Homological Lie brackets on moduli spaces and pushforward operations in twisted K-theory
aa r X i v : . [ m a t h . K T ] J a n Homological Lie brackets on moduli spaces andpushforward operations in twisted K-theory
Markus Upmeier
Abstract.
Enumerative geometry studies the intersection theory of virtualfundamental classes in the homology of moduli spaces. These usually dependon auxiliary parameters and are then related by wall-crossing formulas.We construct a homological graded Lie bracket on the homology of modulispaces which can be used to express universal wall-crossing formulas. For thiswe develop a new topological theory of pushforward operations for principalbundles with orientations in twisted K -theory. We prove that all rational push-forward operations are obtained from a single new projective Euler operation,which also has applications to Chern classes in twisted K -theory. Contents
1. Introduction and results 12. Pushforward operations and stability 43. Background on stacks and twisted K-theory 114. Projective Euler operations 145. Homological Lie brackets on moduli spaces 22References 24
1. Introduction and results
We begin in §2 by studying a new algebraic topology tool, called twisted opera-tions or pushforward operations , the bundle versions of cohomology operations [ ].Let G be a topological group. For every principal G -bundle π : P → B with asuitable orientation class θ, a twisted operation Ξ m | n of type ( m, n ) determines amap, natural in P, θ, Ξ m | nP,θ : H m ( P ) −→ H n ( B ) . (1.1)In general, these are hard to classify. This simplifies once we introduce stability where the maps are defined for all m with fixed degree k = n − m. Then theclassification becomes a tractable problem in parameterized stable homotopy theory.For moduli spaces, the case of interest is when G has the homotopy type B U(1) of a classifying space for complex line bundles and the orientation is a class θ ∈ K P ( B ) in twisted K -theory. The G -action arises by scaling morphisms by phasein the category of the moduli problem ( e.g. coherent sheaves, connections, quiverrepresentations). In this context, we discover the following new operation. Key words and phrases.
Graded Lie bracket, wall-crossing, moduli space, cohomology oper-ation, pushforward, twisted K -theory. Theorem . For any principal G -bundle π : P → B and orientation θ ∈ K P ( B ) in twisted K -theory of rank r = | θ | there is a projective Euler operation π θ ! : H ∗ ( P ; Q ) −→ H ∗ +2 r +2 ( B ; Q ) , (1.2) which is uniquely determined by the following properties. (a) Naturality: For a pullback diagram of principal G -bundles ¯ P P ¯ B B Φ¯ π πφ and the pullback orientation ¯ θ = φ ∗ ( θ ) ∈ K ¯ P ( ¯ B ) we have ¯ π ¯ θ ! ◦ Φ ∗ = φ ∗ ◦ π θ ! . (1.3)(b) Stability: For the product of a principal G -bundle with S we have H ∗ ( P ; Q ) H ∗ +1 ( P × S ; Q ) H ∗ +2 r +2 ( B ; Q ) H ∗ +2 r +3 ( B × S ; Q ) . π θ ! ( π × id S ) θ ×S (1.4)(c) Normalization: For the trivial bundle B × G the projective Euler operation H ∗ ( B × G ; Q ) ∼ = H ∗ ( B ; Q )[ ξ ] → H ∗ +2 r +2 ( B ; Q ) is given by π θ ! ( ξ k ) = c k + r +1 ( ϑ ) . (1.5) Here ξ ∈ H ( G ; Q ) ∼ = Q is the generator and for a trivial bundle we identifythe orientation θ with an ordinary K -theory class ϑ ∈ K ( B ) . (d) Base-linearity: For all α ∈ H ∗ ( B ; Q ) and β ∈ H ∗ ( P ; Q ) we have π θ ! ( π ∗ ( α ) ∪ β ) = α ∪ π θ ! ( β ) . (1.6)(e) Duality: For the dual principal G -bundle ˘ π : ˘ P → B ( with action reversed ) and the opposite orientation ˘ θ ∈ K ˘ P ( B ) we have ˘ π ˘ θ ! = ( − r +1 π θ ! . (1.7)(f) Composition: For principal G -bundles π : P → B, π : P → B use theanti-diagonal G -action on the fiber product P × B P and write P ⊗ P = P for its quotient with G -action ( p ⊗ p ) g = ( p g ) ⊗ p = p ⊗ ( p g ) as in π ∗ ( P ) ∼ = P × B P ∼ = π ∗ ( P ) P P ⊗ P P B. κ κ κ π π π (1.8) Given θ k ∈ K P k ( B ) , k , for the pushforwards around (1.8) we have: ( π ) θ ! ◦ ( κ ) π ∗ θ + π ∗ θ ! − ( π ) θ ! ◦ ( κ ) π ∗ θ + π ∗ θ ! = ( − | θ | ( π ) θ ! ◦ ( κ ) π ∗ ˘ θ + π ∗ θ ! (1.9)(g) For the underlying K -theory class ˜ θ ∈ K ( P ) of the orientation we have π ∗ ◦ π θ ! ( α ) = X ℓ > ℓ ! ( t ⋄ ℓ ⋄ α ) ∪ c ℓ + r +1 (˜ θ ) , (1.10) using the action ‘ ⋄ ’ defined in (4.12) below. OMOLOGICAL LIE BRACKETS AND PUSHFORWARD OPERATIONS 3 (h)
The Dixmier–Douady class η P ∈ H ( B ) characterizing π : P → B satisfies π θ ! ( α ) ∪ η P = 0 , ∀ α ∈ H ∗ ( P ; Q ) . (1.11)By dual arguments, there is also a projective Euler operation π ! θ : H ∗ ( B ; Q ) → H ∗− r − ( P ; Q ) in homology, but we prefer to state (d),(g),(h) for the cohomologicalversion. The above theorem was conjectured by Joyce [ ]. The projective Euleroperation is the fundamental new operation in the following sense. We can construc-tion further operations by composing (1.2) with automorphisms of H ∗ ( P ; Q ) . Thering Q J t K acts on H ∗ ( P ; Q ) by (4.12) below and Q [ c , c , · · · ] acts by multiplicationwith the Chern classes of the K -theory class ˜ θ ∈ K ( P ) underlying the orientation.This determines on the group Π ∗ of stable pushforward operations a graded mod-ule structure over the semi-direct product algebra S = Q J t K ⋊ Q [ c , c , · · · ] . We cannow state our main classification result.
Theorem . Suppose G has the homotopy type of B U(1) . Let Π k be theabelian group of degree k stable pushforward operations H ∗ ( P ; Q ) −→ H ∗ + k ( B ; Q ) for principal G -bundles with orientations in twisted K -theory of rank r = | θ | . (a) When r = 0 the even part Π ev is generated as a module over the semi-directproduct algebra S = Q J t K ⋊ Q [ c , c , · · · ] by the projective Euler operation.All odd operations in Π odd ∼ = Q are rational multiples of a degree operation η ∈ Π characterized by η (1 P ) = η P ∈ H ( B ; Q ) . (b) When r = 0 we have η = 0 ( rationally trivial case ) and there is a degree operation s ∗ θ : H ∗ ( P ; Q ) → H ∗ ( B ; Q ) generating the S -module Π ∗ with s ∗ θ ◦ π ∗ = r · id H ∗ ( B ) . Remark . For G the projective unitary group P U( H ) = U( H ) / U(1) of aseparable complex Hilbert space, a twisted K -theory class is represented by anequivariant family of Fredholm operators { θ p | p ∈ P } on H satisfying θ pg = θ p ⊗ L g , ∀ p ∈ P, g ∈ G, (1.12)where L denotes the universal complex line bundle over P U( H ) , see [ ]. Its deter-minant line bundle is a complex line bundle Λ( θ ) → P and satisfies Λ( θ pg ) (1.12) = Λ( θ p ⊗ L g ) = Λ( θ p ) ⊗ ( L g ) ⊗ ind θ p . When ind θ p = 0 , it thus descends to a complex line bundle on B whose Chern classis given by π θ ! (1 P ) . This class was constructed by Atiyah–Segal [ , Prop. (8.8)].The difficulties encounted in their approach to constructing other Chern classes intwisted K -theory find a natural explanation in terms of projective Euler opera-tions. When ind θ p = 1 the determinant bundle is G -equivariant and its classifyingmap f Λ( θ ) : P → G determines a global trivialization ( π, f Λ( θ ) ) : P → B × G. Theoperation s ∗ θ is simply the pullback along the corresponding global section of P. For our main result on homological Lie algebras, suppose that ( M, Φ) is anH-space with a compatible action of an abelian group G ≃ B U(1) . Write M α ⊂ M for the connected component of α ∈ π ( M ) . The H-space operation induces maps (Φ α,β ) ∗ : H ∗ (( M α × M β ) /G ; Q ) −→ H ∗ ( M α + β /G ; Q ) . Regarding π α,β : ( M α × M β ) /G → M α /G × M β /G as a G -bundle, we have a twisted K -theory and we choose orientations θ α,β ∈ K ( M α × M β ) /G ( M α /G × M β /G ) with σ ∗ α,β ( θ α,β ) = ˘ θ β,α , (Φ α,β × id M γ /G ) ∗ ( θ α + β,γ ) = θ α,γ + θ β,γ . MARKUS UPMEIER
We then have the projective Euler operation from Theorem 1.1, ( π α,β ) ! θ α,β : H ∗ ( M α /G × M β /G ; Q ) −→ H ∗− − | θ α,β | (( M α × M β ) /G ; Q ) . Theorem . Let ( M, Φ) be an H-space with orientations θ α,β and let ǫ α,β be signs satisfying the conditions of Assumption . Then the pairing H a ( M α /G ; Q ) × H b ( M β /G ; Q ) [ , ] −→ H a + b − | θ α,β |− ( M α + β /G ; Q ) defined by [ ζ, η ] = ǫ α,β ( − a | θ β,β | (Φ α,β ) ∗ ( π α,β ) ! θ α,β ( ζ ⊠ η ) (1.13) determines a graded Lie algebra structure on H ∗ ( M/G ; Q ) = M α ∈ π ( M ) H ∗ ( M α /G ; Q ) , where ζ ∈ H a ( M α /G ) gets the shifted degree | ζ | ′ = a + 2 − | θ α,α | . Theorem 1.4 was proven by Joyce [ ] in the rationally trivial case, where itcan be obtained from a vertex algebra structure on H ∗ ( M ) . However, the derivedmoduli space of quiver representations and most other cases are not rationallytrivial. This motivates our development of the projective Euler operation.The purpose of the graded Lie bracket is to give the homology of moduli spacesthe necessary algebro-topological structure to express wall-crossing formulas. Inthe conjectural picture of [ , §4] there are virtual fundamental classes [ M α ( τ )] ,α ∈ π ( M ) , for τ an auxiliary parameter such as a stability condition. There isthen a wall-crossing formula of the form [ M α (˜ τ )] = X U ( α , . . . , α n ; τ, ˜ τ ) (cid:2)(cid:2) · · · (cid:2) [ M α ( τ )] , [ M α ( τ )] (cid:3) , · · · (cid:3) , [ M α n ( τ )] (cid:3) with combinatorial rational coefficients, summing over all α + . . . + α n = α. Anadvantage of this approach as compared to cohomological Hall algebras is that itdoes not need compactness for (1.13) and consequently applies in great generality.It applies, for example, to algebraic Donaldson invariants of projective surfaceswhere the conjectures in [ , §4] are currently being proven by Joyce. For themoduli space of quiver representations, we conjecture that the universal envelopingalgebra of our Lie algebra is ‘dual’ to the cohomological Hall algebra of [ ].In moduli problems, the requirements of Theorem 1.4 are satisfied for the fol-lowing reason. For every additive C -linear dg-category A one can construct a modulistack M A parameterizing the objects of A . Taking direct sums of objects definesan operation M A × M A → M A and makes the topological realization M top A intoan H-space. Scaling morphisms by phase defines an operation of the quotient stack [ ∗ (cid:12) U(1)] , thus endowing M top A with a B U(1) -action. For a pair of objects of A there is an Ext-complex which determines a ‘symmetric obstruction theory’ θ α,β . Note that the homology of the quotient H ∗ ( M A /B U(1)) rather than H ∗ ( M A ) contains the virtual classes of the moduli problem, see [ , Conjecture 4.2].
