Canonical orientations for moduli spaces of G 2 -instantons with gauge group SU(m) or U(m)
aa r X i v : . [ m a t h . DG ] F e b Canonical orientations for moduli spaces of G -instantons with gauge group SU( m ) or U( m ) Dominic Joyce and Markus Upmeier
Abstract
Suppose (
X, g ) is a compact, spin Riemannian 7-manifold, with Diracoperator / D g : C ∞ ( X, / S) → C ∞ ( X, /
S). Let G be SU( m ) or U( m ), and E → X be a rank m complex bundle with G -structure. Write B E forthe infinite-dimensional moduli space of connections on E , modulo gauge.There is a natural principal Z -bundle O / D g E → B E parametrizing orienta-tions of det / D g Ad A for twisted elliptic operators / D g Ad A at each [ A ] in B E .A theorem of Walpuski [30] shows O / D g E is trivializable.We prove that if we choose an orientation for det / D g , and a flag struc-ture on X in the sense of [17], then we can define canonical trivializationsof O / D g E for all such bundles E → X , satisfying natural compatibilities.Now let ( X, ϕ, g ) be a compact G -manifold, with d( ∗ ϕ ) = 0. Then wecan consider moduli spaces M G E of G -instantons on E → X , which aresmooth manifolds under suitable transversality conditions, and derivedmanifolds in general, with M G E ⊂ B E . The restriction of O / D g E to M G E is the Z -bundle of orientations on M G E . Thus, our theorem inducescanonical orientations on all such G -instanton moduli spaces M G E .This contributes to the Donaldson–Segal programme [11], which pro-poses defining enumerative invariants of G -manifolds ( X, ϕ, g ) by count-ing moduli spaces M G E , with signs depending on a choice of orientation. Contents Introduction
This is the third of four papers: Upmeier [27], Joyce–Tanaka–Upmeier [21], thispaper, and Cao–Gross–Joyce [7], on orientability and canonical orientations forgauge-theoretic moduli spaces. The first [27] proves the Excision Theorem (seeTheorem 2.15 below), which relates orientations on different moduli spaces.The second [21] develops the general theory of orientations of moduli spaces,and applies it in dimensions 3,4,5 and 6. This paper studies orientations ofmoduli spaces in dimension 7. It uses results from [21, 27], but is self-containedand can be read independently. The sequel [7] concerns dimension 8.Let X be a compact connected manifold, G be SU( m ) or U( m ) for m > g be the Lie algebra of G , and E → X be a rank m complex vector bundlewith a G -structure, so that E is associated to a principal G -bundle Q → X inthe vector representation. Let Ad E be the associated bundle of Lie algebras,the bundle of skew-Hermitian endomorphisms of E , trace-free if G = SU( m ). Definition 1.1. A E ⊂ Ω ( Q, g ) is the space of connections on Q , equippedwith its affine Fr´echet structure modelled on Ω ( X, Ad E ). The gauge group G = Aut( Q ) acts continuously on A E by pullback. The quotient space B E := A E / G is the moduli space of connections on E , as a topological space with thequotient topology. As in [10, p. 133], a connection ∇ ∈ A E is irreducible if thestabilizer group of ∇ under the G -action on A E equals the centre Z ( G ). Write A irr E ⊂ A E for the subset of irreducible connections, and B irr E = A irr E / G ⊂ B E forthe moduli space of irreducible connections.Suppose now that ( X, g ) is an odd-dimensional compact Riemannian spinmanifold with real spinor bundle / S → X . The real Dirac operator coupled to theinduced connections on Ad E defines a family of self-adjoint elliptic operators / D g Ad A : C ∞ ( X, / S ⊗ R Ad E ) −→ C ∞ ( X, / S ⊗ R Ad E ) , ∀ A ∈ A E . (1.1)Let det / D g Ad E be the determinant line bundle of this family, a real line bundleover A E , and let ¯ O / D g E := (cid:0) det / D g Ad E \ { zero section } (cid:1)(cid:14) R > be the associated ori-entation double cover, a principal Z -bundle ¯ O / D g E → A E , where Z = {± } . As A E is contractible, ¯ O / D g E is trivializable, and we have two possible orientations.For X a compact spin 7-manifold and G = SU( m ) the argument of Walpuskiin [30, Prop. 6.3] shows that the gauge group acts trivially on the set of trivi-alizations of ¯ O / D g E , and [21, Ex. 2.13] implies that this also holds for G = U( m ).Hence ¯ O / D g E descends to a principal Z -bundle O / D g E → B E , and orientations maybe constructed equivalently over A E or B E . See [21] for more details.Now suppose ( X, ϕ, g ) is a compact G -manifold, as in [15, § § ∗ ϕ ) = 0. As in Donaldson–Thomas [12] and Donaldson–Segal [11], a connec-tion A on E is called a G - instanton if its curvature F A satisfies F A ∧ ∗ ϕ = 0 . G -instantons is elliptic, and therefore the modulispace M G E of irreducible G -instantons on E modulo gauge is a smooth man-ifold (of dimension 0) under suitable transversality assumptions, and a derivedmanifold (of virtual dimension 0) in the sense of [16, 18–20] in the general case.Examples and constructions of G -instantons are given in [23–25, 29–31].As in [21, § O / D g E → B E to M G E ⊂ B E is the principal Z -bundle of orientations of M G E , as a (derived) manifold. Thus M G E is ori-entable, and an orientation of O / D g E → B E determines an orientation of M G E .Such orientations are important for the programme of [11, 12].In the present paper, we solve the problem of defining canonical orientationsfor M G E . As for moduli spaces of anti-self-dual instantons in dimension four,where orientations depend on an orientation of H ( X ) ⊕ H ( X ) ⊕ H ( X ) (seeDonaldson [9] and Donaldson–Kronheimer [10, Prop. 7.1.39]), this will dependon some additional algebro-topological data, a so-called flag structure .When ( X, ϕ, g ) is a compact G -manifold one can define an interesting classof minimal 3-submanifolds Y in X called associative -folds [15, § M ass , as (derived) manifolds. In the spirit of [11, 12], the firstauthor [17] discussed defining enumerative invariants of ( X, ϕ, g ) counting asso-ciative 3-folds. To determine signs, he defined canonical orientations on modulispaces M ass , using the new idea of flag structures [17, § G -instantons, as a sequence of G -instantons( E, A i ) ∞ i =1 can ‘bubble’ along an associative 3-fold Y as i → ∞ . So the problemsof defining canonical orientations on moduli spaces of associative 3-folds and of G -instantons should be related. In [17, Conj. 8.3], the first author conjecturedthat one should define canonical orientations for moduli spaces of G -instantonsusing flag structures. This paper proves that conjecture.We shall follow a general procedure to orient gauge-theoretic moduli spacesusing excision explained in [21, § orientation Z -torsor Or E of an SU( m )-bundle E .Up to an orientation for the untwisted Diracian, this is the set of orientationson the determinant line bundle of (1.1). For an SU( m )-bundle E → X andSU( m )-bundle E → X we have canonical isomorphisms (Proposition 2.14)Or E ⊕ E ∼ = Or E ⊗ Z Or E , (1.2)Or C m ∼ = Z , (1.3)where (1.3) corresponds to the ‘standard orientations’ of [21, § heorem 1.2. A flag structure F on a compact spin -manifold X determines,for every SU( m ) -bundle E → X and m ∈ N , a canonical orientation o F ( E ) ∈ Or E (1.4) satisfying the following axioms, by which o F ( E ) is uniquely determined: (a) (Normalization.) For E = C m trivial, let o flat ( E ) ∈ Or E be the image of ∈ Z under the isomorphism (1.3) . Then o F ( E ) = o flat ( E ) . (1.5) (b) (Stabilization.) Under the isomorphism Or E ⊕ C ∼ = Or E ⊗ Z Or C ∼ = Or E , using (1.2) and (1.3), we have o F ( E ⊕ C ) = o F ( E ) . (1.6) (c) (Excision.) Let E ± → X ± be SU( m ) -bundles over a pair of compact spin -manifolds with flag structures F ± . Let ρ ± be SU( m ) -frames of E ± outside compact subsets of open U ± ⊂ X ± . Let Φ : E + | U + → φ ∗ ( E − | U − ) be an SU( m ) -isomorphism covering a spin diffeomorphism φ : U + → U − .Assume Φ ◦ ρ + = φ ∗ ρ − outside a compact subset of U + . Under the excisionisomorphism of Theorem we then have Or( φ, Φ , ρ + , ρ − ) (cid:0) o F + ( E + ) (cid:1) = (cid:0) F + | U + /φ ∗ ( F − | U − ) (cid:1) ( α + ) · o F − ( E − ) , (1.7) where α + ∈ H ( U + ; Z ) is the homology class Poincar´e dual to the relativeChern class c ( E + | U + , ρ + ) ∈ H ( U + ; Z ) .Moreover, the following additional properties hold: (i) (Direct sums.) Let E → X be an SU( m ) -bundle and E → X an SU( m ) -bundle. Under the isomorphism (1.2) we then have o F ( E ⊕ E ) = o F ( E ) ⊗ o F ( E ) . (1.8) (ii) (Families.) Let P be a compact Hausdorff space, X a compact spin -manifold, and E → X × P an SU( m ) -bundle. The union of all torsors Or( E | X ×{ p } ) for each p ∈ P is a double cover of P, of which the map p o F ( E | X ×{ p } ) defines a continuous section. In particular, canonicalorientations are deformation invariant.Now let E → X be a rank m complex vector bundle with U( m ) -structure.Then ˜ E = E ⊕ Λ m E ∗ is a rank m + 1 complex vector bundle with SU( m + 1) -structure, and [21, Ex. 2.13] defines a canonical isomorphism of Z -torsors Or E ∼ = Or ˜ E . Hence the first part gives canonical orientations o F ( E ) ∈ Or E forall U( m ) -bundles E → X . These satisfy the analogues of (a) – (c) and (ii) , butmay not satisfy (i) . emark 1.3. The problem with extending (i) to U( m )-bundles in the last part,is that if E , E → X are U( m )- and U( m )-bundles then the left hand side of(1.8) comes from the orientation for the SU( m + m + 1)-bundle ( E ⊕ E ) ⊕ Λ m + m ( E ⊕ E ) ∗ , but the right hand side comes from the orientation for theSU( m + m + 2)-bundle ( E ⊕ Λ m E ∗ ) ⊕ ( E ⊕ Λ m E ∗ ), which is different.The orientations o F ( E ) for U( m )-bundles defined in the last part may notsatisfy (i). For example, let X = CP × S , which has two flag structures F + , F − , and take E = π ∗ CP ( O ( k )) and E = π ∗ CP ( O ( l )) for k, l ∈ Z odd.Using (1.7) we find that changing from F + to F − changes the sign of all threeof o F ± ( E ) , o F ± ( E ) , o F ± ( E ⊕ E ), so (1.8) holds for only one of F + , F − .It may still be possible to choose orientations o F ( E ) for all U( m )-bundles E → X satisfying (a),(b),(i),(ii), and perhaps (c), by a different method.One application of this theorem is to the problem of defining orientations formoduli spaces of G -instantons M G E . As the moduli space is zero-dimensional,there are many arbitrary orientations, so the point of the problem is to come upwith a natural assignment, in particular one that is stable under deformations ofthe G -structure. Following [21, § Corollary 1.4.
