Covariant constancy of quantum Steenrod operations
CCovariant constancy of quantum Steenrod operations
Paul Seidel, Nicholas Wilkins Introduction
Quantum Steenrod operations, originally introduced by Fukaya [7], have recently appeared ina variety of contexts: their properties have been explored in [18] (which also contains the firstnontrivial computations); they can be used to study arithmetic aspects of mirror symmetry [14];and in Hamiltonian dynamics, they are relevant for the existence of pseudo-rotations [16, 2, 17].Nevertheless, computing quantum Steenrod operations remains a challenging problem in all butthe simplest cases. Using methods similar to [18], this paper establishes a relation betweenquantum Steenrod operations and the quantum connection. As a consequence, the contributionof rational curves of low degree (very roughly speaking, of degree < p if one is interested inquantum Steenrod operations with F p -coefficients) can be computed using only ordinary Steenrodoperations and Gromov-Witten invariants. This is consonant with other indications that thegeometrically most interesting part of quantum Steenrod operations may come from p -fold coveredcurves. Even though our method does not reach that part, it yields interesting results in manyexamples (some are carried out here, and there are more in [14]). Throughout this paper, M is a closed symplectic manifold which is weakly monotone [8](in [10, Definition 6.4.1], this is called semi-positive). Fix an arbitrary coefficient field F . Theassociated Novikov ring Λ is the ring of series(1.1) γ = (cid:80) A c A q A , where the exponents are A ∈ H sphere ( M ; Z ) = im( π ( M ) → H ( M ; Z )) such that either A = 0or (cid:82) A ω M >
0; and among those A such that (cid:82) A ω M is bounded by a given constant, only finitelymany c A may be nonzero. We think of this as a graded ring, where | q A | = 2 c ( A ) (the notationbeing that c ( A ) is the pairing between c ( M ) and A ). Write I max ⊂ Λ for the ideal generatedby q A for nonzero A , so that Λ /I max = F .For each a ∈ H ( M ; Z ) there is an F -linear differentiation operation ∂ a : Λ → Λ,(1.2) ∂ a q A = ( a · A ) q A . Write I diff ⊂ I max for the ideal generated by q A , where A (cid:54) = 0 lies in the kernel of the map H sphere ( M ; Z ) (cid:44) → H ( M ; Z ) (cid:16) Hom ( H ( M ; Z ) , F ). In other words, the generators are preciselythose nontrivial monomials whose derivatives (1.2) are zero. (If F is of characteristic zero and H ( X ; Z ) sphere is torsion-free, then I diff = 0; but that’s not the case we’ll be interested in.) Remark 1.1.
Clearly, ∂ a only depends on a ⊗ ∈ H ( M ; Z ) ⊗ F . One could define suchoperations for all elements in H ( M ; F ) , and prove a version of our results in that context. We a r X i v : . [ m a t h . S G ] F e b PAUL SEIDEL, NICHOLAS WILKINS have refrained from doing so, since it adds a technical wrinkle (having to represent classes in H ( M ; F ) geometrically) without giving any striking additional applications. We will exclusively consider genus zero Gromov-Witten invariants. The three-pointedGromov-Witten invariant in a class A ∈ H sphere ( M ; Z ) can be written as a bilinear operation(1.3) ∗ A : H ∗ ( M ; F ) ⊗ −→ H ∗− c ( A ) ( M ; F ) , (cid:90) M ( c ∗ A c ) c = (cid:104) c , c , c (cid:105) A . One extends this to H ∗ ( M ; Λ), and then packages all the ∗ A into the small quantum product(1.4) γ ∗ γ = (cid:88) A ( γ ∗ A γ ) q A . Let t be another formal variable, of degree 2. The quantum connection on H ∗ ( M ; Λ)[[ t ]] consistsof the operations(1.5) ∇ a γ = t∂ a γ + a ∗ γ, where ∗ has been extended t -linearly. By the divisor axiom in Gromov-Witten theory, we havethat for any a , a ∈ H ( M ; Z ) and c , c ∈ H ∗ ( M ; F ),(1.6) ( a · A ) (cid:90) M ( a ∗ A c ) c = (cid:104) a , a , c , c (cid:105) A = ( a · A ) (cid:90) M ( a ∗ A c ) c . This implies that the operations (1.5) for different a commute: the connection is flat.We will consider endomorphisms Σ of H ∗ ( M ; Λ)[[ t ]] which are Λ[[ t ]]-linear and covariantly con-stant, which means that they satisfy(1.7) ∇ a Σ − Σ ∇ a = 0 . This is a system of linear first order differential equations. By looking at the equations for each q A coefficient of Σ, one sees that: Lemma 1.2.