2. Pushforward operations and stability2.1. Unstable pushforward operations.
Definition . Let G be a topological group acting on a topological space F. (a) An F -orientation of a principal G -bundle π B : P → B is a homotopy classof sections θ ∈ [ B, P ⊗ G F ] Γ of the associated fiber bundle P ⊗ G F (whose el-ements are equivalence classes [ p, f ] where [ pg, f ] = [ p, gf ] for all g ∈ G ). Amorphism ( φ, Φ) : P → P of principal G -bundles preserves F -orientations θ k ∈ [ B k , P k ⊗ G F ] Γ if φ ∗ ( θ ) = (Φ ⊗ G id F ) ◦ θ . OMOLOGICAL LIE BRACKETS AND PUSHFORWARD OPERATIONS 5 (b) A pushforward operation Ξ m | n of type ( m, n ) assigns to each principal G -bundle π B : P → B with F -orientation θ a possibly non-linear map Ξ m | nP,θ : H m ( P ) −→ H n ( B ) . (2.1)For every morphism ( φ, Φ) : P → P preserving F -orientations θ , θ werequire a commutative naturality diagram H m ( P ) H m ( P ) H n ( B ) H n ( B ) . Φ ∗ Ξ m | nP ,θ Ξ m | nP ,θ φ ∗ (2.2)(c) A pushforward operation Ξ m | n as in (b) is pointed if Ξ m | nP,θ (0 P ) = 0 B for all P → B and θ, and linear if each Ξ m | nP,θ is a linear map.We next review the background on stable homotopy theory necessary for theconstruction and classification of pushforward operations. We represent pushfor-ward operations as cohomology classes on universal spaces as follows. Definition . As in [ , Def. 2.1], for G a topological group a G -spectrum { F m , ϕ m | m > } is a sequence of pointed G -spaces F m and G -equivariant based connecting maps ϕ m : Σ( F m ) → F m +1 on the reduced suspension. When all adjoints ϕ † m : F m → Ω( F m +1 ) are homeomorphisms, we call { F m , ϕ m } an Ω - G -spectrum .For a pointed G -space K and G -spectrum { F m , ϕ m } we can construct the smashproduct G -spectrum F m ∧ K with connecting maps ϕ m ∧ id K . Example . For G = { } there is the Eilenberg–Mac Lane Ω -spectrum { H m , η m } with terms characterized by π k ( H m ) = 0 for k = m and π m ( H m ) = Q . Asin [ ] there are natural isomorphisms H m ( X ) ∼ = [ X, H m ] to the cohomology groupsunder which the connecting maps become the suspension isomorphism. Putting X = H m we obtain the tautological class m ∼ = [id H m ] ∈ H m ( H m ) . Example . The mapping spectrum M Gm = Map( G, H m ) with connectingmaps ( η † m ) ∗ : Map( G, H m ) → Map( G, Ω H m +1 ) ∼ = Ω Map( G, H m +1 ) is an Ω - G -spectrum, where our convention for the action of g ∈ G on ϕ ∈ M Gm is ρ M Gm : G × M Gm −→ M Gm , ρ M Gm ( g, ϕ )( x ) = ϕ ( xg ) . (2.3) Proposition . Let P → B be a principal G -bundle. There is a naturalisomorphism H m ( P ) ∼ = −→ (cid:2) B, P ⊗ G M Gm (cid:3) Γ , [ α ] [ s α ] , (2.4) which maps α : P → H m to the section s α of the associated bundle defined by s α ( b ) = [ p, α p ] , for an arbitrary choice p ∈ π − B ( b ) and α p : G → H m , g α ( pg ) . Proof.
A section s : B → P ⊗ G M Gm amounts to a G -equivariant map P → M Gm , which is equivalently a G -invariant map P × G → H m . These are completelydetermined by their value on P × { } . Similarly for homotopies through sections.Therefore, (cid:2)
B, P ⊗ G M Gm (cid:3) Γ is the set of homotopy classes [ P, H m ] ∼ = H m ( P ) . (cid:3) Proposition . (a) Pushforward operations Ξ m | n for principal G -bundleswith F -orientation correspond bijectively to cohomology classes Ξ m | n cla ∈ H n (cid:0) EG ⊗ G ( M Gm × F ) (cid:1) . (2.5) For a principal G -bundle P → B with classifying map φ : B → BG,F -orientation θ : B → EG ⊗ G F , and [ α ] ∈ H m ( P ) corresponding to asection s α : B → EG ⊗ G M Gm by (2.4) , the class (2.5) is characterized by Ξ m | nP,θ ([ α ]) = ( s α , θ ) ∗ (cid:0) Ξ m | n cla (cid:1) . (2.6) MARKUS UPMEIER (b)
Write i : EG ⊗ G F → EG ⊗ G ( M Gm × F ) for the inclusion of the constantmap ∗ M Gm . Then Ξ m | n is pointed if and only if i ∗ (cid:0) Ξ m | n cla (cid:1) = 0 . Equivalently, Ξ m | n cla belongs to the subgroup e H n ( EG ⊗ G ( M Gm ∧ F + )) . (c) When P = B × G is trivialized, ( s α , θ ) factors over the fiber M Gm × F ofthe base-point ∗ BG and so Ξ m | nP,θ ([ α ]) = ( s α , θ ) ∗ (cid:0) Ξ m | n cla | M Gm × F (cid:1) . Proof. (a) There is a universal example given by the principal G -bundle P m := EG × ( M Gm × F ) π Bm −−−→ B m := EG ⊗ G ( M Gm × F ) , class [ ι m ] ∈ H m ( P m ) corresponding to the section s ι m = id EG ⊗ G π M Gm by (2.4)for the projection π M Gm : M Gm × F → M Gm , and the universal F -orientation θ m =id EG ⊗ G π F : P m → EG ⊗ G F. As ( s ι m , θ m ) = id B m , property (2.6) forces us todefine (2.5) by Ξ m | n cla = Ξ m | nP m ,θ m ([ ι m ]) . We then claim (2.6) for every P → B with F -orientation θ and [ α ] ∈ H m ( P ) . The F -orientation gives a G -equivariant map θ † : P → F. Pick an isomorphism
Φ : P → φ ∗ ( EG ) over φ : B → BG and define Ψ : P → P m , Ψ( p ) = (Φ( p ) , α p , θ † ( p )) . Then we have a commutative diagram
P P m B B mπ B Ψ π Bm ( s α ,θ ) preserving the F -orientations θ, θ m . Then (2.2) implies the commutativity of [ α ] ∈ H m ( P ) H m ( P m ) ∋ [ ι m ] H n ( B ) H n ( B m ) ∋ Ξ m | n cla . Ξ m | nP,θ Ξ m | nPm,θm Ψ ∗ ( s α ,θ ) ∗ Since ι m ◦ Ψ = α we have [ α ] = Ψ ∗ ([ ι m ]) and therefore (2.6) follows. Conversely,each cohomology class Ξ m | n cla defines a pushforward operation by (2.6).(b) Suppose i ∗ (Ξ m | n cla ) = 0 . Using the constant map α to represent the zero class,we take ( s α , θ ) = i ◦ θ in (2.6), so Ξ m | nP,θ ([ α ]) = ( s α , θ ) ∗ (Ξ m | n cla ) = θ ∗ i ∗ (Ξ m | n cla ) = 0 . Conversely, suppose Ξ m | n is pointed. For α ] ∈ H m ( P m ) we can use i = ( s α , θ m ) in (2.6), so i ∗ (Ξ m | n cla ) = (ˆ s α , θ m ) ∗ (Ξ m | n cla ) = Ξ m | nP m ,θ m (0) = 0 . Part (c) is obvious. (cid:3)
Definition . Let Ξ m +1 | n +1 be a pointed pushforward operation for princi-pal G -bundles with F -orientation θ of type ( m + 1 , n + 1) . The desuspension is thefollowing pointed pushforward operation σ (Ξ) m | n of type ( m, n ) . Let P → B be aprincipal G -bundle and consider P × S → B × S with its natural F -orientation θ × S . The long exact sequence of the pair ( P × S , P × { } ) splits e H m +1 (Σ( P + )) ∼ = H m ( P ) H m +1 ( P × S ) H m +1 ( P ) 00 e H n +1 (Σ( B + )) ∼ = H n ( B ) H n +1 ( B × S ) H n +1 ( B ) 0 . ς P σ (Ξ) m | nP,θ Ξ m +1 | n +1 P ×S ,θ ×S Ξ m +1 | n +1 P,θ ς B (2.7) OMOLOGICAL LIE BRACKETS AND PUSHFORWARD OPERATIONS 7
Here ς X : H ∗ ( X ) ∼ = e H ∗ +1 (Σ( X + )) −→ H ∗ +1 ( X × S ) denotes suspension followedby inclusion. The right square commutes by (2.2). Define σ (Ξ) m | nP,θ by restrictionbetween the indicated groups, using exactness and Ξ m +1 | n +1 P,θ (0) = 0 . Definition . A stable pushforward operation Ξ •|• + k of degree k ∈ Z forprincipal G -bundles with F -orientation is a sequence of pointed pushforward oper-ations Ξ m | m + k of type ( m, m + k ) with σ (Ξ) m | m + k = Ξ m | m + k . To classify stable pushforward operations, we review some homotopy theory.
Definition . Let P → B be a principal G -bundle over a finite-dimensionalCW-complex and { F m , ϕ m } a G -spectrum. The associated fiber bundles E m = P ⊗ G F m → B have natural base-point sections s E m : B → E m and are connectedby morphisms ǫ m = id E ⊗ G ϕ m : Σ f ( E m ) = P ⊗ G Σ( F m ) → E m +1 on the fiberwisesuspension, adjoint to homeomorphisms E m → Ω f ( E m +1 ) into the space of basedloops staying in one fiber. Hence Σ f ( E m ) → B is a fiber bundle with base-pointsection, and the usual suspension is the quotient q m : Σ f ( E m ) → Σ( E m ) by col-lapsing this section. As in May–Sigurdsson [ ], we call { E m , ǫ m } the associatedparameterized spectrum .We have two morphisms of pairs ǫ m : (Σ f ( E m ) , Σ( F m )) → ( E m +1 , F m +1 ) and q m : (Σ f ( E m ) , Σ( F m )) → (Σ( E m ) , Σ( F m )) for each m > , which determine com-mutative ladders of long exact sequences: H ∗ +1 ( E m +1 , F m +1 ) H ∗ +1 ( E m +1 ) H ∗ +1 ( F m +1 ) H ∗ +1 (Σ f ( E m ) , Σ( F m )) H ∗ +1 (Σ f ( E m )) H ∗ +1 (Σ( F m )) H ∗ +1 (Σ( E m ) , Σ( F m )) H ∗ +1 (Σ( E m )) H ∗ +1 (Σ( F m )) H ∗ ( E m , F m ) H ∗ ( E m ) H ∗ ( F m ) ǫ ∗ m ǫ ∗ m ϕ ∗ m q ∗ m q ∗ m id ∼ = ∼ = ∼ = From the long exact sequence of the pairs (Σ f ( E m ) , B ) and (Σ f ( E m ) / Σ( F m ) , B ) the two vertical maps q ∗ m are invertible when ∗ > dim( B ) . In fact, there alwaysexist sections of q ∗ m making the ladder commute, but we shall not need this as ourinterest is in large cohomological degrees. We obtain structure maps H ∗ +1 ( E m +1 ) −→ H ∗ ( E m ) , H ∗ +1 ( F m +1 ) −→ H ∗ ( F m ) , (2.8)and passing to the inverse limit leads to a sequence · · · lim H m + k ( E m , F m ) lim H m + k ( E m ) lim H m + k ( F m ) · · · (2.9)which is exact when working over Q or assuming a Mittag–Leffler condition.The geometric interpretation of these inverse limit groups is as follows. Proposition . Let P → B be a principal G -bundle and E m = P ⊗ G F m beas above. Represent Ξ k cla ∈ lim H m + k ( E m ) in the inverse limit by classes Ξ m | m + k cla ∈ H m + k ( E m ) that are mapped onto each other by (2.8) . For all m + k > dim B wemay represent Ξ m | m + k cla by maps ξ m | m + k : E m → H m + k taking the base section s E m : B → E m to the base-point of H m + k , so that we may apply the fiberwise loopspace functor to ξ m | m + k ( viewing H m + k as a trivial bundle over B ) . Then the MARKUS UPMEIER following diagram is homotopy commutative: E m H m + k Ω f ( E m +1 ) Ω( H m +1+ k ) ǫ † m ξ m | m + k η † m + k Ω f ( ξ m +1 | m +1+ k ) (2.10) Proof.
