ) n/a

12 First theorem Here is Chataur and Menichi s result again: Theorem (Chataur and Menichi 2007) Suppose G is a compact Lie group. Then H (LBG) is the value on S 1 in an HCFT of degree dim(g). Our first theorem is similar: Theorem (Hepworth and L) Suppose G is a compact Lie group. Then H (LBG) is the value on S 1 in an HCFT of degree dim(g).

13 Comparison The HCFT we construct extends Chataur and Menichi s HCFT. Godin C M H L type of cobordisms open closed closed only open closed unit for Φ (S 1 ) counit for Φ (S 1 ) unit for Φ (I ) n/a counit for Φ (I ) n/a Q: Is it possible to turn any of the s into s the H L-column? A: No. Closest possible analogue to Godin s theory. An extension of C M to an open closed theory (without units or counits) has also been constructed by Guldberg. Addition of counits is the harder part requires new techniques!

14 Part 2: Beyond HCFTs Idea: instead of surfaces and diffeomorphisms, work with homotopy graphs and homotopy equivalences.

15 H-graphs and h-graph cobordisms Work towards the definition a novel kind of field theory A Homological H-Graph Field Theory Definition An h-graph X is a space homotopy equivalent to a finite graph. Examples: S 1, S 1 S 1, I, connected compact surfaces with non-empty boundary. Definition An h-graph cobordism S : X Y is a diagram X S Y of h-graphs satisfying certain conditions. Example: An ordinary cobordism Σ: X Y between 1-manifolds with the property that all components of Σ meet X.

16 Definition of HHGFT For an h-graph cobordism S : X Y, denote haut(s) = { self-homotopy equivalences of S fixing X and Y pointwise }. Rough Definition A Homological H-Graph Field Theory (HHGFT) Φ of degree d is an assignment ( h-graph X ) ( graded vector space Φ (X) ) ( ) h-graph cob ( H (BhAut(S)) Φ (X) Φ S : X Y +dχ(s,x) (Y ) ) compatible with disjoint unions and composition of cobordisms.

17 Second theorem An HHGFT restricts to an HCFT: S 1 and I are h-graphs An ordinary cobordism Σ: X Y is an h-graph cobordisms (as long as X meets every component of Σ) We have a natural map Diff(Σ) haut(σ) Theorem (Hepworth and L) The HCFT from our first theorem extends to an HHGFT Φ G. On h-graphs, the theory is given by Φ G (X) = H map(x, BG).

18 Some consequences and benefits New cobordisms new operations Example: : S 1 I new operation Φ (S 1 ) Φ (I ) New factorizations of existing cobordisms Example: = Operations parametrized by homologies of automorphism groups of free groups (with boundaries)

19 Automorphism groups of free groups with boundary Definition The automorphism group of free group on n generators with k boundary circles and s boundary points is A s n,k = π 0hAut(Γ s n,k ; ) where Γ s n,k = n k and =. s A 1 n,0 Aut(F n) A 2 n,0 F n Aut(F n ) = Hol(F n ) A 1 0,k is a central extension by Zk of the pure symmetric automorphism group of F k.

20 Operations parametrized by H (BA s n,k ) n A s n,k = π 0hAut(Γ s n,k ; ); Γs n,k = s k ; = If, we can turn Γ s n,k into an h-graph cobordism by dividing into incoming and outgoing parts. The HHGFT now gives operations parametrized by H BhAut(Γ s n,k ; ). The components of haut(γ s n,k ; ) are contractible. Therefore haut(γ s n,k ; ) As n,k, and we get operations parametrized by H (BA s n,k ).

21 Calculations 2 1 Let S n : pt pt be the h-graph cobordism. Σ n injects into haut(s n ) as permutations of the n strings. Let ϕ G n be the composite n ϕ G n : H BΣ n H BG H BhAut(S n ) H BG ΦG (S n) H +shift BG. Theorem (Hepworth and L) The map ϕ Z/2 2 : H BΣ 2 { H B(Z/2) H B(Z/2) a b if the degree of a is positive is given by a b 0 if the degree of a is 0 Recall that H BΣ 2 is a ring: H BΣ 2 Γ(x), x = 1. Canonical iso Z/2 Σ 2 makes H B(Z/2) into a H BΣ 2 -module.

22 Calculations (continued) Theorem (Hepworth and L) The map ϕ Z/2 2 : H BΣ 2 { H B(Z/2) H B(Z/2) a b if the degree of a is positive is given by a b 0 if the degree of a is 0 Gives an infinite family of non-trivial higher string topology operations one for each non-zero a H BΣ 2, a > 0. For a > 1, these operations cannot arise from any HCFT operation.

23 Calculations (continued) L: Further calculations of ϕ G n : H BΣ n H BG H +shift BG: for G = (Z/2) k, D 4k+2 and all n for G = T k, SU(2) and small n Get interesting operations for all these G. For example: Theorem (L) The map ϕ SU(2) 2 : H BΣ 2 H BSU(2) H +3 BSU(2) is given by a k b a k+3 b Here a k denotes the non-trivial class in H k BΣ 2 Z/2, k 0. H BSU(2) is made into a H BΣ 2 -module using the map { A if σ = id Σ 2 SU(2) SU(2), (σ, A) A if σ = (12)

24 Application to homology of Hol(F N ) and Aut(F N ) The calculations give lots of examples of non-trivial string topology operations associated with S n : pt pt. More generally, get lots of non-trivial operations for composites S n1 S nk : pt pt. Corollary Get non-trivial classes in H q BhAut(S n1 S nk ) H q BHol(F N ) for various q s. Here N = Σ k i=1 (n i 1). Corollary H q 1 (BAut(F N ); F N 2 ) 0 for these q. For G abelian, the elements in H q BHol(F N ) survive to H q BAff N (Z), where Aff N (Z) = Hol(Z N ) = Z N GL N (Z).

25 Other examples of HHGFTs? Are there other examples of HHGFTs? Conjecture Godin s HCFT in string topology of manifolds extends to an HHGFT.