Let ( X, ϕ, g ) be a compact G -manifold with d( ∗ ϕ ) = 0 , andchoose an orientation of det / D g for the untwisted Diracian and a flag structure F on X . Then we can define a canonical orientation for the moduli space M G E of G -instantons on X whenever E → X is an SU( m ) - or U( m ) -bundle. Donaldson and Segal [11] propose defining enumerative invariants of (
X, ϕ, g )by counting M G E , with signs , and adding correction terms from associative 3-folds in X . To determine the signs we need an orientation of M G E . Thus,Corollary 1.4 contributes to the Donaldson–Segal programme.It is natural to want to extend Theorem 1.2 and Corollary 1.4 to modulispaces of connections on principal G -bundles Q → X for Lie groups G otherthan SU( m ) and U( m ), but this is not always possible. Section 2.4 gives anexample of a compact, spin 7-manifold X for which O / D g Q → B Q is not orientablewhen Q = X × Sp( m ) → X is the trivial Sp( m )-bundle, for all m > Outline of the paper
We begin in § E of an SU( m )-bundle E → X , is introduced along with its basic properties. We recall from [27]the excision technique from index theory in the context of orientations. It canbe regarded as extending the functoriality of orientation torsors from globallydefined isomorphisms to local ones. Section 3 briefly recalls flag structures, and § X be a compact spin 7-manifold with flag structure F , and E → X an SU( m )-bundle. We show that we can find:5i) A compact 3-submanifold Y ⊂ X .(ii) An SU( m )-trivialization ρ : C m | X \ Y ∼ = −→ E | X \ Y .(iii) An embedding ι : Y ֒ → S , so Y ′ = ι ( Y ) is a 3-submanifold of S .(iv) An isomorphism Ψ : N Y → ι ∗ ( N Y ′ ) between the normal bundles of Y in X and Y ′ in S , preserving orientations and spin structures.(v) Tubular neighbourhoods U of Y in X and U ′ of Y ′ in S , and a spindiffeomorphism ψ : U → U ′ with ψ | Y = ι and d ψ | N Y = Ψ.Define an SU( m )-bundle E ′ → S by E ′ | S \ Y ′ ∼ = C m , E ′ | U ′ ∼ = ψ ∗ ( E | U ),identified over U ′ \ Y ′ by ( ψ | U \ Y ) ∗ ( ρ ), with Ξ : E | U ∼ = −→ ψ ∗ ( E ′ | U ′ ). Thenwe have an excision isomorphism Or( ψ, Ξ , ρ, ρ ′ ) : Or E → Or E ′ .Now every SU( m )-bundle on S is stably trivial, so Theorem 1.2(a),(b)determine a unique orientation o flat ( E ′ ) ∈ Or E ′ . Following Theorem 1.2(c)we define an orientation o FY,ρ,ι, Ψ ( E ) ∈ Or E by o FY,ρ,ι, Ψ ( E ) = (cid:0) F | U /ψ ∗ ( F S | U ′ ) (cid:1) [ Y ] · Or( ψ, Ξ , ρ, ρ ′ ) − ( o flat ( E ′ )) , (1.9)where F S is the unique flag structure on S .Observe that if Theorem 1.2(a)–(c) hold, they force o F ( E ) = o FY,ρ,ι, Ψ ( E ).Thus, if orientations o F ( E ) exist satisfying Theorem 1.2(a)–(c), then theyare uniquely determined, as claimed.(B) We prove that o FY,ρ,ι, Ψ ( E ) is independent of the choices in (A)(i)–(v):(i) Independence of U, U ′ , ψ for fixed Y, ρ, ι,
Ψ is obvious from propertiesof excision isomorphisms.(ii) Independence of Ψ for fixed
Y, ρ, ι is nontrivial. Given two differ-ent choices Ψ , Ψ and ψ , ψ , we compute the signs comparing how ψ , ψ act on orientations of bundles trivial away from Y , and how ψ , ψ act on flag structures near Y , and show these signs are thesame, so the combined effect of both signs in (1.9) cancels out.This is the main point where flag structures are used in the proof.(iii) Independence of ι : Y ֒ → S for fixed Y, ρ is easy, as any two suchembeddings are isotopic through embeddings.(iv) Independence of
Y, ρ is again nontrivial, and is proved by analyzinga bordism Z ⊂ X × [0 ,
1] between two choices Y , Y ⊂ X .We can now define o F ( E ) = o FY,ρ,ι, Ψ ( E ) for all X, F and E → X .(C) We verify the o F ( E ) in (B) satisfy Theorem 1.2(a)–(c) and (i)–(ii).(D) We extend from SU( m )-bundles to U( m )-bundles, which is easy. Acknowledgements.
This research was partly funded by a Simons Collabora-tion Grant on ‘Special Holonomy in Geometry, Analysis and Physics’. Thesecond author was funded by DFG grant UP 85/3-1 and by grant UP 85/2-16f the DFG priority program SPP 2026 ‘Geometry at Infinity.’ The authorswould like to thank Yalong Cao, Aleksander Doan, Sebastian Goette, JacobGross, Andriy Haydys, Johannes Nordstr¨om, Yuuji Tanaka, Richard Thomasand Thomas Walpuski for helpful conversations.
For finite-dimensional vector spaces, the top exterior power has the fundamentalproperty that a short exact sequence0 / / U f / / V g / / W / / canonical isomorphismΛ top U ⊗ Λ top W ∼ = Λ top V. Lemma 2.1.
For finite-dimensional vector spaces V and W we have Λ top ( V ⊗ W ) ∼ = (Λ top V ) ⊗ dim W ⊗ (Λ top W ) ⊗ dim V . The determinant of a homomorphism f : V → V of finite-dimensional vectorspaces is an element of (Λ top V ) ∗ ⊗ Λ top V . This is isomorphic to (Λ top Ker f ) ∗ ⊗ Λ top Coker f , by the fundamental property applied to0 / / Ker f / / V / / V / / Coker f / / . Definition 2.2.
Let F : H → H be a Fredholm operator between Hilbertspaces. The determinant line of F is det F := Λ top Ker F ⊗ (Λ top Coker F ) ∗ . Proposition 2.3.
For every commutative diagram of bounded operators / / F / / F (cid:15) (cid:15) G G (cid:15) (cid:15) / / H H (cid:15) (cid:15) / / / / F / / G / / H / / with exact rows and F, G, H
Fredholm there is a canonical isomorphism det G ∼ = det F ⊗ det H. (2.1) Proof.
Snake lemma and the fundamental property in finite dimensions.7 efinition 2.4.
Let T be a paracompact Hausdorff space. A T -family of Fred-holm operators { F t : H t → H t } t ∈ T is a homomorphism F : H → H of Hilbertspace bundles over T whose restriction to every fibre is Fredholm. The deter-minant line bundle of F is det F := F t ∈ T det F t .To see that det F is locally trivial, pick t ∈ T and s ( t ) : C k → H t so that F t ⊕ s ( t ) is surjective. Extend s to a neighbourhood of t . Proposition 2.3 for( F, F ⊕ s, C k → { } ) gives det F = det( F ⊕ s ) = Λ top Ker( F ⊕ s ) ∗ . Since F ⊕ s is surjective near t , Ker( F ⊕ s ) is a subbundle there. Example 2.5.
Let D be a family of elliptic differential or pseudo-differentialoperators over a compact manifold X . These determine Fredholm operators byregarding them as acting on Sobolev spaces. The determinant line bundle isindependent of the degree of the Sobolev space, since by elliptic regularity thekernels of D and D ∗ consist of smooth sections. Here, D ∗ denotes the formallyadjoint differential operator and we recall Ker D ∗ ∼ = Coker D .For a family of differential operators the manifold and vector bundle maydepend on t ∈ T , as long as they form a fibre bundle [6]. Lemma 2.6.
Let { F t : H t → H t } t ∈ T and { F t : H t → H t } t ∈ T be homotopicthrough T -families of Fredholm operators. Then det F ∼ = det F .Proof. By definition, a homotopy is a ( T × [0 , H ( t, s ). The determinant line bundle of H restricts over T × { s } to det F s for s = 0 ,
1. The inclusions of the endpoints of T × [0 ,
1] are homotopic and as T is paracompact Hausdorff, the pullbacks det H | T ×{ } and det H | T ×{ } areisomorphic.Up to this point the discussion applies to operators over both the real or thecomplex numbers. From now on we need real operators. Definition 2.7.
The orientation cover of a T -family of real Fredholm operators { F t : H t → H t } t ∈ T is Or F := (det F \ { zero section } ) / R > . An orientation forthe determinant of the family is a global section of Or F .As det F is locally trivial, Or F is a double cover of T , so for T connectedthere are either two orientations or none. An advantage of orientation covers istheir deformation invariance. The argument for Lemma 2.6 now gives: Lemma 2.8.
Let { F t : H t → H t } t ∈ T and { F t : H t → H t } t ∈ T be homotopicthrough T -families of real Fredholm operators. Then we have a canonical fibretransport isomorphism Or F ∼ = Or F . In particular, the orientation cover of a T -family of real elliptic operators D depends only on the principal symbol. 8 .2 Orientation torsors and excision We now simplify the discussion by restricting to Diracians twisted by connec-tions. On the level of orientations only the underlying vector bundles matter:
Definition 2.9.
Let (
X, g ) be an odd-dimensional compact spin manifold withreal spinor bundle / S. Let E → X be a vector bundle with SU( m )-structure,and let Ad E be the associated bundle of Lie algebras. The twisted Diracians / D g Ad A : C ∞ ( X, / S ⊗ R Ad E ) −→ C ∞ ( X, / S ⊗ R Ad E ) , A ∈ A E , determine a family / D g Ad E of real elliptic operators parametrized by the space A E of SU( m )-connections on E . Let / D g Ad C m , be the Diracian twisted by thetrivial bundle Ad C m and zero connection. The orientation torsor of E → X isOr E := C ∞ (cid:0) A E , ¯ O / D g E (cid:1) ⊗ Z Or (cid:0) det( / D g Ad C m , ) (cid:1) ∗ . (2.2)Similarly, for a paracompact Hausdorff space P and a P -family of SU( m )-bundles, meaning an SU( m )-bundle E → X × P smooth in the X directions, weget a double cover Or E → P by taking global sections only in the X -direction.By Lemma 2.8, Or E does not depend on g up to canonical isomorphism.More formally, one may take global sections also in this contractible variable. Remark 2.10.
Let Q be the principal SU( m )-frame bundle of E . In the ter-minology of [21], when B Q is n-orientable, the orientation torsor Or E is the setof global sections of the n-orientation bundle ˇ O / D g Q → B Q . Remark 2.11. As / D g Ad C m , is symmetric, the second factor in (2.2) is canoni-cally trivial. However, when m is even, this orientation is sensitive to the metricand changes discontinuously according to the spectral flow of / D g . We prefer tokeep track of an extra choice of orientation for the untwisted Diracian / D g . By(2.1) it induces a trivialization of Or (cid:0) det( / D g Ad C m , ) (cid:1) . The second factor in (2.2)has been introduced to simplify the formulation of the excision principle below. Remark 2.12.