The long exact sequence of the pair ( E m , B ) splits into short exactsequences and we have H m + k ( E m , B ) ∼ = H m + k ( E m ) when m + k > dim B. As H m + k ( E m , B ) is the set of pointed homotopy classes E m /B → H m + k , we mayrepresent ξ m | m + k as asserted. A loop γ ∈ Ω f ( E m +1 ) through a fiber E m +1 | b based at s E m +1 ( b ) is then mapped under ξ m +1 | m +1+ k to a based loop Ω f ( ξ m +1 ) = ξ m +1 | m +1+ k ◦ γ. The composite ξ m +1 | m +1+ k ◦ ǫ m : Σ f ( E m ) → H m +1+ k factorsthrough q m : Σ f ( E m ) → Σ( E m ) to a map ξ m +1 | m +1+ k : Σ( E m ) → H m +1+ k . Bydefinition of the inverse limit, η m + k ◦ ( ξ m +1 | m +1+ k ) † ≃ ξ m | k , and the commutativ-ity up to homotopy follows. (cid:3) We can now sharpen the classification of Proposition 2.6 in the stable case.
Theorem . As in Definition , let F be a G -space and write M Gm forthe mapping spectrum of Example . Define the G -spectrum F m = M Gm ∧ F + (2.11) and let E m = EG ⊗ G F m be the associated parameterized spectrum over BG.
Then: (a)
The class of the desuspension σ (Ξ) m | n cla ∈ e H n ( E m ) of a pointed pushforwardoperation is the image of Ξ m +1 | n +1cla ∈ e H n +1 ( E m +1 ) under (2.8) . (b) Stable pushforward operation of degree k ∈ Z for principal G -bundles with F -orientation correspond bijectively to classes Ξ k cla ∈ lim H m + k ( E m ) . (c) Every stable pushforward operation is Q -linear. Proof. (a) Recall the universal example P m → B m from the proof of Propo-sition 2.6. There is a commutative diagram of principal G -bundles P m × S P m +1 B m × S B m +1 , π Bm Γ m π Bm +1 γ m with Γ m : P m × S = EG × M Gm × F × S → P m +1 , ( e, ϕ, f, t ) ( e, ϕ ( η m ( − ∧ t ) , f )) defined using the connecting map η m : H m ∧S → H m +1 . Then Γ m is G -equivariant,so induces a quotient map γ m . Moreover, ( γ m , Γ m ) preserves the F -orientations θ m × S , θ m +1 so that naturality (2.2) gives γ ∗ m (cid:0) Ξ m +1 | n +1cla (cid:1) = γ ∗ m (cid:0) Ξ m +1 | n +1 P m +1 ,θ m +1 ([ ι m +1 ]) (cid:1) = Ξ m +1 | n +1 P m ×S ,θ m ×S (cid:0) Γ ∗ m [ ι m +1 ] (cid:1) . We have Γ ∗ m ([ ι m +1 ]) = [ ι m +1 ◦ Γ m ] = ς P m ([ ι m ]) so from (2.7) this equals Ξ m +1 | n +1 P m ×S ,θ m ×S (cid:0) ς P m ([ ι m ]) (cid:1) = ς B m (cid:0) σ (Ξ) m | nP m ,θ m ([ ι m ]) (cid:1) = ς B m (cid:0) σ (Ξ) m | n cla (cid:1) . Hence γ ∗ m (cid:0) Ξ m +1 | n +1cla (cid:1) = ς B m (cid:0) σ (Ξ) m | n cla (cid:1) . By comparing the connecting maps γ m and ǫ m we find a commutative diagram which implies our claim: Ξ m +1 | n +1cla ∈ H n +1 ( E m +1 ) H n ( E m ) ∋ σ (Ξ) m | n cla H n +1 ( B m +1 ) H n +1 ( B m × S ) H n ( B m ) ⊂ (2.8) ⊂ γ ∗ m ς Bm OMOLOGICAL LIE BRACKETS AND PUSHFORWARD OPERATIONS 9
This proves (a) from which Part (b) easily follows.(c) Set F = EG ⊗ G F + , G m = EG ⊗ G M Gm with connecting maps γ m : Σ( G m ) → G m +1 , so E m = F ∧ BG G m in terms of the fiberwise smash product over BG.
Write ξ m = ξ m | m + k for maps representing Ξ m | m + k as in Proposition 2.10. There is acommutative diagram F ∧ BG G m × BG G m F ∧ BG Ω f ( G m +1 ) × BG Ω f ( G m +1 ) F ∧ BG Ω f ( G m +1 ) E m E m × BG E m Ω f ( E m +1 ) × BG Ω f ( E m +1 ) H m + k × H m + k Ω( H m +1+ k ) × Ω( H m +1+ k ) Ω( H m +1+ k ) H m + k (obvious) id F ∧ γ † m × γ † m ∆ F (obvious) id F ∧ ℓ e ∆ F (2.10) Ω f ( ξ m +1 ) ◦ i ξ m id F ∧ γ † m (2.10) ǫ † m × ǫ † m ξ m × ξ m Ω f ( ξ m +1 ) × Ω f ( ξ m +1 ) η † m + k × η † m + k ℓ η † m + k written in terms loop composition ℓ, the inclusion of constant loops i : F ∧ BG Ω f ( G m +1 ) → Ω f ( E m +1 ) , the ‘diagonal’ ∆ F : F ∧ BG ( G m × BG G m ) → ( F ∧ BG G m ) × BG ( F ∧ BG G m ) , and e ∆ F ( f ∧ ( γ , γ ) = ( f ∧ γ , f ∧ γ ) . The commutativityof the far right square uses (2.10) and the fact that ǫ † m = (id F ∧ γ † m ) ◦ i. Let α, β ∈ H m ( P ) . Then α + β is represented by loop composition using η † m : H m → Ω( H m +1 ) , which gives that ( θ, s α + β ) is represented by B ( θ,s α ,s β ) −−−−−−→ F × BG G m × BG G m id F × γ † m × γ † m −−−−−−−−−→ F × BG Ω f ( G m +1 ) × BG Ω f ( G m +1 ) id F × ℓ −−−−→ F × BG Ω f ( G m +1 ) id F × γ † m ←−−−−− E m , where the last arrrow is a homeomorphism. Observe that this includes the toprow of the previous diagram. The claim that ξ m ◦ ( θ, s α + β ) represents the sumof ξ m ( θ, s α ) and ξ m ( θ, s β ) now follows from the commutativity of the previousdiagram, whose lower row can be used to represent addition in H m + k ( B ) . (cid:3) Recall that { H m , η m | m > } is a ring spectrum. For all m , m > wetherefore have products maps, associative up to homotopy, M m ,m : H m × H m −→ H m + m , ( h , h ) h ∗ m ,m h . (2.12)Continue with the setup of Theorem 2.11. Define ρ m ,m : E m × H m → E m + m ,ρ m ,m ([ e, ( f, ϕ )] , h ) = [ e, ( f, ϕ ∗ m ,m h )] , where e ∈ EG, f ∈ F, ϕ ∈ M Gm , h ∈ H m . With this notation, it is not hard to show the following:
Proposition . A stable pushforward operation Ξ •|• + k is H ∗ ( B ) -linear,meaning Ξ m + m | m + m + kP,θ ( π ∗ B ( α ) ∪ β ) = α ∪ Ξ m | m + kP,θ ( β ) for all α ∈ H m ( B ) ,β ∈ H m ( P ) , precisely when the associated classes satisfy ρ ∗ m ,m (cid:0) Ξ m + m | m + m + k cla (cid:1) = Ξ m | m + k cla × m . (2.13)By Theorem 2.11(b) stable pushforward operations are bijective to classes in lim H m + k ( E m ) . To compute these, the inverse limit groups of the fiber spectra H m + k ( F m ) will play an important role, so we describe these next. Theorem . Let F be a G -space. Define the G -spectrum F m = M Gm ∧ F + as in (2.11) and consider the evaluation map ev : G + ∧ M Gm −→ H m . Fix a basis { t i } i ∈ I by homogeneous elements of H ∗ ( G ) . Then we have: (a) e H ∗ ( M Gm ) is a free graded-commutative Q -algebra on x ( m ) i = ev ∗ ( m ) /t i , where i ranges over all indices with | t i | m. The structure maps µ † m = σ − M Gm ◦ µ ∗ m : e H ∗ +1 ( M Gm +1 ) −→ e H ∗ +1 (Σ( M Gm )) ∼ = ←− e H ∗ ( M Gm ) defined using the suspension isomorphism σ GM m satisfy µ † m (cid:0) x ( m ) i (cid:1) = x ( m ) i for all i ∈ I with | t i | m, µ † m (cid:0) x ( m ) i (cid:1) = 0 for m < | t i | m + 1 , and µ † m (cid:0) P ( x ( m )0 , x ( m )1 , . . . ) (cid:1) = 0 for all monomials P of degree > . (b) e H ∗ ( F m ) ∼ = e H ∗ ( M Gm ) ⊗ H ∗ ( F ) has structure maps ϕ † m = µ † m ⊗ id H ∗ ( F ) . (c) The inverse limit Γ ∗ = lim e H m + ∗ ( F m ) ∼ = H ∗ ( F ) h x i | i ∈ I i is isomorphicto the free graded H ∗ ( F ) -module on generators x i of degree −| t i | . Proof. (a) Represent the class x ( m ) i by a pointed map φ ( m ) i : M Gm → H m −| t i | . In fact, choosing maps t i : S | t i | + k → H k ∧ G + representing t i , we can represent the ( | t i | + k ) -fold suspension of φ ( m ) i as S | t i | + k ∧ M Gm t i ∧ id MGm −−−−−−→ H k ∧ G + ∧ M Gm id Hk ∧ ev −−−−−−→ H k ∧ H m M k,m −−−→ H k + m . (2.14)Taken together, these define for each m > a map φ ( m ) = (cid:0) φ ( m ) i (cid:1) : M Gm −→ Y i ∈ I : | t i | m H m −| t i | . It is a consequence of the Künneth theorem that φ ( m ) is a weak equivalence.Rationally, H ∗ ( H m −| t i | ) is known to be the free graded-commutative algebra onthe single generator m −| t i | . Hence, by Künneth, e H ∗ ( M Gm ) is freely generated by (cid:0) φ ( m ) i (cid:1) ∗ ( m −| t i | ) = x ( m ) i for all i ∈ I with | t i | m. It remains to compute µ † m . Recall that σ H m −| ti | ( m −| t i | ) = η ∗ m −| t i | ( m +1 −| t i | ) ∈ e H m +1 −| t i | (Σ( H m −| t i | )) . Inthe representation (2.14) one checks the homotopy commutativity of Σ( M Gm ) Σ( H m −| t i | ) M Gm +1 H m +1 −| t i | Σ φ ( m ) i µ m η m −| ti | φ ( m +1) i (2.15)when | t i | m and that φ ( m +1) i ◦ µ m is null-homotopic if m | t i | m + 1 . Hence µ ∗ m (cid:0) x ( m +1) i (cid:1) = µ ∗ m (cid:0) φ ( m +1) i (cid:1) ∗ ( m +1 −| t i | ) = (cid:0) Σ φ ( m ) i (cid:1) ∗ η ∗ m −| t i | ( m +1 −| t i | )= (cid:0) Σ φ ( m ) i (cid:1) ∗ ( σ H m −| ti | ( m −| t i | )) = σ M Gm ( x ( m ) i ) for | t i | m and similarly µ ∗ m (cid:0) x ( m +1) i (cid:1) = 0 for m | t i | m + 1 . Finally, µ † m vanishes on products based on the fact that cup products on suspensions vanish: µ ∗ m (cid:0) x ( m +1) i ∪ x ( m +1) j (cid:1) = µ ∗ m (cid:0) φ ( m +1) i , φ ( m +1) j (cid:1) ∗ ( m +1 −| t i | × m +1 −| t j | ) (2.15) = (cid:0) Σ φ ( m ) i , Σ φ ( m ) j (cid:1) ∗ ( η m −| t i | × η m −| t j | ) ∗ ( m +1 −| t i | × m +1 −| t j | )= (cid:0) Σ φ ( m ) i , Σ φ ( m ) j (cid:1) ∗ (cid:0) σ H m −| ti | ( m −| t i | ) × σ H m −| tj | ( m −| t j | ) (cid:1) = σ M Gm (cid:0) ( φ ( m ) i ) ∗ ( m −| t i | ) (cid:1) ∪ σ M Gm (cid:0) ( φ ( m ) j ) ∗ ( m −| t j | ) (cid:1) = 0 . OMOLOGICAL LIE BRACKETS AND PUSHFORWARD OPERATIONS 11 (b) This follows from the following commutative diagram, using [ , VII,7.11], e H ∗ ( F m +1 ) e H ∗ (Σ( F m )) e H ∗ ( F m ) e H ∗ ( M Gm +1 ) ⊗ H ∗ ( F ) e H ∗ (Σ( M Gm )) e H ∗ ( M Gm ) ⊗ H ∗ ( F ) . ϕ ∗ m σ Fm × ∼ = µ ∗ m ⊗ id H ∗ ( F ) × × ∼ = σ MGm ⊗ id H ∗ ( F ) (c) Elements z in the inverse limit are sequences z ( m ) = X i : | t i | m f ( m ) i x ( m ) i + X j e f ( m ) j Q ( m ) j ( x ( m )0 , x ( m )1 , . . . ) , with Q j a monomial in x ( m ) i of degree > , satisfying ϕ † m ( z ( m +1) ) = z ( m ) . Thisforces the second summand to vanish and f i = f ( m ) i to be independent of m. Wewrite x i for the element of the inverse limit represented by the sequence x ( m ) i . Itfollows that we have a unique expression z = P f i x i , as required. (cid:3)