For anti-self-dual moduli spaces in dimension four the Diracianis replaced by d ⊕ d ∗ + : C ∞ (Λ T ∗ X ⊕ Λ T ∗ X ) → C ∞ (Λ T ∗ X ), as in Donaldson–Kronheimer [10]. For these Or E is canonically trivial and the untwisted operatoris responsible for the dependence of orientations on H ( X ) ⊕ H ( X ) ⊕ H ( X ). Definition 2.13.
For E = C m we can evaluate at the zero connection andcanonically identify the orientation torsor with Z . We write o flat ( C m ) ∈ Or C m for this canonical base-point. The behaviour of orientation bundles under direct sums is studied in the com-panion paper [21, Ex. 2.11]. From there we recall the following:9 roposition 2.14.
Let E be an SU( m ) -bundle, E an SU( m ) -bundle over acompact odd-dimensional spin manifold X . We have a canonical isomorphism λ E ,E : Or E ⊗ Z Or E −→ Or E ⊕ E . (2.3) These have the following properties: (i) (Families.)
Let P be compact Hausdorff, E → X × P an SU( m ) -bundle,and E → X × P an SU( m ) -bundle, regarded as P -families of bundles.Then the collection of all maps λ E | X ×{ p } ,E | X ×{ p } for each p ∈ P becomesa continuous map of double covers over P . (ii) (Associative.) λ E ,E ⊕ E ◦ (id Or E ⊗ λ E ,E ) = λ E ⊕ E ,E ◦ ( λ E ,E ⊗ id Or E ) (iii) (Commutative.) Or(flip) ◦ λ E ,E = λ E ,E ◦ flip : Or E ⊗ Or E → Or E ⊕ E . (iv) (Unital.) λ C m , C m (cid:0) o flat ( C m ) ⊗ o flat ( C m ) (cid:1) = o flat ( C m + m ) .Moreover, in (2.5) we will see that the isomorphisms (2.3) are natural. We shall adopt the product notation u · v := λ E ,E ( u ⊗ v ). Proof.
We briefly recall the argument of [21, Ex. 2.11]. For the adjoint bundlesAd( E ⊕ E ) ∼ = Ad( E ) ⊕ Ad( E ) ⊕ R ⊕ Hom C ( E , E ), so by (2.1)¯ O / D g E ⊕ E ∼ = ¯ O / D g E ⊗ Z ¯ O / D g E ⊗ Z Or (cid:0) det R / D g (cid:1) ⊗ Z Or (cid:0) det R ( / D g Hom C ( E ,E ) ) (cid:1) . As the Diracian twisted by Hom C ( E , E ) is complex linear, its kernels andcokernels are complex vector spaces and Or (cid:0) det R ( / D g Hom C ( E ,E ) ) (cid:1) is canonicallytrivial. This, combined with the same for C m , C m in place of E , E , gives(2.3). The same proof works for families. Associativity is [21, (2.12)] andcommutativity is [21, (2.11)], noting that indices vanish in odd dimensions. Seeley’s excision principle [26, Th. 1 on p. 198] (also called transplanting) isone of the key techniques in the K -theory proof of the Atiyah–Singer indextheorem [4, § § G = SU( m ): Theorem 2.15 (Excision) . Let E ± → X ± be SU( m ) -bundles over compactconnected spin manifolds. Let U ± ⊂ X ± be open and let ρ ± be SU( m ) -framesof E ± defined outside compact subsets of U ± . Let φ : U + → U − be a spindiffeomorphism covered by an SU( m ) -isomorphism Φ : E + | U + → E − | U − with Φ ◦ ρ + = φ ∗ ρ − outside some compact subset of U + . This data induces an excision isomorphism Or( φ, Φ , ρ + , ρ − ) : Or E + −→ Or E − . (2.4) These excision isomorphisms have the following properties: i) (Functoriality.) Let E × → X × be an SU( m ) -bundle, U × ⊂ X × open, ψ : U − → U × a spin diffeomorphism, ρ × an SU( m ) -frame defined outsidea compact subset of U × , and Ψ an SU( m ) -isomorphism covering ψ thatidentifies ρ − and ρ × outside a compact subset of U − . Then Or( ψ, Ψ , ρ − , ρ × ) ◦ Or( φ, Φ , ρ + , ρ − ) = Or( ψ ◦ φ, Ψ ◦ Φ , ρ + , ρ × ) . Moreover,
Or(id , id , ρ + , ρ − ) = id Or E . (ii) (Families.) Let E ± → X ± × P be SU( m ) -bundles, where P is a compactHausdorff space. Let ρ ± be SU( m ) -frames of E ± outside compact subsetsof open U ± ⊂ X ± . Let Φ : E + | U + → E − | U − be an SU( m ) -isomorphismcovering a continuous P -family of spin diffeomorphisms φ : U + → U − .Assume Φ ◦ ρ + = φ ∗ ρ − outside a compact subset of U + . Then the collec-tion of all maps (2.4) for each p ∈ P becomes a continuous map of doublecovers over P .In particular, when E ± is pulled back from X ± along the projection, theisomorphism (2.4) is unchanged under deformation of the rest of the data U ± , ρ ± , φ, Φ . (iii) (Empty set.) If U ± = ∅ then Or( φ, Φ , ρ + , ρ − ) = id Z under the isomor-phisms Or E ± ∼ = Z induced by Definition and E + ρ + = C m ρ − = E − . (iv) (Direct sums.) For k = 1 , let E ± k → X ± be SU( m k ) -bundles and let ρ ± k be SU( m k ) -frames of E ± k outside compact subsets of U ± ⊂ X ± .Let φ : U + → U − be a spin diffeomorphism covered by SU( m k ) -isomorph-isms Φ k : E + k → E − k for k = 1 , . Then we have a commutative diagram Or E +1 ⊗ Or E +2 (2.3) (cid:15) (cid:15) Or( φ, Φ ,ρ ± ) ⊗ Or( φ, Φ ,ρ ± ) / / Or E − ⊗ Or E − (2.3) (cid:15) (cid:15) Or E +1 ⊕ E +2 Or( φ, Φ ⊕ Φ ,ρ ± ⊕ ρ ± ) / / Or E − ⊕ E − . (2.5) (v) (Restriction.) Let ˜ φ : ˜ U + → ˜ U − be a spin diffeomorphism extending φ toopen supersets U ± ⊂ ˜ U ± ⊂ X ± , let ˜Φ be an SU( m ) -isomorphism over ˜ φ extending Φ , and assume ˜Φ ◦ ρ + = ˜ φ ∗ ρ − outside a compact subset of U + .Then Or( φ, Φ , ρ + , ρ − ) = Or( ˜ φ, ˜Φ , ρ + , ρ − ) . Here we recall from [22, p. 86] that a spin diffeomorphism is an orientation-preserving diffeomorphism φ : X + → X − together with a choice of lift of theinduced map on GL + ( R )-frame bundles to the topological spin bundles.If E ± → X are SU( m )-bundles and Φ : E + → E − an SU( m )-isomorphism,we may take X + = X − = U + = U − = X , and φ = id X , and ρ + = ∅ = ρ − tobe defined over the empty set. Then we use the shorthandOr(Φ) = Or(id X , Φ , ∅ , ∅ ) : Or E + −→ Or E − . .3 Global automorphisms Theorem 2.15 includes as the special case U ± = X ± the more obvious functo-riality for globally defined diffeomorphisms φ : X + → X − and Φ. The theoremcan be regarded as extending this functoriality to open manifolds and compactlysupported data. The effect of a globally defined diffeomorphism can be studiedusing the following construction. Definition 2.16.
The mapping torus of a diffeomorphism ψ : X → X is thequotient X ψ of X × [0 ,
1] by the equivalence relation ( x, ∼ ( ψ ( x ) , Proposition 2.17.
The mapping torus has the following properties: (i) If X is compact, then X ψ is compact. (ii) X ψ is a fibre bundle over S with typical fibre X . (iii) If X is oriented and ψ is orientation preserving, then X ψ is oriented. (iv) When X has a spin structure and ψ is a spin structure preserving diffeo-morphism we get a topological spin structure on X ψ . (v) Let E → X be a vector bundle and Ψ : E → E an automorphism covering ψ . Then the mapping torus E Ψ is a vector bundle over X ψ . Let X be an odd-dimensional compact spin manifold and Ψ : E → ψ ∗ ( E ) an SU( m ) -isomorphism of an SU( m ) -bundle E → X coveringa spin diffeomorphism ψ : X → X . Then Or( ψ, Ψ , ∅ , ∅ ) = ( − δ ( ψ, Ψ) · id Or E for δ ( ψ, Ψ) := Z X ψ ˆ A ( T X ψ ) (cid:0) ch( E ∗ Ψ ⊗ E Ψ ) − rk( E Ψ ) (cid:1) . Proof.
By choosing a connection A ∈ A E and any smooth path A t , t ∈ [0 , A to A = Ψ ∗ A we may regard E Ψ → X ψ as an S -family of SU( m )-bundles with connection. Pick a metric on X ψ . Using the induced metrics g t on X t we can form the S -family of Diracians / D g t Ad A t twisted by Ad E andthe S -family / D g t Ad C m , twisted by the flat connection. Then Or (cid:0) / D g t Ad A t (cid:1) ⊗ Z Or (cid:0) / D g t Ad C m , (cid:1) ∗ is a double cover of S with monodromy Or( ψ, Ψ , ∅ , ∅ ). On theother hand, since dim X is odd, the Diracians are self-adjoint and the mon-odromy coincides with the spectral flow around the loop [3, Th. 7.4].As explained by Atiyah–Patodi–Singer in [3, p. 95], the spectral flow arounda loop agrees with the index of a single operator on the mapping torus, using [2,Th. 3.10]. For the family / D g t Ad A t we get the positive Diracian / D + on X ψ twistedby Ad E Ψ . To compute the index of a single operator we may complexify andcan then apply the cohomological index formula of Atiyah–Singer [5] to getind( / D +Ad E Ψ ) = Z X ψ ˆ A ( T X ψ )ch(Ad E Ψ ⊗ R C ) = Z X ψ ˆ A ( T X ψ ) (cid:0) ch( E ∗ Ψ ⊗ E Ψ ) − (cid:1) , E Ψ ⊗ R C ) ⊕ C ∼ = E ∗ Ψ ⊗ E Ψ . Applying the same argument to the family / D g t Ad C m , and subtracting yields the desired result. Proposition 2.19.