3. Background on stacks and twisted K-theory3.1. Classifying spaces.
We consider only topological spaces with reasonablepoint-set topology (paracompact Hausdorff).
Definition . A principal G -bundle E → B with contractible total space iscalled a model for the classifying space of the topological group G. A classifyingmap for a principal G -bundle P → X is a pair of a continuous map f P : X → B and an isomorphism F P : P → f ∗ P ( E ) . Theorem . Let E → B be a classifying space of G and P → X a principal G -bundle. Every classifying map of a subcomplex A ⊂ X can be extended over X. In particular, the space C P of classifying maps for P is contractible. Proof.
Since the associated fiber bundle P ⊗ G E → X has contractible fibers,a theorem of Dold [ , Cor. 2.8] implies that every section on A can be extendedover X. Moreover, sections of P ⊗ G E are equivalently classifying maps. (cid:3) In particular, isomorphic bundles P , P → X have homotopic classifying maps.Moreover, Theorem 3.2 implies that any two models for the classifying space arehomotopy equivalent, and we write EG → BG for this well-defined homotopy type. Theorem . There exists a functorial construction of classifying space EG → BG for topological groups G. Thus, every continuous homomorphism φ : G → H induces a continuous map Bφ : BG → BH covered by a φ -equivariant bundle map Eφ : EG → ( Bφ ) ∗ ( EH ) , functorial in the obvious sense. For a pair of topologi-cal groups the maps B ( G × H ) → BG × BH and E ( G × H ) → EG × EH arehomeomorphisms and B { } = { } for the trivial group. Proof.
The nerve of G regarded as a one object topological category ∗ (cid:12) G isa simplicial space N ( ∗ (cid:12) G ) . The topological realization ( ) top of simplicial spacesis natural and preserves products. We can also regard G as a category G (cid:12) ∗ withonly identity morphisms. Let G (cid:12) G be the category with objects G and a uniquemorphism between any two objects. Then G (cid:12) G is contractible, N ( G (cid:12) ∗ ) top ∼ = G, and the topological realization of N ( G (cid:12) ∗ ) → N ( G (cid:12) G ) → N ( ∗ (cid:12) G ) is defined as G → EG → BG, which is indeed a model for the classifying space, see Segal [ ].Evidently this construction is strictly functorial and preserves products. (cid:3) In particular, the classifying space BA of a topological abelian group A isagain of this kind. Define iterated classifying spaces B k A = B ( B k − A ) , B A = A by recursion. We shall use this construction for A = U(1) . Theorem . Let → G i → G p → G → be a short exact sequence oftopological groups with p a fibration. Then we have a homotopy fiber sequence G G G BG BG BG . i r δ Bi Bp (3.1) The construction of δ is natural for maps of short exact sequences. For a centralextension, (3.1) may be extended one step further to the right by B G . Proof.
Since p is a fibration, G is the homotopy fiber of p . To define theboundary map in (3.1), fix a model EG , a contractible space with a free and proper G -action. Restricting the action to G shows EG = EG and quotienting by G and by G yields a fiber bundle G δ ֒ → BG ։ BG . As δ ◦ p factors over EG , thereis a null-homotopy δ ◦ p ≃ const . This provides a map G → F into the homotopyfiber of δ , which is itself part of a fibration Ω BG ֒ → F ։ G . Hence we maycompare this fibration with G ֒ → G ։ G , and the five lemma combined with thelong exact sequence of homotopy groups shows that G → F is a weak equivalence.Next, the associated bundle ( EG × EG ) /G is a fiber bundle over BG withcontractible fiber EG , so Dold’s theorem [ , (2.8)] implies ( EG × EG ) /G ≃ BG . Local triviality of EG over BG shows that ( EG × EG ) /G is also a fiberbundle over BG with fiber ( EG × G ) /G = EG /G = BG . We conclude thatthe last three steps in (3.1) also form a homotopy fiber sequence. Finally, when G is central, BG is again a group and Bp is a principal BG -bundle. We may thenextend (3.1) by the classifying map of Bp into B ( BG ) . (cid:3) While the formalism of §3.1 is conve-nient from a theory point of view, geometric examples arise as in the proof ofTheorem 3.3. We now generalize this approach. Following [ ], a stack is a topo-logical groupoid X = ( X , X ) whose object and arrow sets are topological spaces.Also, the source and target maps X → X , composition µ : X × X X → X , andidentity X → X are continuous. Morphisms of stacks are functors that are con-tinuous on objects and arrows. The topological realization X top is the topologicalrealization of the nerve simplicial space associated to X . This construction preservesproducts ( X × Y ) top ∼ = X top × Y top , morphisms of stacks F : X → Y induce con-tinuous maps F top : X top → Y top , and natural transformations induce homotopies.We can regard topological spaces B as stacks B (cid:12) ∗ and then ( B (cid:12) ∗ ) top = B. Definition . A central extension of a stack X = ( X , X ) is a principal U(1) -bundle L → X with morphism M : π ∗ ( L ) ⊗ U(1) π ∗ ( L ) → µ ∗ ( L ) such that L | f ⊗ L | g ⊗ L | h L | f ⊗ L | g ◦ h L | f ◦ g ⊗ L | h L | f ◦ g ◦ h id Lf ⊗ M g,h M f,g ⊗ id Lh M f,g ◦ h M f ◦ g,h (3.2)commutes for each triple of composable arrows f, g, h in X . The tensor product L ⊗ L , the pullback F ∗ ( L ) along a continuous functor F, and the dual ˘ L of a central extension are defined in the obvious ways. Proposition . Let L = ( X , L ) be the stack determined by a central exten-sion. Then the topological realization of the morphisms [ ∗ (cid:12) U(1)] → L → X is aprincipal B U(1) -bundle G = B U(1) → L top → X top . The tensor product, pullback,and dual carry over to the corresponding operations on principal G -bundles ( thetensor product P ⊗ G P is the associated bundle, the dual has its action inverted ) . Proof.
Principal bundles are defined in terms of products, restrictions to opensubsets, and isomorphisms, all of which are preserved by ( ) top . (cid:3) OMOLOGICAL LIE BRACKETS AND PUSHFORWARD OPERATIONS 13
Example . For an action of a topological group G on a space S, we havethe quotient stack S (cid:12) G = ( S, S × G ) . Let → U(1) → E → G → be a centralextension with E → G a principal U(1) -bundle. Then S (cid:12) E → S (cid:12) G is a centralextension of the quotient stack. For example, we have:(a) For a Lie group G we have the adjoint representation Ad : G → Aut( g ) , which takes values in SO( g ) for G compact connected. The complex spingroup is Spin c ( n ) = Spin( n ) × {± } U(1) and forms a central extension −→ U(1) −→ Spin c ( n ) −→ SO( n ) −→ . Then E = Ad ∗ (Spin c ( n )) → G defines a central extension of G. Hence everyprincipal G -bundle S → M on a topological space M determines a centralextension of the stack M ≃ S (cid:12) G, measuring the extent to which theassociated bundle of Lie algebras g S = S ⊗ G g → M has a spin c structure.Topologically, this is the pullback of B U(1) → B Spin c ( n ) → B SO( n ) from(3.1) along a classifying map f g S : M → B SO( n ) . (b) Let E → G be a central extension of a Lie group ( e.g. U( n ) → PU( n ) ) and P → M a principal E -bundle over a smooth manifold. The subgroup U(1) of the gauge group G P = Aut( P ) acts trivially on the space of connections A P , yielding a central extension A P (cid:12) G P → A P (cid:12) ( G P / U(1)) . (c) Let G be a Lie group and P → M a principal G -bundle over a smoothmanifold. Then a central extension E P → G P of the gauge group (for exam-ple, from a projective representation of G P ) determines a central extension A P (cid:12) E P → A P (cid:12) G P . When M = S and P is trivial, G P is the loop group whose central extensions are well-studied, see Pressley–Segal [ ]. In thiscase, A P (cid:12) G P is equivalent to G (cid:12) G. In general, A P (cid:12) G P is equivalentto the quotient stack Map ◦ ( M, BG ) P (cid:12) G of base-point preserving maps inthe connected component of f P , see [ , (5.1.4)]. Definition . Let G be a topological group of the homotopy type B U(1) . As BG is an Eilenberg–Mac Lane space K ( Z , , the classifying map f P : B → BG identifies principal G -bundles P → B up to isomorphism with classes η P ∈ H ( B ) = [ B, K ( Z , . We call P → B rationally trivial if the image of η P in ra-tional cohomology H ( B ; Q ) vanishes. This terminology comes from the fact that H ∗ ( P ) ∼ = H ∗ ( B × G ) for rationally trivial bundles, by the Serre spectral sequence.For B a simple Lie group, H ( B ) ∼ = Z and η P ∈ Z is called the level . Thisprovides more examples of central extensions. Karoubi has introduced twisted K -theory for study-ing K -theoretic Poincaré duality, see [ ]. The approach taken here was initiatedby Rosenberg [ ]. Our presentation follows Atiyah–Segal [ ] and Freed–Hopkins–Teleman [ ]. For the rest of the paper, assume the following setup: Assumption . (a) G is a topological group that is a classifying space B U(1) for complex line bundles with unit the trivial bundle and whose groupoperation µ G : G × G → G classifies the tensor product of line bundles.(b) F is a topological space homotopy equivalent to the classifying space B U of stable complex vector bundles. Up to homotopy, taking direct sums ofvector bundles defines an operation ⊕ : F × F → F, the external tensorproduct gives ⊗ : F × F → F, and duals give an involution ˘( ) : F F. (c) There is a group action ρ F : G × F → F representing the tensor product ofa line bundle with a vector bundle. Up to homotopy, we then have that ρ F preserves the direct sum operation. The operations extend to F × Z classifying virtual vector bundles of rank r ∈ Z . Example . Let G be the projective unitary group P U( H ) = U( H ) / U(1) of a separable Hilbert space acting by conjugation on the space F × Z of Fredholmoperators, where r ∈ Z is given by the index. As U( H ) is contractible by Kuiper’stheorem, we have P U( H ) = B U(1) . Example . Another important example is G = [ ∗ (cid:12) U(1)] top and F × Z the topological realization of the stack F of Fredholm complexes H H . . . H n , D D D n − D i +1 ◦ D i = 0 , n ∈ N , (3.3)where all D i are bounded operators of separable Hilbert spaces H i and the cohomol-ogy Ker( D i ) / Im( D i +1 ) at each step is finite-dimensional. Here r ∈ Z correspondsto the Euler characteristic of (3.3). A morphism of Fredholm complexes is a com-mutative ladder of bounded operators and the stack [ ∗ (cid:12) U(1)] acts by scaling these.The stack is topologized in the operator norm. The direct sum of Fredholm com-plexes determines a morphism Φ F : F × F → F . The dual is obtained by takingadjoints of all operators in the Fredholm complex in (3.3).An advantage of Example 3.11 is that G is abelian on the nose. Definition . Fix
G, F as in Assumption 3.9. Let π : P → B be a principal G -bundle. Twisted K -theory is the set of homotopy classes of sections K P ( B ) =[ B, P ⊗ G ( F × Z )] Γ of the associated fiber bundle, an abelian group under directsums. A continuous map φ : ¯ B → B with isomorphism Φ : ¯ P → φ ∗ ( P ) to a principal G -bundle ¯ π : ¯ P → ¯ B induces a functorial pullback homomorphism ( φ, Φ) ∗ : K P ( B ) −→ K ¯ P ( ¯ B ) . (3.4)Suppose now in addition that G is commutative. Let π k : P k → B be principal G -bundles for k = 1 , . As G is abelian, the associated bundle π ⊗ π : P ⊗ G P → B has a natural G -action and we have a bilinear tensor product pairing K P ( B ) ⊗ K P ( B ) −→ K P ⊗ P ( B ) . (3.5)The involution on F determines an involution K P ( B ) → K ˘ P ( B ) , θ ˘ θ on twisted K -theory. Here ˘ P → B denotes the dual principal G -bundle, with the same under-lying space but its G -action inverted.An ( F × Z ) -orientation of π thus amounts to a class θ ∈ K P ( B ) . Regardingsections as maps P → F × Z , write ˜ θ ∈ K ( P ) for the underlying K -theory class .