For ψ, Ψ as in Proposition and dim X = 7 we have δ ( ψ, Ψ) ≡ Z X ψ c ( E Ψ ) ≡ Z X ψ p ( T X ψ ) c ( E Ψ ) mod 2 . (2.6) Hence
Or(Ψ) := Or(id , Ψ , ∅ , ∅ ) = id for every SU( m ) -automorphism Ψ : E → E ( this was obtained by Walpuski in [30, Prop. 6.3]) . Therefore Or( ψ, Ψ , ∅ , ∅ ) =Or( ψ, Ψ , ∅ , ∅ ) whenever Ψ and Ψ cover the same spin diffeomorphism.Proof. The proof is similar to that of Walpuski [30, Prop. 6.3]. We have δ ( ψ, Ψ) = Z X ψ m + 66 c ( E Ψ ) − m c ( E Ψ ) + m p ( T X ψ ) c ( E Ψ ) . By the Atiyah–Singer index theorem [5] A := Z X ψ ˆ A ( T X ψ ) (cid:0) ch( E Ψ ) − m (cid:1) = Z X ψ c ( E Ψ ) − c ( E Ψ )12 + p ( T X ψ ) c ( E Ψ )24is an index and hence an integer. Then (2.6) follows from δ ( ψ, Ψ) − m · A = Z X ψ c ( E Ψ ) ,δ ( ψ, Ψ) − (12 + 2 m ) A = 2 Z X ψ c ( E Ψ ) + 12 p ( T X ψ ) c ( E Ψ ) . Finally, when ψ = id X we have X ψ = X × S and p ( T X ψ ) = pr ∗ X p ( T X ). Onthe spin 7-manifold X the cohomology class p ( T X ) is divisible by four.
Example 2.20.
Let E → X be an SU( m )-bundle over a compact spin 7-manifold with second Chern class Poincar´e dual to a 3-submanifold Y ⊂ X .Let Ψ : E → E be an SU( m )-isomorphism covering a spin diffeomorphism ψ : X → X satisfying ψ | Y = id Y . Then we may regard Y × S ⊂ X ψ .Formula (2.6) is the self-intersection (mod 2) of the class α in H ( X ψ )Poincar´e dual to c ( E Ψ ). We have α = [ Y × S ] + β for some β ∈ H ( X ),where X is included into X ψ at some fixed point of [0 , β • β = 0 in X ψ , we get δ ( ψ, Ψ) ≡ Z X ψ c ( E Ψ ) ≡ α • α ≡ [ Y × S ] • [ Y × S ] mod 2 . This again shows that δ (id , Ψ) ≡ ψ = id.13 .4 An example of a non-orientable moduli space B Q Suppose (
X, g ) is a compact, spin Riemannian 7-manifold, and G is any Liegroup, and Q → X is a principal G -bundle. Then generalizing § A Q of connections on Q and B Q = A Q / G of connections on Q modulo gauge, and a principal Z -bundle ¯ O / D g Q → A Q parametrizing orientationson det / D g Ad A for A ∈ A Q .In § G = SU( m ) or U( m ), and then Walpuski [30, Prop. 6.3]and [21, Ex. 2.13] show that G acts trivially on the set of global sections of ¯ O / D g Q ,so that ¯ O / D g Q descends to a principal Z -bundle O / D g Q → B Q , which is orientable.But what about other Lie groups G ?This section will give an example of ( X, g ) for which when G = Sp( m ) for m > Q = X × Sp( m ) → X is the trivial Sp( m )-bundle, G acts non-triviallyon the set of global sections of ¯ O / D g Q , so that although ¯ O / D g Q does in fact descendto a principal Z -bundle O / D g Q → B Q , this is non-orientable (i.e. it has no globalsections). Hence the analogue of [30, Prop. 6.3] is false for Sp( m )-bundles.A result on stabilizing H m -bundles [21, Ex. 2.16] implies that if O / D g Q → B Q is non-orientable for Q = X × Sp(2) the trivial Sp(2)-bundle, then the sameholds for Q = X × Sp( m ) for m >
2. So we consider only G = Sp(2).To show that O / D g Q → B Q is non-orientable, it is enough to find a smooth loop γ : S → B Q such that the monodromy of O / D g Q around γ is −
1. As in [7, § γ : S → B Q is equivalent to a principal Sp(2)-bundle R → X × S which is trivial on X × { } , together with a partial connection on R in the X directions, and any such R may be written as the mapping torus R f of a smoothmap f : X → Sp(2), obtained by taking the trivial bundle on X × [0 ,
1] andidentifying endpoints using the gauge transformation f : X → Sp(2).As in Walpuski [30, § O / D g Q around γ is ( − SF( γ ) , where SF( γ ) is the spectral flow of the family of el-liptic operators (cid:0) / D g Ad γ ( t ) (cid:1) t ∈S , which may be computed as an index SF( γ ) =ind( / D +Ad( R f ) ) of the positive Dirac operator / D + on X × S twisted by any con-nection on Ad( R f ).Thus, to show that O / D g Q → B Q is non-orientable on X , we should find acompact spin 7-manifold X and a smooth f : X → Sp(2) such that ind( / D +Ad( R f ) )is odd. We will do this in Example 2.24, after some initial computations. Lemma 2.21.
For an
Sp(2) -bundle R over an -dimensional base we have ch(Ad( R ) ⊗ R C ) = 10 − c ( R ) + 32 c ( R ) − c ( R ) . (2.7) Proof.
This can be computed using Chern roots, meaning it suffices to establish(2.7) in the case that the H -bundle ( R × H ) / Sp(2) → X associated to R isthe direct sum of quaternionic line bundles.14 roposition 2.22. Let X be a compact spin -manifold and R → X × S an Sp(2) -bundle. Then the index has the parity of the Euler number of R : ind / D +Ad( R ) ≡ Z X ×S c ( R ) mod 2 . Proof.
Using (2.7) we find thatind / D +Ad( R ) + 6 · ind / D + R = Z X ×S p ( T X ) c ( R )2 + 2 c ( R ) − c ( R ) . As p ( T X ) is divisible by four, the first summand on the right is even.Let R f be the mapping torus bundle over X ×S of a smooth f : X → Sp(2).Then the Euler number of R f is the degree of X f −→ Sp(2) π −→ Sp(2) / Sp(1) = S . For non-orientability, we seek X and f such that this degree is odd. Example 2.23.
Let X = S and f : S → Sp(2) be smooth. Then the degreeof π ◦ f is always divisible by 12 and therefore O / D g Q → B Q is orientable, for Q the trivial Sp(2)-bundle over S . To see this, consider the long exact sequenceof homotopy groups of the fibration π : · · · / / π Sp(2) π ∗ / / π S = Z ∂ / / π Sp(1) / / π Sp(2) / / · · · As π Sp(1) = Z and π Sp(2) = { } , the cokernel of π ∗ is Z . Hence ori-entability holds for the moduli space of Sp(2)-connections on S . Example 2.24.
For (cid:0) a bc d (cid:1) ∈ Sp(2) and q ∈ Sp(1), define M := (cid:18) | a | + bq ¯ b a ¯ c + bq ¯ dc ¯ a + dq ¯ b | c | + dq ¯ d (cid:19) . Then M ∈ Sp(2). Replacing ( a, b, c, d, q ) by ( ar, bs, cr, ds, ¯ sqs ) for r, s ∈ Sp(1)does not change the matrix M . Hence the formula defines a map f : X = Sp(2) × Sp(1) × Sp(1)
Sp(1) −→ Sp(2) , where the diagonal subgroup ( r, s ) ∈ Sp(1) × Sp(1) ⊂ Sp(2) acts on q ∈ Sp(1) byconjugating with the second factor q sq ¯ s . It is easy to see that deg( π ◦ f ) = 1.It follows that O / D g Q → B Q is non-orientable, where Q = X × Sp(2) → X is thetrivial Sp(2)-bundle over X .In the sequel [7, Ex. 1.14] we will use Example 2.24 to find non-orientablemoduli spaces of Sp( m )-connections B Q for m > X × S .15 Flag structures
We recall the following from [17, § X is not assumed to be compact. Definition 3.1.
Let X be an oriented 7-manifold, and consider pairs ( Y, s ) ofa compact, oriented 3-submanifold Y ⊂ X , and a non-vanishing section s of thenormal bundle N Y of Y in X . We call ( Y, s ) a flagged submanifold in X .For non-vanishing sections s, s ′ of N Y define d ( s, s ′ ) := Y • (cid:8) t · s ( y ) + (1 − t ) · s ′ ( y ) (cid:12)(cid:12) t ∈ [0 , , y ∈ Y (cid:9) ∈ Z , (3.1)using the intersection product ‘ • ’ between a 3-cycle and a 4-chain whose bound-ary does not meet the cycle, see Dold [8, (13.20)].Let ( Y , s ) , ( Y , s ) be disjoint flagged submanifolds with [ Y ] = [ Y ] in H ( X ; Z ). Choose an integral 4-chain C with ∂C = Y − Y . Let Y ′ , Y ′ be smallperturbations of Y , Y in the normal directions s , s . Then Y ′ ∩ Y = Y ′ ∩ Y = ∅ as s , s are non-vanishing, and Y ′ ∩ Y = Y ′ ∩ Y = ∅ as Y , Y are disjointand Y ′ , Y ′ are close to Y , Y . Define D (( Y , s ) , ( Y , s )) to be the intersectionnumber ( Y ′ − Y ′ ) • C in homology over Z . Here we regard[ C ] ∈ H ( X, Y ∪ Y ; Z ) , [ Y ′ ] , [ Y ′ ] ∈ H ( Y ′ ∪ Y ′ , ∅ ; Z ) . Note that since Y ′ , Y ′ are small perturbations and Y , Y are disjoint we have( Y ∪ Y ) ∩ ( Y ′ ∪ Y ′ ) = ∅ . This is independent of the choices of C and Y ′ , Y ′ .In [17, Prop.s 3.3 & 3.4] the first author shows that if ( Y , s ) , ( Y , s ) , ( Y , s )are disjoint flagged submanifolds with [ Y ] = [ Y ] = [ Y ] in H ( X ; Z ) then D (( Y , s ) , ( Y , s )) ≡ D (( Y , s ) , ( Y , s ))+ D (( Y , s ) , ( Y , s )) mod 2 , (3.2)and if ( Y ′ , s ′ ) is any small deformation of ( Y, s ) with
Y, Y ′ disjoint then D (( Y, s ) , ( Y ′ , s ′ )) ≡ . (3.3) Definition 3.2. A flag structure on X is a map F : (cid:8) flagged submanifolds ( Y, s ) in X (cid:9) −→ {± } , satisfying:(i) F ( Y, s ) = F ( Y, s ′ ) · ( − d ( s,s ′ ) .(ii) If ( Y , s ) , ( Y , s ) are disjoint flagged submanifolds in X with [ Y ] = [ Y ]in H ( X ; Z ) then F ( Y , s ) = F ( Y , s ) · ( − D (( Y ,s ) , ( Y ,s )) . This is a well behaved condition by (3.2)–(3.3).(iii) If ( Y , s ) , ( Y , s ) are disjoint flagged submanifolds then16 ( Y ∐ Y , s ∐ s ) = F ( Y , s ) · F ( Y , s ) . Flag structures restrict to open subsets in the obvious way.Here is [17, Prop. 3.6]:
Proposition 3.3.
Let X be an oriented -manifold. Then: (a) There exists a flag structure F on X . (b) If F, F ′ are flag structures on X then there exists a unique group mor-phism H ( X ; Z ) → {± } , denoted F ′ /F, such that F ′ ( Y, s ) = F ( Y, s ) · ( F ′ /F )[ Y ] for all ( Y, s ) . (3.4) (c) Let F be a flag structure on X and ǫ : H ( X ; Z ) → {± } a morphism,and define F ′ by (3.4) with F ′ /F = ε . Then F ′ is a flag structure on X .Hence the set of flag structures on X is a torsor over Hom (cid:0) H ( X ; Z ) , Z (cid:1) . Example 3.4.