4. Projective Euler operations
In this section, we classify rational stable pushforward operations and constructprojective Euler operations. Over the rationals, this reduces to an algebraic prob-lem, as explained in §4.1, where the algebraic structure in constructed. The proofof Theorem 1.2 and the classification result is given in §4.2. The further propertiesof the projective Euler operations and Theorem 1.1 is proven in §4.3.
Let
G, F be as in Assump-tion 3.9 and fix the rank r ∈ Z . Definition . Define a principal G -bundle over S by clutching trivial bun-dles over S = CP along γ | CP , and let i : S → BG be its classifying map. Recallthe G -spectrum { F m , ϕ m } from (2.11) and let i ∗ ( E m ) = i ∗ ( EG ) ⊗ G F m be thepullback spectrum over S . The group of stable pushforward classes is defined as Π ∗ = lim e H m + ∗ ( i ∗ ( E m )) (4.1)with inverse limit along the structure maps (2.8) for P = i ∗ ( EG ) → S . OMOLOGICAL LIE BRACKETS AND PUSHFORWARD OPERATIONS 15
Proposition . In rational cohomology, Π ∗ ∼ = lim e H m + ∗ ( EG ⊗ G F m ) . Henceby Theorem classes in Π ∗ correspond to stable pushforward operations. Proof.
Since BG is homotopy equivalent to an Eilenberg–Mac Lane space, H ∗ ( BG ) ∼ = Q is generated by i ∗ [ S ] , see [ , §3]. For each m the fiber bundle map i ∗ ( EG ) ⊗ G F m → EG ⊗ G F m induces a morphism of Serre spectral sequences,which is rationally an isomorphism on the fiber and base and hence induces anisomorphism of the E -pages. It follows that the map between the targets of thespectral sequence is an isomorphism as well. (cid:3) The objective of this section is to determine Π ∗ . We summarize the classifi-cation of the previous section and make the differential explicit. Fix a homotopyequivalence γ : CP ∞ → G and set t i = γ ∗ ([ CP i ]) ∈ H i ( G ) for i > , which is dualto the basis ξ i ∈ H i ( G ) , where ξ is the Chern class of the universal line bundle.Recall from [ , Thm. 14.5] that H ∗ ( F ) is a free commutative algebra Q [ y , y , . . . ] on the Chern classes y j of the universal bundle. Equally, we may use the Cherncharacter components z j , defined recursively by the Newton identities z k = ( − k − y k ( k − X k − i =1 ( − i + k − i ! k ! y k − i z i , ∀ k > . (4.2) Theorem . By Theorem , the vector spaces Γ ∗ = lim e H m + ∗ ( F m ) ∼ = H ∗ ( F ) h x , x , · · ·i are free modules over H ∗ ( F ) = Q [ y , y , . . . ] on generators x i ofdegree − i. Here the exact sequence (2.9) simplifies to · · · Γ ∗− Π ∗ Γ ∗ · · · . δ δ (4.3) Since Γ ∗ is concentrated in even degrees, the exact sequence splits. For the groupof stable pushforward operations we therefore have Π ∗ ∼ = ( Ker (cid:0) δ : Γ ∗ −→ Γ ∗− (cid:1) ( ∗ even ) , Coker (cid:0) δ : Γ ∗− −→ Γ ∗− (cid:1) ( ∗ odd ) . (4.4) The connecting homomorphism δ : Γ ∗ → Γ ∗− is Q -linear and is given by δ ( C ⊠ x i ) = ∂ ( C ) ⊠ x i + C ⊠ ( i + 1) x i +1 , ∀ C ∈ H ∗ ( F ) , i > , (4.5) for the unique algebra derivation ( setting y j = 0 for j < ) ∂ : H ∗ ( F ) −→ H ∗− ( F ) with ∂ ( y j ) = ( r − j + 1) y j − . (4.6) Equivalently, ∂ ( z j ) = z j − for j > and ∂ ( z ) = r. Proof.
For fiber bundles over S we have i ∗ ( E m ) /F m ≃ Σ ( F m ) + . Beforepassing to the inverse limit (4.3), the exact sequence (2.9) for each fixed m includingthis identification H ∗ ( E m , F m ) ∼ = H ∗− ( F m ) is called the Wang sequence. Since E m is obtained from two trivial fiber bundles by a clutching map g m : S × F m → F m , the differential in the Wang sequence is δ m : H ∗ ( F m ) −→ H ∗ +1 ( E m , F m ) ∼ = H ∗− ( F m ) , z g ∗ m ( z ) (cid:14) [ S ] , see Atiyah–Segal [ , (3.8)]. Here g m = ρ F m ( g × id F m ) for g : S → G and the action ρ F m : G × F m → F m that combines (2.3) and ρ F : G × F → F. It is a general factthat δ m is a derivation.Recall that H ∗ ( F m ) ⊂ H ∗ ( M Gm × F ) . Write π F , π M Gm for the two projections of M Gm × F and µ : G × G → G for the group operation. Then ev ◦ µ = ev ◦ (id G × ρ M Gm ) . Moreover, g ∗ ([ S ]) = t ∈ H ( G ) and one easily computes µ ∗ ( t i × t ) = ( i + 1) t i +1 . Using standard properties of the slant product, δ m ( π ∗ Map ( x ( m ) i )) = ( g × π Map ) ∗ ρ ∗ M Gm ( x ( m ) i ) / [ S ]= (id G × π Map ) ∗ ρ ∗ M Gm (cid:0) ev ∗ ( m ) /t i (cid:1) /t = π ∗ Map (cid:0) (id G × ρ M Gm ) ∗ ev ∗ ( m ) / ( t i × t ) (cid:1) = π ∗ Map (cid:0) ev ∗ ( m ) /µ ∗ ( t i × t ) (cid:1) = ( i + 1) π ∗ Map (ev ∗ ( m ) /t i +1 ) = ( i + 1) π ∗ Map ( x ( m ) i +1 ) . For a complex line bundle L and K -theory class V the Chern classes satisfy c j ( L ⊠ V ) = P jℓ =0 (cid:0) rk V − ℓj − ℓ (cid:1) c ( L ) j − ℓ × c ℓ ( V ) . This implies ρ ∗ F ( y j ) = X jℓ =0 (cid:18) r − ℓj − ℓ (cid:19) ξ j − ℓ × y ℓ , (4.7)where ξ = c ( E U(1)) ∈ H ( G ) and y = 1 . Hence δ m (cid:0) π ∗ F ( y j ) (cid:1) = ( g × π F ) ∗ ρ ∗ F ( y j ) (cid:14) [ S ]= (id G × π F ) ∗ ρ ∗ F ( y j ) (cid:14) t = π ∗ F (cid:0) ρ ∗ F ( y j ) (cid:14) t (cid:1) = ( r − j + 1) π ∗ F ( y j − ) in H ∗ ( M Gm × F ) . For an equivalent formula in the classes (4.2) see [ , (3.9)]. (cid:3) Note that the splitting (4.4) does not happen unstably, making the exact se-quence less useful. The splitting has the following consequence.
Corollary . Even degree stable pushforward operations Ξ k are determineduniquely by their values on the trivial bundle P = B × G with arbitrary F -orientation θ. Writing Ξ k cla = P i > C i ⊠ x i ∈ Π k with C i ∈ H k +2 i ( F ) , we have Ξ •|• + kB × G,θ ( ξ i ) = C i (cid:0) ˜ θ | B ×{ } (cid:1) . (4.8) Here we view C i as a characteristic class for virtual vector bundles which we eval-uate at the restriction of the associated K -theory class ˜ θ ∈ K ( P ) . Proof.
For uniqueness, it suffices to prove (4.8). Indeed, by Theorem 4.3an even degree stable pushforward operation is classified by an element Ξ k cla = P i > C i ⊠ x i in Π k . Moreover, the characteristic classes C i ∈ H k +2 i ( F ) are deter-mined entirely by their values on all K-theory classes ϑ ∈ K ( B ) , corresponding toarbitrary F -orientations of the trivial bundle. The formula (4.8) is a consequenceof Proposition 2.6: for a trivial bundle φ is constant so s α and θ can be identifiedwith maps α = ξ i : B → H m and ϑ : B → F, with ϑ = ˜ θ | B ×{ } . Hence ( s α , θ ) ∗ (cid:0) Ξ m | m + k cla (cid:1) = X i > θ ∗ ( C i ) ∪ s ∗ α ( x i ) = X i > ϑ ∗ ( C j ) ∪ ( α/t i ) = C ∗ j (˜ θ | B ×{ } ) . (cid:3) Corollary . Even degree stable pushforward operations are H ∗ ( B ) -linear. Proof.
We check (2.13). For k even, write Ξ k cla = P i > C i ⊠ x i so that thecomponents in the inverse limit are Ξ m + m | m + m + k cla = P i > C i ⊠ x ( m + m ) i and Ξ m | m + k cla = P i > C i ⊠ x ( m ) i . Using M ∗ m ,m ( m + m ) = m × m for the pullbackalong (2.12) we find ρ ∗ m ,m ( x ( m + m ) i ) = [(ev × id H m ) ∗ M ∗ m ,m ( m + m )] /t i = [ev ∗ ( m ) × m ] /t i = [ev ∗ ( m ) /t i ] × m = x ( m ) i × m . OMOLOGICAL LIE BRACKETS AND PUSHFORWARD OPERATIONS 17
Also, one checks ρ ∗ m ,m ( y j ) = y j . Now (2.13) follows from the observation thateach x ( m + m ) i appears linearly in the expression for Ξ m + m | m + m + k . (cid:3) For k even, the group of rational stable pushforward operation Π k = Ker( δ ) consists of all elements Ξ = X i > ( − i i ! P i ⊠ x i , P i ∈ H i + k ( F ) , ∂P i = P i − , P − = 0 . (4.9)These can be depicted as P P P · · · . ∂ ∂ ∂ ∂ (4.10) Definition . The projective Euler operation is the rational stable pushfor-ward operation of degree r + 2 corresponding under the association of Theorem 4.3to Ξ PE = P i > y i + r +1 ⊠ x i ∈ Π r +2 . We write π θ ! = Ξ •|• +2 r +2PE . When r = 0 definethe generating class Ξ gen = P i > ( − i i ! z i ⊠ x i ∈ Π , where we set z = r. The next step in the classification is to introduce further algebraic structure.There are two ways to contruct new pushforward operations from a given one.
Definition . Let Ξ •|• + k ∈ Π k be a stable pushforward operation.(a) Apply the Chern character to the associated K -theory class ˜ θ ∈ K ( P ) toconstruct a new stable pushforward operation z j (Ξ) ∈ Π j + k by H ∗ ( P ) −→ H ∗ +2 j + k ( B ) , α Ξ P,θ (cid:0) α ∪ z j (˜ θ ) (cid:1) . (4.11)(b) Using ρ P : P × G → P and slant product by t ∈ H ( G ) , define an action Q J t K × H ∗ ( P ) −→ H ∗− ( P ) , t ⋄ α = ρ ∗ P ( α ) /t . (4.12)We can then construct a new operation t (Ξ) ∈ Π k − by H ∗ ( P ) −→ H ∗− k ( B ) , α Ξ P,θ ( t ⋄ α ) . (4.13) Proposition . For k even, let Ξ = P i > ( − i i ! P i ⊠ x i ∈ Π k as in (4.10) . (a) The pushforward operation t (Ξ) is represented by the shifted sequence P P · · · . ∂ ∂ ∂ ∂ (4.14)(b) The pushforward operation z j (Ξ) is represented by P z j − P z j − + . . . + ( − j P j z · · · , ∂ ∂ (4.15) with i th term P jk =0 ( − k (cid:0) i + kk (cid:1) P i + k z j − k . Proof.
Clearly the stability of the pushforward operations is inherited by(4.11) and (4.13). We verify (4.15). By (4.8) it suffices to do this for the trivialbundle P = B × G, where we can factor the orientation as θ = ρ F ◦ ( θ | B ×{ } × id G ) . From ch j ( L ⊠ V ) = P ∞ k =0 c ( L ) k k ! × ch j − k ( V ) and by H ∗ ( B ) -linearity, we find z j (Ξ)( ξ i ) = Ξ (cid:0) ξ i ∪ z j (˜ θ ) (cid:1) = Ξ (cid:16) ξ i ∪ X k > ξ k k ! ∪ π ∗ B (cid:0) z j − k (˜ θ | B ×{ } ) (cid:1)(cid:17) = X k > z j − k (˜ θ | B ×{ } ) k ! ∪ Ξ( ξ i + k ) = X k > z j − k (˜ θ | B ×{ } ) k ! ( − i + k ( i + k )! P i + k . This implies (4.15). The proof of (4.14) is analogous, based on t j ⋄ ξ i = (cid:0) ij (cid:1) ξ i − j . (cid:3) Remark . The commutator of (4.14) with (4.15) is t ( z j (Ξ)) − z j ( t (Ξ)) = z j − (Ξ) . Hence Π ∗ is a graded module over the ring S = Q J t K ⋊ H ∗ ( F ) withunderlying group Q J t K ⊗ H ∗ ( F ) but twisted multiplication X i > p i t i · X j > q j t j = X k > X i,j > (cid:18) ik − j (cid:19) p i ∂ i + j − k ( q j ) t k , ∀ p i , q j ∈ H ∗ ( F ) . The coefficient of each t k is in fact a finite sum. Classifying pushforward operationsup to (4.14) and (4.15) amounts to finding a generating set of the S -module Π ∗ . To calculate (4.4), we first introduce an auxil-iary ring R = Q [ z , z , z , . . . ] with a bigrading | z k | = (1 , k ) and derivation ∂ ( z k ) = z k − ( ∀ k > , ∂ ( z ) = 0 . Introduce an algebra homomorphism ǫ : R → Q by ǫ ( z k ) = 0 for k > , and the Q [ z ] -linear map (since ∂ is elementwise nilpotent, the infinite sum makes sense) γ = X k > ( − k z k ∂ k . Clearly ǫ ◦ ∂ = 0 on generators and ∂ ◦ γ = 0 is also easy to check. Write R d,e ⊂ R for the subset of elements of bidegree ( d, e ) . Then ∂ and γ have bidegree (0 , − and (1 , , and we have a chain complex (with Q [0] put in bidegree (0 , ) R d − ,e R d,e R d,e − Q [0] d,e − . γ ∂ ǫ (4.16) Lemma . For all ( d, e ) = (0 , the sequence (4.16) is exact. Proof.
Using R ,e = { } for e = 0 and that R ,e = Q · z e is one-dimensional,the cases ( d, e ) ∈ { (1 , , (1 , , (1 , } and ( d, e ) = (0 , e ) with e > are easilychecked by direct inspection. Moreover, when d < or e < the entire sequencevanishes. We proceed by induction on the total degree d + e > , d, e > . Definean algebra epimorphism φ : R → R by φ ( z k ) = z k − for k > and φ ( z ) = 0 . Then φ has bidegree (0 , − d ) and there is a diagram of chain complexes with exact rows: R d − ,e R d − ,e R d − ,e − d R d − ,e R d,e R d,e − d R d − ,e − R d,e − R d,e − d −
00 0 Q [0] d,e − Q [0] d,e − d −
00 0 0 z γ γ φ − γ ◦ ∂∂ z ∂ φ ∂z ǫ φ ǫ Hence we have a snake lemma long exact sequence · · · H ∗ ( column 1 ) H ∗ ( column 2 ) H ∗ ( column 3 ) · · · . Column 1 is exact by induction for ( d, e ) = (1 , and Column 3 is exact by inductionwhen ( d, e ) = (1 , and d > since then ∂ : R d − ,e − d → R d,e − d is surjective.Therefore the middle column is exact also. (cid:3) The additional variable z corresponds to the rank r ∈ Z . Define an algebraepimorphism ψ : R → H ∗ ( F ) by ψ ( z ) = r and ψ ( z k ) = z k for k > . Setting
OMOLOGICAL LIE BRACKETS AND PUSHFORWARD OPERATIONS 19 R e = L d > R d,e , the map ψ : R e → R e preserves the degree and has the gradedideal h z − r i as kernel. For r ∈ Z set γ r = r · id H ∗ ( F ) + X k > ( − k z k ∂ k . (4.17)For e > we have another commutative diagram with exact rows: h z − r i e − R e − H e − ( F ) 00 h z − r i e R e H e ( F ) 00 h z − r i e R e H e ( F ) 00 0 0 γ ψγ γ r ∂ ψ∂ ∂ψ The middle column is exact by Lemma 4.10. The left column is also exact. Forexample, if ∂ ( Q ( z − r )) = ( ∂Q )( z − r ) for Q ∈ R e then ∂Q = 0 and byLemma 4.10 we can write Q = γ ( P ) for some P ∈ R e − . Hence γ ( P ( z − r )) = Q ( z − r ) . Similarly for surjectivity of ∂. As before, the snake lemma implies theexactness of the right vertical sequence. In case e = 0 we note that ∂ is surjectivefor r = 0 and else has image the polynomials in H ∗ ( F ) with zero constant term: H k − ( F ) H k ( F ) H k ( F ) 0 exact for r = 0 H k − ( F ) H k ( F ) H k ( F ) Q exact for r = 0 γ r ∂γ r ∂ ǫ (4.18)We now complete the proof of Theorem 1.2. When r = 0 , the map ∂ is surjec-tive. Then δ is surjective as well, since for D i , i > , we can always solve δ (cid:16)X i > C i ⊠ x i (cid:17) = X i > D i ⊠ x i ⇐⇒ ∂C i = iC i − + D i , C − = 0 recursively for i = 0 , , · · · . Hence Π odd = 0 by (4.4) when r = 0 . Suppose r = 0 and k odd. Taking a homogeneous right hand side D i ∈ H i + k − ( F ) , we still haveautomatic solvability when k > , but for k = 3 the equation ∂C = D has a one-dimensional cokernel. For example, η = 1 ⊠ x is not in the image of δ, and definesa generator of Π . There is an apparent difficulty in solving ∂C i = iC i − + D i for k < and i = (3 − k ) / , as ǫ ( iC i − + D i ) is potentially non-zero. But we canalways arrange for this by subtracting i − ǫ ( D i ) ∈ Q from C i − , which leaves ∂C i − unchanged. Hence δ is surjective in these cases also, and = Π − = · · · . Finally, suppose that k is even and Ξ = P i > i ! P i ⊠ x i ∈ Π k . We claim thereexist f i ∈ H ∗ ( F ) , i > , with Ξ = (cid:0)P i > f i t i (cid:1) (Ξ PE ) in terms of the S -action on Π ev applied to the projective Euler operation. As in (4.10), it is convenient to thinkof elements of Π ev as solution sequences ∂P i = P i − . Write S • for the sequencebelonging to the projective Euler operation when r = 0 and the generating classwhen r = 0 . To prove our claim, we show that for every solution sequence P • with = P ∈ H k ( F ) (otherwise factor over t k first) we can write P • = z j ( Q • ) + t ( R • ) , if k > , (4.19) P • = a · S • + t ( R • ) , if k = 0 , , (4.20)for some solution sequences Q • , R • and a ∈ Q . Given this, the proof is a simpledouble induction: First, work modulo the ideal generated by t and prove the claimby induction on k. Then work modulo t and again use induction on k, and so forth. Both (4.19) and (4.20) follow