Every oriented 7-manifold X with H ( X ; Z ) = 0 has a uniqueflag structure. More generally, a basis [ Y i ] of the image of H ( X ; Z ) → H ( X ; Z )for submanifolds Y i ⊂ X with chosen normal sections s i induce a unique flagstructure F with F ( Y i , s i ) = 1. For example, S has a unique flag structure F S , and Y × S has a preferred flag structure F with F ( Y × { x } , Y × v ) = 1for any x ∈ S and 0 = v ∈ T x S . Definition 3.5.
Let ψ : X ′ → X be an orientation-preserving diffeomorphism.The pullback of the flag structure F on X is ( ψ ∗ F )( Y ′ , s ′ ) := F ( ψ ( Y ′ ) , d ψ ◦ s ′ ).The pushforward is defined to be the pullback along ψ − .When X ′ = X we can compare a flag structure to its pullback along ψ : Proposition 3.6.
Let X be an oriented -manifold and Y ⊂ X a compactoriented -submanifold. Suppose ψ : X → X is an orientation-preserving dif-feomorphism with ψ | Y = id Y . Then ( F/ψ ∗ F )[ Y ] = ( − [ Y ×S ] • [ Y ×S ] for anyflag structure F on X, where [ Y ×S ] • [ Y ×S ] is the self-intersection of Y ×S in the mapping torus X ψ .Proof. Pick s : Y → N Y non-vanishing. Let Γ( y, t ) := (1 − t ) s ( y ) + t · d ψ ◦ s ( y )for y ∈ Y and t ∈ [0 , F/ψ ∗ F )[ Y ] = F ( Y, s ) · F ( Y, d ψ ◦ s ) − = ( − d ( s, d ψ ◦ s ) = ( − Y • Im(Γ) . The normal bundle of Y ×S in X ψ is the mapping torus of d ψ : N Y → N Y , so wecan regard Γ as a normal section of Y × S in X ψ by ˆΓ( y, t ) := [Γ( y, t ) , t ]. Therebeing no intersection points at t = 0 , Y • Im(Γ) = ( Y × S ) • Im(ˆΓ).Finally, by taking s to zero ˆΓ is homologous to Y × S in X ψ . This section proves our main result Theorem 1.2, following the outline in § .1 (A) The construction of orientations o FY,ρ,ι, Ψ ( E ) We first extend o flat ( C m ) ∈ Or C m from Definition 2.13 to stably trivial bundles. Definition 4.1.
Let E → X be a stably trivial SU( m )-bundle over a compactspin 7-manifold. Then we find an SU( m + k )-isomorphism Φ : E ⊕ C k → C m + k over id X . Using (2.3) and Or(Φ) we can identify Or E with Z . That is, thereexists a unique o flat ( E ) ∈ Or E satisfyingOr(Φ) (cid:0) o flat ( E ) · o flat ( C k ) (cid:1) = o flat ( C m + k ) , (4.1)using the product notation defined after Proposition 2.14. Proposition 4.2.
These orientations have the following properties: (i) (Well-defined.)
The definition of o flat ( E ) is independent of k and Φ . (ii) (Families.) Let P be compact Hausdorff and E → X × P stably trivial.Then p o flat ( E | X ×{ p } ) is a continuous section of the double cover Or E . (iii) (Functoriality.) Let E be an SU( m ) -bundle and E an SU( m ) -bundleover X, both stably trivial. Let ℓ , ℓ ∈ N and let Ψ : E ⊕ C ℓ → E ⊕ C ℓ be an SU -isomorphism. Then Or(Ψ)( o flat ( E ⊕ C ℓ )) = o flat ( E ⊕ C ℓ ) . (iv) (Stability.) o flat ( E ⊕ C ℓ ) = o flat ( E ) · o flat ( C ℓ ) . (v) (Direct sums.) Let E , E → X be SU( m ) - and SU( m ) -bundles, bothstably trivial. Then o flat ( E ⊕ E ) = o flat ( E ) · o flat ( E ) .Proof. (i) Given ℓ ∈ N and Φ : E ⊕ C k → C m + k we use the properties of Propo-sition 2.14 and find o flat ( C m + k + ℓ ) = o flat ( C m + k ) · o flat ( C ℓ ) (unital)= Or(Φ) (cid:0) o flat ( E ) · o flat ( C k ) (cid:1) · o flat ( C ℓ ) (by (4.1))= Or(Φ ⊕ id C ℓ ) (cid:2)(cid:0) o flat ( E ) · o flat ( C k ) (cid:1) · o flat ( C ℓ ) (cid:3) (by (2.5))= Or(Φ ⊕ id C ℓ ) (cid:2) o flat ( E ) · (cid:0) o flat ( C k ) · o flat ( C ℓ ) (cid:1)(cid:3) (assoc.)= Or(Φ ⊕ id C ℓ ) (cid:0) o flat ( E ) · o flat ( C k + ℓ ) (cid:1) (unital) . This proves independence of k . The independence of Φ follows from Proposi-tion 2.19. Once o flat is well-defined, (iii)–(iv) are clear.(ii) We know already that Or(Φ) and λ are continuous maps of double coversover P and that o flat ( C k ) and o flat ( C m + k ) are continuous sections.18v) Let Φ i : E ⊕ C k i → C m i + k i be SU( m i + k i )-isomorphisms. Then o flat ( C m + m + k + k )= o flat ( C m + k ) · o flat ( C m + k ) (unital)= Or(Ψ ) (cid:0) o flat ( E ) · o flat ( C k ) (cid:1) · Or(Ψ ) (cid:0) o flat ( E ) · o flat ( C k ) (cid:1) (by (4.1))= Or(Ψ ⊕ Ψ ) (cid:0) o flat ( E ) · o flat ( C k ) · o flat ( E ) · o flat ( C k ) (cid:1) (by (2.5))= Or(Ψ ⊕ Ψ ) Or(id E ⊕ flip ⊕ id C k ) (cid:0) o flat ( E ) · o flat ( E ) · o flat ( C k ) · o flat ( C k ) (cid:1) (comm.).On the other hand, by using the trivialization (Φ ⊕ Φ ) ◦ (id E ⊕ flip ⊕ id C k )of E ⊕ E ⊕ C k + k in the definition (4.1) we haveOr(Φ ⊕ Φ ) ◦ Or(id E ⊕ flip ⊕ id C k )( o flat ( E ⊕ E ) · o flat ( C k + k ))= o flat ( C m + m + k + k ) . Combining the last two equations and using unitality implies the result.
Example 4.3.
Since π (SU( m )) = 0 for m ≥
4, every SU( m )-bundle E over S is stably trivial. SU( m ) -bundles outside codimension 4 The next definition explains how to trivialize an SU( m )-bundle E → X outsidea submanifold Y ⊂ X of codimension 4. Definition 4.4.
Let X be a compact, oriented manifold of dimension n , with n
11, and E → X be a rank m complex vector bundle with SU( m )-structure,for m >
1. Write C m − = X × C m − for the trivial vector bundle over X with fibre C m − , and Hom( C m − , E ) → X for the bundle of complex vectorbundle morphisms over X , and Hom ( k ) ( C m − , E ) for the determinantal varietyof homomorphisms s : C m − → E x of rank m − − k for k = 0 , . . . , m −
1, whichis a submanifold of Hom( C m − , E ) of real codimension 2 k ( k + 1).A morphism s : C m − → E is a section s : X → Hom( C m − , E ). We call s generic if it is transverse to each Hom ( k ) ( C m − , E ). This is an open densecondition on such s . If s is generic then s − (cid:0) Hom ( k ) ( C m − , E ) (cid:1) is a submanifoldof X of dimension n − k ( k + 1), and so is empty if k > n Y = s − (cid:0) Hom (1) ( C m − , E ) (cid:1) , the degeneracy locus of s . Then Y isan embedded submanifold of X of dimension n −
4. It is closed in X , as theclosure lies in the union of s − (cid:0) Hom ( k ) ( C m − , E ) (cid:1) for k >
1, but these areempty for k >
1. It is oriented as X is, and the fibres of Hom( C m − , E ) → X and Hom (1) ( C m − , E ) → X are complex manifolds and so oriented.As X \ Y = s − (cid:0) Hom (0) ( C m − , E ) (cid:1) , we see that s | X \ Y : C m − | X \ Y → E | X \ Y is injective. Let s ∗ | X \ Y : E | X \ Y → C m − | X \ Y be its Hermitian adjoint,with respect to the Hermitian metrics on E and C m − . Then s ∗ ◦ s | X \ Y :19 m − | X \ Y → C m − | X \ Y is invertible with positive eigenvalues, and thus has aninverse square root ( s ∗ ◦ s ) | − / X \ Y : C m − | X \ Y → C m − | X \ Y . Consider s | X \ Y ◦ ( s ∗ ◦ s ) | − / X \ Y : C m − | X \ Y −→ E | X \ Y . This is an injective linear map of complex vector bundles which is isometric forthe Hermitian metrics on C m − | X \ Y and E | X \ Y .As E has an SU( m )-structure, there is a unique isomorphism of SU( m )-bundles ρ : C m | X \ Y → E | X \ Y , such that ρ | C m − | X \ Y = s | X \ Y ◦ ( s ∗ ◦ s ) | − / X \ Y ,regarding C m − as a vector subbundle of C m = C m − ⊕ C .Thus, for any SU( m )-bundle E → X we can find a codimension 4 submani-fold Y ⊂ X and an SU( m )-framing ρ : C m | X \ Y ∼ = −→ E | X \ Y of E outside Y .From the definition of Chern classes in terms of degeneracy cycles in Griffithsand Harris [13, p. 412-3], we see that the homology class [ Y ] ∈ H n − ( X ; Z ) isPoincar´e dual to the second Chern class c ( E ) ∈ H ( X ; Z ). More generally, if U is any open neighbourhood of Y in X then [ Y ] ∈ H n − ( U ; Z ) is Poincar´e dualto to the compactly-supported second Chern class c ( E | U , ρ | U \ Y ) ∈ H ( U ; Z ). We will need the following variation on Whitney’s Embedding Theorem:
Theorem 4.5 (Haefliger [14, p. 47]) . Let Y be a compact -manifold. Then: (i) There is an embedding
Y ֒ → S . (ii) Any two embeddings
Y ֒ → S are isotopic through embeddings. Wall has shown the following:
Theorem 4.6 (Wall [28, p. 567]) . Let Z be a compact connected -manifoldwith non-empty boundary. Then there exists an embedding Z ֒ → S . o FY,ρ,ι, Ψ ( E ) Definition 4.7.