from (4.18): As ∂P = 0 we may choose a preim-age P = γ ( Q ) in Q ∈ R. Pick j minimal with ∂ j ( Q ) = 0 and set Q j − = Q,Q j − = ∂Q j − , . . . , Q = ∂Q . If k > then automatically ǫ ( Q ) = 0 and we mayinductively choose preimages Q j , Q j +1 , . . . of Q under ∂. Hence Q • is a solution se-quence. Then z j ( Q • ) is solution sequence beginning with P , so P • − z j ( Q • ) = t ( R • ) for another solution sequence R • . This proves (4.19). If k = 1 we have P = ay forsome a ∈ Q . Hence P • − a · S • has a zero leftmost term and can therefore be written t ( R • ) for a solution sequence R • . When r = 0 , k = 0 cannot occur, as P = ∂ ( P ) and the image of ∂ is the complement of the constants. If r = 0 and k = 0 then P = b ∈ Q and P • − br S • has a zero leftmost term, so can be written t ( R • ) . (cid:3) According to Proposition 4.2, Theorem 4.3,and Corollary 4.4 the class Ξ PE = P i > y i + r +1 ⊠ x i determines a unique stablepushforward operation π θ ! = (Ξ PE ) P,θ satisfying (1.5). Hence for a principal G -bundle P → B with section s : B → P the projective Euler operation satisfies π θ ! (cid:0) γ ∗ s ( ξ k ) (cid:1) = c k + r +1 ( s ∗ ˜ θ ) , (4.21)where γ P,s = γ s : P −→ G, γ s ( sg ) = g and ˜ θ is the underlying K -theory class of θ. It remains to verify Theorem 1.1(d)–(h). Part (d) follows from Corollary 4.5.For the dualization property (e) it suffices by Corollary 4.4 to treat the caseof a trivial principal G -bundle with section s : B → P. The map γ ˘ P , ˘ s for the dualbundle ˘ π : ˘ P → B with same section ˘ s = s is γ ˘ P , ˘ s = i ◦ γ P,s for the inversion i : G → G. For the dual orientation we have c k + r +1 (˘ s ∗ (˘ θ )) = ( − k + r +1 c k + r +1 ( s ∗ θ ) . Also γ ∗ ˘ P , ˘ s ( ξ k ) = γ ∗ P,s ( i ∗ ( ξ ) k ) = ( − k γ ∗ P,s ( ξ k ) , so ˘ π ˘ θ ! (cid:0) γ ∗ P,s ( ξ k ) (cid:1) = ( − k ˘ π ˘ θ ! (cid:0) γ ∗ ˘ P , ˘ s ( ξ k ) (cid:1) = ( − k c k + r +1 (cid:0) ˘ s ∗ (˘ θ ) (cid:1) = ( − r +1 c k + r +1 ( s ∗ θ ) = ( − r +1 π θ ! ( γ ∗ s ( ξ k )) . The dualization property (1.7) follows.For the composition property (f), note that as in Theorem 4.3 a rational evenstable pushforward operation for principal ( G × G ) -bundles is also uniquely deter-mined by its values on trivial bundles. Suppose therefore that π : P → B and π : P → B are principal G -bundles with sections s : B → P and s : B → P . These determine sections s = s ⊗ s : B → P ⊗ P = P , r : P → P × B P , ( s g ⊗ s ) ( s g, s ) , r : P → P × B P , p ( p , s ) , and r : P → P × B P ,p ( s , p ) fitting into a commutative diagram: P × B P P P = P ⊗ P P B κ κ κ r π π r π r s s s For ( s , s ) : B → P × B P define also γ ( s ,s ) ( s g , s g ) = ( g , g ) . Then H ∗ ( P × B P ) is generated as an H ∗ ( B ) -module by γ ∗ ( s ,s ) ( ξ i ⊠ ξ j ) . We now calculate the first term of (1.9) at γ ∗ ( s ,s ) ( ξ i ⊠ ξ j ) . Write r k = | θ k | for the ranks. From γ ( s ,s ) = ( γ r , γ s ◦ κ ) , we have γ ∗ ( s ,s ) ( ξ i ⊠ ξ j ) = γ ∗ r ( ξ i ) ∪ κ ∗ ( γ ∗ s ξ j ) so applying ( κ ) κ ∗ θ + κ ∗ θ ! and using linearity over κ ∗ from (1.6), gives c i + r + r +1 (cid:0) r ∗ κ ∗ θ + r ∗ κ ∗ θ (cid:1) ∪ γ ∗ s ( ξ j ) . (4.22) OMOLOGICAL LIE BRACKETS AND PUSHFORWARD OPERATIONS 21
Using the Whitney sum formula for Chern classes, naturality of Chern classes, κ ◦ r = s ◦ π , κ ◦ r = ρ P ( s ◦ π , γ s ) , and (4.7), we can rewrite (4.22) as X a + b + c = i + r + r +1 (cid:18) r − cb (cid:19) π ∗ c a ( s ∗ θ ) ∪ π ∗ c c ( s ∗ θ ) ∪ γ ∗ s ( ξ j + b ) . (4.23)Now apply ( π ) θ ! to (4.23) and use linearity over π ∗ to get ( π ) θ ! ( κ ) κ ∗ θ + κ ∗ θ ! (cid:0) γ ∗ ( s ,s ) ( ξ i ⊠ ξ j ) (cid:1) (4.24) = X a + b + c = i + r + r +1 (cid:18) r − cb (cid:19) c a ( s ∗ θ ) ∪ c j + b + r +1 ( s ∗ θ ) ∪ c c ( s ∗ θ )= X a + b + c = i + j + r + r + r +2 (cid:18) r − cb − j − r − (cid:19) c a ( s ∗ θ ) ∪ c b ( s ∗ θ ) ∪ c c ( s ∗ θ ) , where in the last step we subtract j + r + 1 from the index b. Symmetrically, thesubstitution (1 , , i, j, a, b ) → (2 , , j, i, b, a ) yields ( π ) θ ! ( κ ) κ ∗ θ + κ ∗ θ ! (cid:0) γ ∗ ( s ,s ) ( ξ i ⊠ ξ j ) (cid:1) (4.25) = X a + b + c = i + j + r + r + r +2 (cid:18) r − ca − i − r − (cid:19) c a ( s ∗ θ ) ∪ c b ( s ∗ θ ) ∪ c c ( s ∗ θ ) . Finally, we compute the middle route through (1.8). Letting ˆ µ : G × G → G, ˆ µ ( g, h ) = gh − , we have γ ( s ,s ) = (ˆ µ ( γ s ◦ κ , γ r ) , γ r ) . From ˆ µ ∗ ( ξ i ) = ( ξ ⊠ − ⊠ ξ ) i in H ∗ ( G × G ) and the binomial theorem we find γ ∗ ( s ,s ) ( ξ i ⊠ ξ j ) = i X m =0 ( − i − m (cid:18) im (cid:19) κ ∗ γ ∗ s ( ξ m ) ∪ γ ∗ r ( ξ i − m ) ∪ γ ∗ r ( ξ j )= X n + m = i + j ( − i − m (cid:18) im (cid:19) κ ∗ γ ∗ s ( ξ m ) ∪ γ ∗ r ( ξ n ) . (4.26)Using linearity over κ ∗ , the image of (4.26) under ( κ ) κ ∗ ˘ θ + κ ∗ θ ! is X n + m = i + j ( − i − m (cid:18) im (cid:19) γ ∗ s ( ξ m ) ∪ c n + r + r +1 ( r ∗ κ ∗ ˘ θ + r ∗ κ ∗ θ ) . Use the Whitney sum formula, naturality of Chern classes, κ ◦ r = ρ P ( s ◦ π , γ s ) ,κ ◦ r = s ◦ π , and (4.7) to rewrite this as X a + b + m = i + j + r + r +1 X k ( − i − m (cid:18) im − k (cid:19)(cid:18) r − ak (cid:19) π ∗ c a ( s ∗ ˘ θ ) ∪ π ∗ c b ( s ∗ θ ) ∪ γ ∗ s ( ξ m )= X a + b + m = i + j + r + r +1 ( − i − m (cid:18) r − a + im (cid:19) π ∗ c a ( s ∗ ˘ θ ) ∪ π ∗ c b ( s ∗ θ ) ∪ γ ∗ s ( ξ m ) . Here we use the formula P k (cid:0) mk (cid:1)(cid:0) np − k (cid:1) = (cid:0) m + np (cid:1) for binomial coefficients. Recall c a ( s ∗ (˘ θ )) = ( − a c a ( s ∗ θ ) . Applying the map ( π ) θ ! we then get ( π ) θ ! ( κ ) κ ∗ ˘ θ + κ ∗ θ ! (cid:0) γ ∗ ( s ,s ) ( ξ i ⊠ ξ j ) (cid:1) (4.27) = X a + b + m = i + j + r + r +1 ( − i − m (cid:18) r − a + im (cid:19) c a ( s ∗ ˘ θ ) ∪ c b ( s ∗ θ ) ∪ c m + r +1 ( s ∗ θ )= X a + b + c = i + j + r + r + r +2 ( − r + b − j − r − (cid:18) r − a + ic − r − (cid:19) c a ( s ∗ θ ) ∪ c b ( s ∗ θ ) ∪ c c ( s ∗ θ ) , where in the last step we have reindexed c = m + r + 1 . The composition property(1.9) now follows by summing (4.24),(4.25), and (4.27) and noticing that (cid:18) r − cb − j − r − (cid:19) − (cid:18) r − ca − i − r − (cid:19) = ( − b − j − r − (cid:18) r − a + ic − r − (cid:19) , for each term ( a, b, c ) with a + b + c = i + j + r + r + r + 2 in the sum. Indeed,putting n = r − c and k = b − j − r − we can rewrite this as the well-knownidentity (cid:0) nk (cid:1) − (cid:0) nn − k (cid:1) = ( − k (cid:0) k − n − − n − (cid:1) , where (cid:0) nk (cid:1) = 0 for k < . To prove (g), consider the pullback π along itself with its canonical section ∆ ,P × B P PP B. π π ππ ∆ The corresponding trivialization τ : P × G → P × B P composed with π is theprincipal action ρ : P × G → P. Write t k ⋄ α = ρ ∗ ( α ) /t k for the slant product, sorationally t k ⋄ α = k ! t ⋄ · · · ⋄ t ⋄ α. In general, ρ ∗ ( α ) = P k > ( t k ⋄ α ) × ξ k . Since τ − = ( π , γ ∆ ) , this implies π ∗ ( α ) = P k > π ∗ ( t k ⋄ α ) ∪ γ ∗ ∆ ( ξ k ) and so π ∗ ◦ π θ ! ( α ) (1.3) = ( π ) π ∗ θ ! ◦ π ∗ ( α ) (1.5) , (1.6) = X k > ( t k ⋄ α ) ∪ c k + r +1 (˜ θ ) . Finally, (h) is a corollary of Theorem 1.2 since α π θ ! ( α ) ∪ η P defines a stablepushforward operation in Π = 0 . (cid:3)
5. Homological Lie brackets on moduli spaces
We first list the additional data needed for the construction of a Lie bracket.
Assumption . Let
F, G be as in Assumption 3.9. Suppose also that G isabelian. Let M be a topological space. Assume that:(a) There is an operation Φ : M × M → M , that is associative and commuta-tive up to homotopy. The set of connected components π ( M ) is then acommutative monoid with operation α + β = π (Φ)( α, β ) . Write M α ⊂ M for the connected components and Φ α,β = Φ | M α × M β for the restriction.Then Φ α + β,γ ◦ (Φ α,β × id M γ ) ≃ Φ α,β + γ ◦ (id M α × Φ β,γ ) and Φ α,β ◦ σ α,β ≃ Φ β,α where σ α,β : M β × M α → M α × M β swaps the factors.(b) We have a free group action Ψ : M × G → M whose quotient projection M → M/G is a principal G -bundle. The action commutes with Φ , so Ψ(Φ( m , m ) , g ) = Φ(Ψ( m , g ) , Ψ( m , g )) for all m , m ∈ M, g ∈ G. Onthe Cartesian product of G -spaces the diagonal G -action is generally un-derstood. As G is abelian, the quotient ( M × M ) /G has another G -actionof weight (1 , − . The projection ( M × M ) /G → M/G × M/G is then aprincipal G -bundle and Φ descends to Φ /G : ( M × M ) /G → M/G.
OMOLOGICAL LIE BRACKETS AND PUSHFORWARD OPERATIONS 23 (c) There is an orientation θ ∈ K π ( M/G × M/G ) . In other words, there is θ : M × M → F with θ (Ψ( m , g ) , m ) = θ ( m , m ) g = θ ( m , Ψ( m , g − )) for all m , m ∈ M, g ∈ G. Writing θ α,β = θ | M α × M β , we require σ ∗ α,β ( θ α,β ) = ˘ θ β,α , (5.1) (Φ α,β /G × id M γ /G ) ∗ ( θ α + β,γ ) = θ α,γ + θ β,γ , (5.2) (id M α /G × Φ β,γ /G ) ∗ ( θ α,β + γ ) = θ α,β + θ α,γ . (5.3)Hence the Euler form χ ( α, β ) = rk( θ α,β ) is symmetric and biadditive.(d) There are signs ǫ α,β ∈ {± } for all α, β ∈ π ( M ) satisfying ǫ α,β · ǫ β,α = ( − χ ( α,β )+ χ ( α,α ) χ ( β,β ) , (5.4) ǫ α,β · ǫ α + β,γ = ǫ β,γ · ǫ α,β + γ . (5.5) Remark . Usually, the signs ǫ α,β are determined geometrically and arederived from orientations, see [ , §8.3]. These orientation problems are solved inthe series [ – ] using the excision technique of [ ]. Clearly (1.13) is bilinear. For skew symmetry,recall that the cross product is anti-symmetric ζ ⊠ η = ( − ab ( σ α,β ) ∗ ( η ⊠ ζ ) . So ( π α,β ) ! θ α,β ( σ α,β ) ∗ (1.3) = ( σ α,β ) ∗ σ ∗ α,β ( π α,β ) ! σ ∗ α,β ( θ α,β ) = ( σ α,β ) ∗ (˘ π β,α ) !˘ θ β,α (1.7) = ( − χ ( α,β )+1 ( σ α,β ) ∗ ( π β,α ) ! θ β,α combined with Φ α,β ◦ σ α,β ≃ Φ β,α gives [ ζ, η ] = ǫ α,β ( − aχ ( β,β )+ ab + χ ( α,β )+1 (Φ β,α ) ∗ ( π β,α ) ! θ β,α ( η ⊠ ζ ) . This differs from [ η, ζ ] by the sign ǫ β,α ( − bχ ( α,α ) ǫ αβ ( − aχ ( β,β )+ ab + χ ( α,β )+1 (5.4) =( − χ ( α,β )+ χ ( α,α ) χ ( β,β )+ aχ ( β,β )+ bχ ( α,α )+ ab +1 = ( − | ζ | ′ | η | ′ . For the Jacobi identity, let G act on P α,β,γ = ( M α × M β × M γ ) /G with weight (1 , , − and write κ α,β,γ : P α,β,γ → Q α,β,γ = ( M α × M β ) /G × ( M γ ) /G for theprojection modulo this action. Then (Φ α,β × id M γ ) /G : P α,β,γ → ( M α + β × M γ ) /G is G -equivariant, where G acts with weight (1 , − on ( M α + β × M γ ) /G. Setting B α,β,γ = M α /G × M β /G × M γ /G, there is a commutative diagram P α,β,γ ( M α + β × M γ ) /G M α + β + γ /GQ α,β,γ M α + β /G × M γ /GB α,β,γκ α,β,γ (Φ α,β × id Mγ ) /G Φ α + β,γ /Gπ α + β,γ π α,β × id Mγ/G Φ α,β × id Mγ/G to which we apply naturality (1.3) combined with (5.2) to get ( − | ζ | ′ | λ | ′ (cid:2) [ ζ, η ] , λ (cid:3) = ( − | ζ | ′ | λ | ′ ǫ α,β ( − aχ ( β,β ) [(Φ α,β ) ∗ ( π α,β ) ! θ α,β ( ζ ⊠ η ) , λ ]= ( − | ζ | ′ | λ | ′ ǫ α,β ( − aχ ( β,β ) ǫ α + β,γ ( − ( a + b ) χ ( γ,γ ) (Φ α + β,γ ) ∗ ( π α + β,γ ) ! θ α + β,γ (cid:0) (Φ α,β ) ∗ ( π α,β ) ! θ α,β ( ζ ⊠ η ) ⊠ λ (cid:1) = ( − ac + aχ ( β,β )+ bχ ( γ,γ )+ cχ ( α,α )+ χ ( α,α ) χ ( γ,γ ) ǫ α,β ǫ α + β,γ (Φ α + β,γ ) ∗ (Φ α,β × id γ ) ∗ ( κ α,β,γ ) ! θ α,γ + θ β,γ ( π α,β × id γ ) ! θ α,β ( ζ ⊠ η ⊠ λ ) . (5.6) Permuting ( a, b, c ) , ( α, β, γ ) , ( ζ, η, λ ) cyclically in (5.6) gives ( − | η | ′ | ζ | ′ (cid:2) [ η, λ ] , ζ (cid:3) = ( − ba + bχ ( γ,γ )+ cχ ( α,α )+ aχ ( β,β )+ χ ( β,β ) χ ( α,α ) ǫ β,γ ǫ β + γ,α · (Φ β + γ,α ) ∗ (Φ β,γ × id α ) ∗ ( κ β,γ,α ) ! θ β,α + θ γ,α ( π β,γ × id α ) ! θ β,γ ( η ⊠ λ ⊠ ζ ) , (5.7) ( − | λ | ′ | η | ′ (cid:2) [ λ, ζ ] , η (cid:3) = ( − cb + cχ ( α,α )+ aχ ( β,β )+ bχ ( γ,γ )+ χ ( γ,γ ) χ ( β,β ) ǫ γ,α ǫ γ + α,β · (Φ γ + α,β ) ∗ (Φ γ,α × id β ) ∗ ( κ γ,α,β ) ! θ γ,β + θ α,β ( π γ,α × id β ) ! θ γ,α ( λ ⊠ ζ ⊠ η ) . (5.8)We will derive the Jacobi identity ( − | ζ | ′ | λ | ′ (cid:2) [ ζ, η ] , λ (cid:3) + ( − | η | ′ | ζ | ′ (cid:2) [ η, λ ] , ζ (cid:3) + ( − | λ | ′ | η | ′ (cid:2) [ λ, ζ ] , η (cid:3) = 0 . (5.9)from (1.9). Set π = π α,β × id γ : P = Q α,β,γ → B = B α,β,γ and π = σ β,γ,α ( π β,γ × id α ) : P = Q β,γ,α → B β,γ,α ∼ = B, where σ β,γ,α : B → B β,γ,α , σ γ,α,β : B → B γ,α,β and Σ β,γ,α : P α,β,γ → P β,γ,α , Σ γ,α,β : P α,β,γ → P γ,α,β are coordinate permutations.We have diagrams with horizontal G -isomorphisms, where we identify P α,β,γ ∼ = π ∗ ( P ) ∼ = P × B P ∼ = π ∗ ( P ) : P α,β,γ π ∗ ( P ) Q α,β,γ P B α,β,γ B κ α,β,γ id κ π α,β × id γ π id P β,γ,α π ∗ ( ˘ P ) Q β,γ,α P B β,γ,α B κ β,γ,α Σ β,γ,α ˘ κ π β,γ × id α π σ β,γ,α P γ,α,β ( P × B P )˘ Q γ,α,β ( P ⊗ G P )˘ B γ,α,β B κ γ,α,β Σ γ,α,β ˘ κ π γ,α × id β ˘ π σ γ,α,β Set θ = θ α,β = ˘ θ β,α , θ = σ ∗ β,γ,α ( θ β,γ ) , and ˘ θ = σ ∗ γ,α,β ( θ γ,α ) . Now (1.9), withnaturality (1.3) applied to these diagrams and the duality property (1.7), yields − ( π ) θ ! ( κ ) π ∗ θ + π ∗ θ ! + ( π ) θ ! ( κ ) π ∗ θ + π ∗ θ ! + ( − r ( π ) θ ! ( κ ) π ∗ ˘ θ + π ∗ θ ! =( − χ ( α,β )+ χ ( α,γ ) (Σ β,γ,α ) − ∗ ( κ β,γ,α ) θ β,α + θ γ,α ! ( π β,γ × id α ) θ β,γ ! ( σ β,γ,α ) ∗ + ( κ α,β,γ ) θ α,γ + θ β,γ ! ( π α,β × id γ ) θ α,β ! + ( − χ ( α,γ )+ χ ( β,γ ) (Σ γ,α,β ) − ∗ ( κ γ,α,β ) θ γ,β + θ α,β ! ( π γ,α × id β ) θ γ,α ! ( σ γ,α,β ) ∗ This, combined with η ⊠ λ ⊠ ζ = ( − a ( b + c ) ( σ β,γ,α ) ∗ ( ζ ⊠ η ⊠ λ ) ,λ ⊠ ζ ⊠ η = ( − ( a + b ) c ( σ γ,α,β ) ∗ ( ζ ⊠ η ⊠ λ ) , (Φ β + γ,α ) ∗ (Φ β,γ × id α ) ∗ = (Φ α + β,γ ) ∗ (Φ α,β × id γ ) ∗ (Σ β,γ,α ) − ∗ , (Φ γ + α,β ) ∗ (Φ γ,α × id β ) ∗ = (Φ α + β,γ ) ∗ (Φ α,β × id γ ) ∗ (Σ γ,α,β ) − ∗ , substituted into the sum of (5.6), (5.7), and (5.8) proves (5.9). (cid:3) References [1] M. Atiyah and G. Segal,
Twisted K -theory , Ukr. Mat. Visn. 1 (2004), 287–330.[2] M. Atiyah and G. Segal, Twisted K -theory and cohomology , pages 5–43 in Inspired by S.S.Chern: A Memorial Volume in Honor of A Great Mathematician , Nankai Tracts Math. 11,World Sci. Publ., Hackensack, NJ, 2006.[3] A. Dold,
Lectures on algebraic topology , Grundlehren Math. Wiss. 200, Springer-Verlag,Berlin–New York, 1980.[4] A. Dold,
Partitions of unity in the theory of fibrations , Ann. of Math. 78 (1963), 223–255.[5] P. Donovan and M. Karoubi,
Graded Brauer groups and K -theory with local coefficients ,Publ. Math. IHÉS 38 (1970), 5–25.[6] S.K. Donaldson and P.B. Kronheimer, The Geometry of Four-Manifolds , OUP, 1990.[7] S. Eilenberg,
Cohomology and continuous mappings , Ann. of Math 41 (1940), 231–251.
OMOLOGICAL LIE BRACKETS AND PUSHFORWARD OPERATIONS 25 [8] S. Eilenberg and S. MacLane,
On the groups H (Π , n ) . III , Ann. of Math. 60 (1954), 513–557.[9] D.S. Freed, M.J. Hopkins, and C. Teleman, Loop groups and twisted K -theory I , J. Topol. 4(2011), 737—798.[10] J. Gross, D. Joyce, and Y. Tanaka, Universal structures in C -linear enumerative invarianttheories. I , arXiv:2005.05637, 2020.[11] D. Joyce, Ringel–Hall style vertex algebra and Lie algebra structures on the homology ofmoduli spaces , work in progress, http://people.maths.ox.ac.uk/ ∼ joyce/hall.pdf.[12] D. Joyce, Y. Tanaka, and M. Upmeier, On orientations for gauge-theoretic moduli spaces ,Adv. Math. 362 (2020), arXiv:1811.01096.[13] D. Joyce and M. Upmeier,
Canonical orientations for moduli spaces of G -instantons withgauge group SU( m ) or U( m ) , arXiv:1811.02405, 2018.[14] D. Joyce and M. Upmeier, On spin structures and orientations for gauge-theoretic modulispaces , Adv. Math. (in print), arXiv:1908.03524, 2019.[15] D. Joyce and M. Upmeier,
Orientation data for moduli spaces of coherent sheaves overCalabi–Yau -folds , Adv. Math. (in print), arXiv:2001.00113, 2020.[16] M. Kontsevich and Y. Soibelman, Cohomological Hall algebra, exponential Hodge structuresand motivic Donaldson–Thomas invariants , Comm. Number Theory Phys. 5 (2011), 231–352.[17] L.G. Lewis, Jr., J.P. May, M. Steinberger, and J.E. McClure,
Equivariant stable homotopytheory , Lecture Notes in Mathematics 1213, Springer-Verlag, Berlin, 1986.[18] J.P. May and J. Sigurdsson,
Parametrized homotopy theory , Mathematical Surveys andMonographs 132, AMS, Providence, RI, 2006.[19] J.W. Milnor and J.D. Stasheff,
Characteristic classes , Annals of Mathematics Studies 76,Princeton Univ. Press, Princeton, NJ, 1974.[20] A. Pressley, G. Segal,
Loop groups , Oxford Mathematical Monographs, Oxford UniversityPress, New York, 1986.[21] J. Rosenberg,
Continuous-trace algebras from the bundle theoretic point of view , J. Austral.Math. Soc. Ser. A 47 (1989), 368–381.[22] G. Segal,
Classifying spaces and spectral sequences , Publ. Math. IHÉS 34 (1968), 105–112.[23] M. Upmeier,
A categorified excision principle for elliptic symbol families , Quart. J. Math.(2020), doi:10.1093/qmath/haaa063, arXiv:1901.10818.