Suppose X is a compact, oriented, spin 7-manifold with flagstructure F , and E → X is a rank m complex vector bundle with SU( m )-structure. After making some arbitrary choices, we will define an orientation o FY,ρ,ι, Ψ ( E ) in Or E .As in Definition 4.4, choose a generic morphism s : C m − → E , and fromthis construct a compact, oriented 3-submanifold Y ⊂ X and an SU( m )-framing ρ : C m | X \ Y ∼ = −→ E | X \ Y of E outside Y . By Theorem 4.5(i) we may choose anembedding ι : Y ֒ → S . Set Y ′ = ι ( Y ), a 3-submanifold of S . Write N Y , N Y ′ for the normal bundles of Y, Y ′ in X, S .We claim that we may choose an isomorphism Ψ : N Y → ι ∗ ( N Y ′ ) of vectorbundles on Y , which identifies the orientations and spin structures on the totalspaces of N Y , N Y ′ induced by the orientations and spin structures on X, S .20ere when we say that Ψ identifies the spin structures, we mean that it has alift ˆΨ to the spin bundles of N Y , N Y ′ .To see this, choose a spin structure on the oriented 3-manifold Y and trans-port it along ι to Y ′ . Using the spin structures on X, S we get, by 2-out-of-3 [22, Prop. 1.15], spin structures P Spin ( N Y ) → Y and P Spin ( N Y ′ ) → Y ′ on N Y and N Y ′ . As dim Y = dim Y ′ = 3 and Spin(4) is 2-connected, these are trivialprincipal bundles, and therefore we may choose an oriented, spin isomorphismΨ between the normal bundles N Y , N Y ′ .Choose tubular neighbourhoods U ⊂ X and U ′ ⊂ S of Y, Y ′ in X, S ,identified with open ǫ -balls in N Y , N Y ′ for small ǫ >
0. Then Ψ induces adiffeomorphism ψ : U → U ′ identifying orientations and spin structures on U, U ′ , with ψ | Y = ι and d ψ | N Y = Ψ. As in Definition 4.4, [ Y ] ∈ H ( U ; Z ) isPoincar´e dual to c ( E | U , ρ | U \ Y ) ∈ H ( U ; Z ).Define a rank m complex vector bundle E ′ → S with SU( m )-structure by E ′ | S \ Y ′ ∼ = C m and E ′ | U ′ ∼ = ψ ∗ ( E | U ), identified over U ′ \ Y ′ by ( ψ | U \ Y ) ∗ ( ρ ).Write Ξ : E | U → ψ ∗ ( E ′ | U ′ ) for the natural isomorphism and ρ ′ = Ξ ◦ ρ ◦ ψ − forthe natural SU( m )-framing C m | S \ Y ′ ∼ = −→ E ′ | S \ Y ′ . Then Theorem 2.15 gives acanonical excision isomorphism Or( ψ, Ξ , ρ, ρ ′ ) : Or E → Or E ′ in (2.4).By Example 4.3, E ′ → S is stably trivial, so Definition 4.1 gives an orien-tation o flat ( E ′ ) ∈ Or E ′ . As in (1.9), define o FY,ρ,ι, Ψ ( E ) ∈ Or E by o FY,ρ,ι, Ψ ( E ) = (cid:0) F | U /ψ ∗ ( F S | U ′ ) (cid:1) [ Y ] · Or( ψ, Ξ , ρ, ρ ′ ) − ( o flat ( E ′ )) , (4.2)where F S is the unique flag structure on S , and F | U /ψ ∗ ( F S | U ′ ) is as inProposition 3.3(b) for the flag structures F | U and ψ ∗ ( F S | U ′ ) on U . Uniqueness of orientations, subject to our axioms, is explained in the outline ofthe proof in § o FY,ρ,ι, Ψ ( E ) is independent of choices We will prove the orientation o FY,ρ,ι, Ψ ( E ) in Definition 4.7 depends only on X, F and E → X , and not on the other arbitrary choices. o FY,ρ,ι, Ψ ( E ) is independent of U, U ′ , ψ for fixed Y, ρ, ι,
ΨIn the situation of Definition 4.7, let
X, F, E, Y, ρ, ι,
Ψ be fixed, and let U , U ′ , ψ and U , U ′ , ψ be alternative choices for U, U ′ , ψ . Then by properties of tubu-lar neighbourhoods we can find families U t , U ′ t and ψ t : U t → U ′ t dependingsmoothly on t ∈ [0 ,
1] and interpolating between U , U ′ , ψ and U , U ′ , ψ . Foreach t ∈ [0 ,
1] we get an orientation o FY,ρ,ι, Ψ ( E ) t in Definition 4.7 defined us-ing U t , U ′ t , ψ t . The families property Theorem 2.15(ii) of excision isomorphismsimplies that o FY,ρ,ι, Ψ ( E ) t depends continuously on t , and so is constant. Hence o FY,ρ,ι, Ψ ( E ) is independent of the choice of U, U ′ , ψ .21 .2.2 (B)(ii) o FY,ρ,ι, Ψ ( E ) is independent of Ψ for fixed Y, ρ, ι
We will need the following:
Proposition 4.8.
Let Y be a compact n -manifold, N → Y be a rank k real vector bundle with an orientation and spin structure on its fibres, and Φ : N → N be an orientation and spin-preserving automorphism of N covering id Y : Y → Y . Suppose E → N is a rank m complex vector bundle with U( m ) -structure for m > n +2 k with a framing ρ outside a compact subset of N . Thenthere exists a U( m ) -isomorphism Θ : E → Φ ∗ ( E ) over id N with Θ ◦ ρ = Φ ∗ ( ρ ) outside a compact subset of N .When n = 3 and k = 4 , the same holds with SU( m ) in place of U( m ) .Proof. By Atiyah, Bott and Shapiro [1, Th. 12.3(ii)], the orientation and spinstructure on N determine a Thom isomorphism Thom : K ( Y ) → K ( N ), aform of Bott periodicity. By naturality we have a commutative diagram K ( Y ) id ∗ Y (cid:15) (cid:15) Thom / / K ( N ) Φ ∗ (cid:15) (cid:15) K ( Y ) Thom / / K ( N ) . Since the horizontal maps are isomorphisms we see that Φ ∗ = id. Thus we have[ E, ρ ] = [Φ ∗ ( E ) , Φ ∗ ( ρ )] in K ( N ). As we are in the stable range 2 m > k , theK-theory class determines the bundle up to isomorphism, so Θ exists as claimed.For the second part, every Spin(4)-bundle over a compact 3-manifold Y istrivializable, and Ω B SU ≃ Ω B U means that stably there is no differencebetween unitary and special unitary bundles on N ∪ {∞} ∼ = Y + ∧ S . Proposition 4.9. o FY,ρ,ι, Ψ ( E ) is independent of Ψ .Proof. In the situation of Definition 4.7, let
X, F, E, Y, ρ, ι, Y ′ be fixed, and letΨ , Ψ : N Y → ι ∗ ( N Y ′ ) be alternative choices for Ψ. Using the same tubularneighbourhoods U, U ′ for Y, Y ′ in X, S , which do not affect o FY,ρ,ι, Ψ i ( E ) by § ψ , ψ : U → U ′ . Let E ′ , E ′ → S be thecorresponding SU( m )-bundles, with SU( m )-framings ρ ′ , ρ ′ over S \ Y ′ .Pick a spin structure on Y ∼ = Y ′ , which determines spin structures on thefibres N Y , N Y ′ by 2-out-of-3 for spin structures, where Ψ , Ψ preserve these spinstructures. Write φ = Ψ − ◦ Ψ : N Y → N Y , so that φ preserves orientationsand spin structures on the fibres of N Y .Write S ( N Y ⊕ R ) for the sphere bundle of the vector bundle N Y ⊕ R → Y ,so that S ( N Y ⊕ R ) → Y is an S -bundle, containing N Y as an open set, andobtained by adding a point at infinity to each fibre R of N Y → Y , makingthe fibres R ∐ {∞} = S . Then S ( N Y ⊕ R ) is a compact, oriented, spin7-manifold, and Y embeds in S ( N Y ⊕ R ) as the zero section of N Y . Write˜ φ : S ( N Y ⊕ R ) → S ( N Y ⊕ R ) for the diffeomorphism induced by φ : N Y → N Y .As U is a tubular neighbourhood of Y in X it is diffeomorphic to the bundleof open ǫ -balls in N Y , so we can regard U as an open neighbourhood of Y in22 Y and S ( N Y ⊕ R ). Write ˜ E → S ( N Y ⊕ R ) for the rank m complex vectorbundle with SU( m )-structure given by ˜ E | U ∼ = E | U (identifying the open subsets U in X and S ( N Y ⊕ R )), and ˜ E | S ( N Y ⊕ R ) \ Y ∼ = C m | S ( N Y ⊕ R ) \ Y , identified over U \ Y by ρ | U \ Y . Write ˜ ρ : C m | S ( N Y ⊕ R ) \ Y ∼ = −→ ˜ E | S ( N Y ⊕ R ) \ Y for the obviousSU( m )-framing. Then c ( ˜ E ) is Poincar´e dual to [ Y ] ∈ H ( S ( N Y ⊕ R ); Z ).After stabilizing by C l for l > m + l ) >
7, using Proposition 4.8 on N Y ⊂ S ( N Y ⊕ R ) we obtain an isomorphism of SU( m + l )-bundlesΘ : ˜ E ⊕ C l −→ ˜ φ ∗ ( ˜ E ⊕ C l ) ∼ = ˜ φ ∗ ( ˜ E ) ⊕ C l , compatible outside a compact subset of N Y ⊂ S ( N Y ⊕ R ) with the SU( m + l )-framings induced by ˜ ρ . Thus Theorem 2.15 gives an isomorphismOr( ˜ φ, Θ , ∅ , ∅ ) : Or ˜ E ⊕ C l −→ Or ˜ E ⊕ C l . (4.3)Let ˜ F be the unique flag structure on S ( N Y ⊕ R ) with ˜ F | U = F | U , regarding U as an open subset of both S ( N Y ⊕ R ) and X . Then combining Propositions2.19 and 3.6 and Example 2.20, we find that Or( ˜ φ, Θ , ∅ , ∅ ) in (4.3) is multipli-cation by the sign( ˜ F / ˜ φ ∗ ˜ F )[ Y ] = (cid:0) ˜ F | U / ˜ φ | ∗ U ( ˜ F | U ) (cid:1) [ Y ] = (cid:0) F | U / ( ψ − ◦ ψ ) ∗ ( F | U ) (cid:1) [ Y ] , (4.4)since identifying subsets U of X and S ( N Y ⊕ R ) identifies ˜ φ | U with ψ − ◦ ψ .By functoriality of excision there is a commutative diagramOr E ⊕ C l Or( ψ − ◦ ψ , Θ | U ,ρ ⊕ id C l ,ρ ⊕ id C l ) (cid:15) (cid:15) Or(id U , id E ⊕ C l | U ,ρ ⊕ id C l , ˜ ρ ⊕ id C l ) / / Or ˜ E ⊕ C l Or( ˜ φ, Θ , ∅ , ∅ )=multiplication by (4.4) (cid:15) (cid:15) Or E ⊕ C l Or(id U , id E ⊕ C l | U ,ρ ⊕ id C l , ˜ ρ ⊕ id C l ) / / Or ˜ E ⊕ C l , which implies that Or( ψ − ◦ ψ , Θ | U , ρ ⊕ id C l , ρ ⊕ id C l ) is multiplication by (4.4).Similarly, we have a commutative diagramOr E ⊕ C l Or( ψ − ◦ ψ , Θ | U ,ρ ⊕ id C l ,ρ ⊕ id C l )=multiplication by (4.4) (cid:15) (cid:15) Or( ψ , Ξ ⊕ id C l | U ,ρ ⊕ id C l ,ρ ′ ⊕ id C l ) - - Or E ′ ⊕ C l , Or E ⊕ C l Or( ψ , Ξ ⊕ id C l | U ,ρ ⊕ id C l ,ρ ′ ⊕ id C l ) which implies that Or( ψ i , Ξ i ⊕ id C l | U , ρ ⊕ id C l , ρ ′ ⊕ id C l ) for i = 0 , i = 0 , Eλ E, C l ⊗ ( −⊗ Z o flat ( C l )) (cid:15) (cid:15) Or( ψ i , Ξ i ,ρ,ρ ′ ) / / Or E ′ λ E ′ , C l ⊗ ( −⊗ Z o flat ( C l )) (cid:15) (cid:15) Or E ⊕ C l Or( ψ i , Ξ i ⊕ id C l | U ,ρ ⊕ id C l ,ρ ′ ⊕ id C l ) / / Or E ′ ⊕ C l ,
23o that that Or( ψ i , Ξ i , ρ, ρ ′ ) for i = 0 , o FY,ρ,ι, Ψ ( E ) = (cid:0) F | U /ψ ∗ ( F S | U ′ ) (cid:1) [ Y ] · Or( ψ , Ξ , ρ, ρ ′ ) − ( o flat ( E ′ ))= (cid:0) F | U /ψ ∗ ( F S | U ′ ) (cid:1) [ Y ] · (cid:0) F | U / ( ψ − ◦ ψ ) ∗ ( F | U ) (cid:1) [ Y ] · Or( ψ , Ξ , ρ, ρ ′ ) − ( o flat ( E ′ ))= (cid:0) F | U /ψ ∗ ( F S | U ′ ) (cid:1) [ Y ] · Or( ψ , Ξ , ρ, ρ ′ ) − ( o flat ( E ′ )) = o FY,ρ,ι, Ψ ( E ) , using (4.2) in the first and fourth steps, that Or( ψ i , Ξ i , ρ, ρ ′ ) for i = 0 , F ′ /F in Proposition 3.3(b) in thethird. This completes the proof. o FY,ρ,ι, Ψ ( E ) is independent of ι : Y ֒ → S for fixed Y, ρ
In a similar way to § o FY,ρ,ι, Ψ ( E ) is independent of s, Y, ρ Proposition 4.10. o FY,ρ,ι, Ψ ( E ) is independent of s, Y, ρ .Proof. In Definition 4.7, let s , s : C m − → E be alternative generic choicesfor s , and let Y , ρ , ι , Ψ , . . . and Y , ρ , ι , Ψ , . . . be subsequent choices, so wehave orientations o FY ,ρ ,ι , Ψ ( E ) and o FY ,ρ ,ι , Ψ ( E ) in Or E .Choose a generic morphism ˇ s : C m − × [0 , → E × [0 ,
1] over X × [0 , s | X ×{ i } = s i for i = 0 ,
1, and let Z be the degeneracy locus of ˇ s . Thenas in Definition 4.4, Z ⊂ X × [0 ,
1] is a compact embedded 4-submanifold withboundary ∂Z = ( Y × { } ) ∐ ( Y × { } ).By genericness, Z intersects the hypersurface X ×{ t } in X × [0 ,
1] for t ∈ [0 , x i , t i ) for i = 1 , . . . , k , with 0 1. Also the projection π X | Z : Z → X is an immersion except atfinitely many points (˜ x j , ˜ t j ) for j = 1 , . . . , l , where { t , . . . , t k } ∩ { ˜ t , . . . , ˜ t l } = ∅ .Define Y t = (cid:8) y ∈ X : ( y, t ) ∈ Z (cid:9) for each t ∈ [0 , t ∈ [0 , \{ t , . . . , t k } then X × { t } intersects Z transversely, so Y t is a compact em-bedded 3-submanifold of X , which depends smoothly on t . But when t = t i , Y t i is generally singular at x i , and the topology of Y t changes by a surgery as t crosses t i in [0 , t ∈ [0 , \ { t , . . . , t k } we have an orientation o FY t ,ρ t ,ι t , Ψ t ( E ) from Def-inition 4.7 with s t = ˇ s | X ×{ t } and Y t in place of s and Y , where § § ι t , Ψ t , . . . . Locally in t we can make these additional choices depend smoothly on t . Hence Theorem2.15(ii) implies that for t in each connected component of [0 , \ { t , . . . , t k } this o FY t ,ρ t ,ι t , Ψ t ( E ) depends continuously on t , and hence is constant. Thus, to showthat o FY ,ρ ,ι , Ψ ( E ) = o FY ,ρ ,ι , Ψ ( E ), it suffices to prove that o FY ti − ǫ ,ρ ti − ǫ ,ι ti − ǫ , Ψ ti − ǫ ( E ) = o FY ti + ǫ ,ρ ti + ǫ ,ι ti + ǫ , Ψ ti + ǫ ( E ) (4.5)24or all i = 1 , . . . , k , where ǫ > { t , . . . , t k } ∩{ ˜ t , . . . , ˜ t l } = ∅ , if ǫ is small then [ t i − ǫ, t i + ǫ ] contains no˜ t j for j = 1 , . . . , l , so that π X | ··· : Z ∩ ( X × [ t i − ǫ, t i + ǫ ]) → X is an immersion.As it is injective on Y t i , which is compact, making ǫ smaller we can suppose thisis an embedding, so that W i := π X (cid:0) Z ∩ ( X × [ t i − ǫ, t i + ǫ ]) (cid:1) is an embedded4-submanifold in X with boundary ∂W i = Y t i − ǫ ∐ Y t i + ǫ . As the bordism W i involves only a single surgery at ( x i , t i ), each connected component of W i musthave nonempty boundary.By Theorem 4.6 there exists an embedding : W i ֒ → S . Since X and S are both oriented and spin, the normal bundles of W i in X and in S are (non-canonically) isomorphic. Hence we can choose open tubular neighbourhoods V of W i in X and V ′ of W ′ i = ( W i ) in S and a spin diffeomorphism χ : V → V ′ .Let U t i ± ǫ be tubular neighbourhoods of Y t i ± ǫ in V . By § § o FY ti ± ǫ ,ρ ti ± ǫ ,ι ti ± ǫ , Ψ ti ± ǫ ( E ) using ι t i ± ǫ = χ | Y ti ± ǫ , U t i ± ǫ , U ′ t i ± ǫ = ξ ( U t i ± ǫ ) and ψ t i ± ǫ = χ | U ti ± ǫ . Then we have o FY ti − ǫ ,ρ ti − ǫ ,ι ti − ǫ , Ψ ti − ǫ ( E )= (cid:0) F | U ti − ǫ /ψ ∗ t i − ǫ ( F S | U ′ ti − ǫ ) (cid:1) [ Y ] · Or( ψ t i − ǫ , Ξ t i − ǫ | U ti − ǫ , ρ t i − ǫ , ρ ′ t i − ǫ ) − ( o flat ( E ′ t i − ǫ ))= (cid:0) F | V /χ ∗ ( F S | V ′ ) (cid:1) [ Y ] · Or( χ, Ξ t i − ǫ , ρ t i − ǫ | X \ W i , ρ ′ t i − ǫ | S \ W ′ i ) − ( o flat ( E ′ t i − ǫ ))= (cid:0) F | V /χ ∗ ( F S | V ′ ) (cid:1) [ Y ] · Or( χ, Ξ t i + ǫ , ρ t i + ǫ | X \ W i , ρ ′ t i + ǫ | S \ W ′ i ) − ( o flat ( E ′ t i + ǫ ))= (cid:0) F | U ti + ǫ /ψ ∗ t i + ǫ ( F S | U ′ ti + ǫ ) (cid:1) [ Y ] · Or( ψ t i + ǫ , Ξ t i + ǫ | U ti + ǫ , ρ t i + ǫ , ρ ′ t i + ǫ ) − ( o flat ( E ′ t i + ǫ ))= o FY ti + ǫ ,ρ ti + ǫ ,ι ti + ǫ , Ψ ti + ǫ ( E ) . Here the first and fifth steps come from (4.2). In the second and fourth steps weuse ψ t i ± ǫ = χ | U ti ± ǫ , expanding the open sets U t i ± ǫ , U ′ t i ± ǫ to V, V ′ , and shrinkingthe domains X \ Y t i ± ǫ , S \ Y ′ t i ± ǫ of ρ t i ± ǫ , ρ ′ t i ± ǫ to X \ W i , S \ W ′ i .In the third step, with E, V, χ fixed, we deform the SU( m )-framing ρ t | X \ W i : C m | X \ W i → E | X \ W i defined using s t smoothly over t ∈ [ t i − ǫ, t i + ǫ ], and hencealso smoothly deforming the data E ′ t , Ξ t , ρ ′ t | S \ W ′ i constructed using ρ t | X \ W i .Theorem 2.15(ii) implies that the corresponding family of orientations deformscontinuously in t ∈ [ t i − ǫ, t i + ǫ ], so has the same value at t i ± ǫ . This provesequation (4.5), and the proposition. o F ( E ) are well defined Sections 4.2.1–4.2.4 have shown that o FY,ρ,ι, Ψ ( E ) in Definition 4.7 depends onlyon X, F, E , and not on the additional choices s, Y, ρ, ι, Ψ , U, U ′ , ψ . Thus we cannow define canonical orientations o F ( E ) = o FY,ρ,ι, Ψ ( E ) ∈ Or E for all X, F andSU( m )-bundles E → X , as in the first part of Theorem 1.2.25 .3 (C) Verification of the axioms Axiom (1.5) in Theorem 1.2(a) is obvious. Proposition 4.11. Let E , E → X be SU( m ) - and SU( m ) -bundles. Thenunder (1.2) we have o F ( E ⊕ E ) = o F ( E ) · o F ( E ) , proving Theorem .Taking E = C gives the stabilization axiom (1.6) in Theorem .Proof. In the situation of Definition 4.7, pick generic s k : C m k − → E k for k = 1 , 2, and let Y k , ρ k , ι k , Y ′ k , Ψ k , U k , ψ k , U ′ k , . . . be the subsequent choices. Bygenericity we may assume that Y ∩ Y = ∅ and Y ′ ∩ Y ′ = ∅ , and making thetubular neighbourhoods smaller we can take U ∩ U = ∅ and U ′ ∩ U ′ = ∅ .As in § o F ( E k ) = o FY k ,ρ k ,ι k , Ψ k ( E k ) for k = 1 , 2. Also we maywrite o F ( E ⊕ E ) = o FY,ρ,ι, Ψ ( E ⊕ E ), where Y = Y ∐ Y , ρ = ρ | X \ Y ⊕ ρ | X \ Y , ι = ι ∐ ι , Ψ = Ψ ∐ Ψ , and o FY,ρ,ι, Ψ ( E ⊕ E ) is defined using U = U ∐ U , ψ = ψ ∐ ψ , U ′ = U ′ ∐ U ′ , E ′ = E ′ ⊕ E ′ , and ρ ′ = ρ ′ | S \ Y ′ ⊕ ρ ′ | S \ Y ′ .Proposition 4.2(v) gives o flat ( E ′ ) = o flat ( E ′ ) · o flat ( E ′ ) . By applying Or( ψ, Ψ , ρ, ρ ′ ) to this equation and using compatibility of excisionwith λ and with restriction we find thatOr( ψ, Ψ , ρ, ρ ′ ) (cid:0) o flat ( E ′ ) (cid:1) = Or( ψ , Ψ , ρ , ρ ′ ) (cid:0) o flat ( E ′ ) (cid:1) · Or( ψ , Ψ , ρ , ρ ′ ) (cid:0) o flat ( E ′ ) (cid:1) . The proposition then follows from (4.2) by multiplying this equation by (cid:0) F | U /ψ ∗ ( F S | U ′ ) (cid:1) [ Y ∪ Y ]= (cid:0) F | U / ( ψ ) ∗ ( F S | U ) (cid:1) [ Y ] · (cid:0) F | U / ( ψ ) ∗ ( F S | U ) (cid:1) [ Y ] . Proposition 4.12. The excision axiom (1.7) in Theorem holds.Proof. Work in the set up of Theorem 1.2(c). Suppose that Φ ◦ ρ + | U + \ K + = φ ∗ ρ − | U + \ K + holds for K + ⊂ U + compact. Enlarging K + within U + to ˇ K + which is the closure of an open subset of U + , we can choose a smooth morphism s + : C m − → E + on X + with s + | X + \ ˇ K + = ρ + | C m − | X + \ ˇ K + , such that s + isgeneric in the interior of ˇ K + .As in Definition 4.7, let Y + be the degeneracy locus of s + , and constructan SU( m )-framing ˇ ρ + : C m | X + \ Y + → E + | X + \ Y + from s + | X + \ Y + . This satisfiesˇ ρ + | X + \ ˇ K + = ρ + | X + \ ˇ K + as s + | X + \ ˇ K + = ρ + | C m − | X + \ ˇ K + . Choose an embed-ding ι + : Y + ֒ → S , an isomorphism of normal bundles Ψ + : N Y + → ι + ∗ ( N Y ′ )for Y ′ = ι + ( Y + ), tubular neighbourhoods ˇ U + , U ′ of Y + , Y ′ in X + , S withˇ U + ⊆ U + , and a spin diffeomorphism ψ + : ˇ U + → U ′ with ψ + | Y + = ι + and d ψ + | N Y + = Ψ + . As in Definition 4.7 we get from these a vector bun-dle E ′ → S with SU( m )-structure, isomorphism Ξ + : E + | ˇ U + → ψ + ∗ ( E ′ | U ′ )and SU( m )-framing ρ ′ : C m | S \ Y ′ ∼ = −→ E ′ | S \ Y ′ .26sing the isomorphisms φ : U + → U − and Φ : E + | U + → φ ∗ ( E − | U − ), we cantransport ˇ K + , Y + , ˇ ρ + , ι + , Ψ + , ˇ U + , ψ + , Ξ + to ˇ K − , . . . , Ξ − on X − withˇ K − = φ ( ˇ K + ) , Y − = φ ( Y + ) , ˇ ρ − | X − \ ˇ K − = ρ − | X − \ ˇ K − , ˇ ρ − | U − \ Y − = φ ∗ (ˇ ρ + ) ,ι − = ι + ◦ φ | − Y + , Ψ − = Ψ + ◦ d φ | − N Y + , ˇ U − = φ ( ˇ U + ) , ψ − = ψ + ◦ φ | − U + . (4.6)Note that the data Y ′ , N Y ′ , U ′ , E ′ , ρ ′ on S is the same in both + , − cases.Then as in § o F ± ( E ± ) = o F ± Y ± , ˇ ρ ± ,ι ± , Ψ ± ( E ± ) ∈ Or E ± . (4.7)We now haveOr( φ, Φ , ρ + , ρ − ) (cid:0) o F + ( E + ) (cid:1) = (cid:0) F + | ˇ U + /ψ + ∗ ( F S | U ′ ) (cid:1) [ Y + ] · Or( φ, Φ , ρ + , ρ − ) ◦ Or( ψ + , Ξ + , ˇ ρ + , ρ ′ ) − ( o flat ( E ′ ))= (cid:0) F + | ˇ U + /φ | ∗ ˇ U + ( F − | ˇ U − ) (cid:1) [ Y + ] · (cid:0) F − | ˇ U − /ψ −∗ ( F S | U ′ ) (cid:1) [ Y − ] · Or( ψ − , Ξ − , ˇ ρ − , ρ ′ ) − ( o flat ( E ′ ))= (cid:0) F + | U + /φ ∗ ( F − | U − ) (cid:1) ( α + ) · o F − ( E − ) , using (4.2) and (4.7) in the first step, (4.6) and functoriality of Or( − ) and F ′ /F in the second, and using (4.2) and (4.7) and writing α + = [ Y + ] in H ( U + ; Z )in the third. Since α + is Poincar´e dual to c ( E + | U + , ρ + ) ∈ H ( U + ; Z ) as inDefinition 4.7, this proves (1.7). Proposition 4.13. The families property Theorem holds.Proof. Let E → X × P be an SU( m )-bundle. By compactness of X each p ∈ P has an open neighbourhood P with E | X × P ∼ = E | X ×{ p } × P . By (1.7) we have o F ( E | X ×{ p } ) ∼ = o F ( E | X ×{ p } ) for every p ∈ P under the excision isomorphism,which depends continuously on p .This completes the proof of the first part of Theorem 1.2, on SU( m )-bundles. U( m ) -bundles Finally we extend Theorem 1.2 to U( m )-bundles. Clearly the canonical ori-entations o F ( E ) for U( m )-bundles E → X are well defined. They also satisfyTheorem 1.2(a)–(c) and (ii), since mapping the U( m )-bundle E to the SU( m +1)-bundle ˜ E = E ⊕ Λ m E ∗ commutes with all the operations in (a)–(c) and (ii). References [1] M.F. Atiyah, R. Bott and A. Shapiro, Clifford modules , Topology 3 (1964)suppl. 1, 3–38. 272] M.F. Atiyah, V.K. Patodi and I.M. Singer, Spectral asymmetry and Rie-mannian geometry. I , Math. Proc. Cambridge Philos. Soc. 77 (1975), 43–69.[3] M.F. Atiyah, V.K. Patodi and I.M. Singer, Spectral asymmetry and Rie-mannian geometry. III , Math. Proc. Cambridge Philos. Soc. 79 (1976),71–99.[4] M.F. Atiyah and I.M. Singer, The Index of Elliptic Operators: I , Ann. ofMath. 87 (1968), 484–530.[5] M.F. Atiyah and I.M. Singer, The Index of Elliptic Operators: III , Ann. ofMath. 87 (1968), 546–604.[6] M.F. Atiyah and I.M. Singer, The Index of Elliptic Operators: IV , Ann. ofMath. 92 (1970), 119–138.[7] Y. Cao, J. Gross and D. Joyce, Orientability of moduli spaces of Spin(7) -instantons and coherent sheaves on Calabi–Yau -folds , arXiv:1811.09658,2018.[8] A. Dold, Lectures on algebraic topology , Grundlehren der math. Wiss. 200,Springer-Verlag, Berlin–New York, 1980.[9] S.K. Donaldson, The orientation of Yang–Mills moduli spaces and -manifold topology , J. Diff. Geom. 26 (1987), 397–428.[10] S.K. Donaldson and P.B. Kronheimer, The Geometry of Four-Manifolds ,OUP, 1990.[11] S.K. Donaldson and E. Segal, Gauge Theory in Higher Dimensions, II ,Surveys in Diff. Geom. 16 (2011), 1–41. arXiv:0902.3239.[12] S.K. Donaldson and R.P. Thomas, Gauge Theory in Higher Dimensions ,Chapter 3 in S.A. Huggett et al., editors, The Geometric Universe , OxfordUniversity Press, Oxford, 1998.[13] P. Griffiths and J. Harris, Principles of Algebraic Geometry , Wiley, NewYork, 1978.[14] A. Haefliger, Plongements diff´erentiables de vari´et´es dans vari´et´es , Com-ment. Math. Helv. 36 (1961), 47–82.[15] D. Joyce, Compact manifolds with special holonomy , Oxford UniversityPress, 2000.[16] D. Joyce, An introduction to d-manifolds and derived differential geometry ,pages 230–281 in L. Brambila-Paz et al., editors, Moduli spaces , L.M.S.Lecture Notes 411, Cambridge University Press, 2014. arXiv:1206.4207.2817] D. Joyce, Conjectures on counting associative -folds in G -manifolds ,pages 97–160 in V. Mu˜noz et al., editors, Modern Geometry: A Celebrationof the Work of Simon Donaldson , Proc. Symp. Pure Math. 99, A.M.S.,Providence, RI, 2018. arXiv:1610.09836.[18] D. Joyce, Kuranishi spaces as a -category , to appear in J. Morgan, editor, Virtual Fundamental Cycles in Symplectic Topology , book to be publishedby the A.M.S. in 2019. arXiv:1510.07444.[19] D. Joyce, D-manifolds and d-orbifolds: a theory of derived differential ge-ometry , to be published by OUP, 2019. Preliminary version (2012) availableat http://people.maths.ox.ac.uk/~joyce/dmanifolds.html .[20] D. Joyce, Kuranishi spaces and Symplectic Geometry , multiple volume bookin progress, 2017–2027. Preliminary versions of volumes I, II available at http://people.maths.ox.ac.uk/~joyce/Kuranishi.html .[21] D. Joyce, Y. Tanaka and M. Upmeier, On orientations for gauge-theoreticmoduli spaces , arXiv:1811.01096, 2018.[22] H.B. Lawson and M.-L. Michelsohn, Spin geometry , Princeton Math. Series38, Princeton Univ. Press, Princeton, NJ, 1989.[23] G. Menet, J. Nordstr¨om and H.N. S´a Earp, Construction of G -instantonsvia twisted connected sums , arXiv:1510.03836, 2015.[24] H.N. S´a Earp, G -instantons over asymptotically cylindrical manifolds ,Geom. Topol. 19 (2014), 61–111. arXiv:1101.0880.[25] H.N. S´a Earp and T. Walpuski, G -instantons over twisted connected sums ,Geom. Topol. 19 (2015), 1263–1285. arXiv:1310.7933.[26] R.T. Seeley, Integro-differential operators on vector bundles , Trans. Amer.Math. Soc. 117 (1965), 167–204.[27] M. Upmeier, A categorified excision principle for elliptic symbol families ,arXiv:1901.10818, 2019.[28] C.T.C. Wall, All -manifolds imbed in -space , Bull. Amer. Math. Soc. 71(1965), 564–567.[29] T. Walpuski, G -instantons on generalized Kummer constructions , Geom.Topol. 17 (2013), 2345–2388. arXiv:1109.6609.[30] T. Walpuski, Gauge theory on G -manifolds , PhD Thesis, Imperial CollegeLondon, 2013.[31] T. Walpuski, G -instantons, associative submanifolds and Fueter sections ,Comm. Anal. Geom. 25 (2017), 847–893. arXiv:1205.5350. The Mathematical Institute, Radcliffe Observatory Quarter, WoodstockRoad, Oxford, OX2 6GG, U.K.E-mails: [email protected], [email protected]@maths.ox.ac.uk, [email protected].