For covariantly constant endomorphisms, the constant term determines the be-haviour modulo I diff . More formally, (1.8) Σ ∈ End ( H ∗ ( M ; F )) ⊗ I max [[ t ]] = ⇒ Σ ∈ End ( H ∗ ( M ; F )) ⊗ I diff [[ t ]] . From now on, we restrict to coefficient fields F = F p , for a prime p . Our arguments involve( Z /p )-equivariant cohomology with F p -coefficients. For a point, that is(1.9) H ∗ Z /p ( point ; F p ) = H ∗ ( B Z /p ; F p ) = F p [[ t, θ ]] , | t | = 2 , | θ | = 1 . The notation requires some explanation. For p = 2, we have θ = t , so F [[ t, θ ]] is actually a ringof power series in a single variable θ . For p >
2, we have tθ = θt and θ = 0, so that F p [[ t, θ ]] isa ring of power series in two supercommuting variables.For any A ∈ H sphere ( M ; Z ) and any class b ∈ H ∗ ( M ; F p ), one can use ( Z /p )-equivariant Gromov-Witten theory to define an operation(1.10) Q Σ A,b : H ∗ ( M ; F p ) −→ ( H ∗ ( M ; F p )[[ t, θ ]]) ∗ + p | b |− c ( A ) . uantum Steenrod 3 For the trivial class A = 0, this is a form of the classical Steenrod operation St , more precisely(1.11) Q Σ b, ( c ) = St ( b ) c. Remark 1.3.
Our notational and sign conventions follow [14] (except that we suppress the prime p ), which differ from the classical conventions for Steenrod operations. In particular, for p > , (1.12) St ( b ) = ( − | b | ( | b |− p − (cid:0) p − ! (cid:1) | b | t p − | b | b + · · · , where · · · is the part involving cohomology classes of degree > | b | . For | b | even, this simplifies to (1.13) St ( b ) = ( − | b | t p − | b | b + · · · At the other extreme, setting t = θ = 0 in St ( b ) still yields the p -fold (cup) power b p . The Cartanrelation says that (1.14) St (˜ b ) St ( b ) = ( − | b | | ˜ b | p ( p − St (˜ bb ) . Note that many coefficients of St ( b ) vanish, because this operation comes from the cohomologyof the symmetric group. Concretely, if | b | is even, all the potentially nonzero terms in St ( b ) areof the form t k ( p − or t k ( p − − θ ; and if | b | is odd, of the form t ( k +1 / p − or t ( k +1 / p − − θ .That is no longer true for quantum operations. As usual, one adds up (1.10) over all A with weights q A . The outcome is denoted by(1.15) Q Σ b : H ∗ ( M ; F p ) −→ ( H ∗ ( M ; Λ)[[ t, θ ]]) ∗ + p | b | . The non-equivariant ( t = θ = 0) part is the p -fold quantum product with b :(1.16) Q Σ b ( c ) = p (cid:122) (cid:125)(cid:124) (cid:123) b ∗ · · · ∗ b ∗ c + ( terms involving t, θ ) . The case b = 1 is trivial:(1.17) Q Σ = id . The relation with the more standard formulation of the quantum Steenrod operation is that(1.18)
QSt ( b ) = Q Σ b (1) . It is convenient to formally extend (1.15). First, turn it into an endomorphism of H ∗ ( M ; Λ)[[ t, θ ]],linearly in the variables q A and ( t, θ ) (with appropriate Koszul signs). Next, extend the b -variableto β ∈ H ∗ ( M ; Λ), by setting(1.19) Q Σ β = (cid:80) A q pA Q Σ b A for β = (cid:80) A b A q A .Then, the composition of these operations is described by(1.20) Q Σ ˜ b ◦ Q Σ b = ( − | b | | ˜ b | p ( p − Q Σ ˜ b ∗ b . Note that for b = 1, (1.16) implies that Q Σ is an automorphism of H ∗ ( M ; Λ)[[ t, θ ]], and (1.20)that it is idempotent. Hence, it must be the identity, so those two properties imply (1.17). The quantum connection can be extended to H ∗ ( M ; Λ)[[ t, θ ]] by making it θ -linear. Ourmain result is: PAUL SEIDEL, NICHOLAS WILKINS
Theorem 1.4.
For any b ∈ H ∗ ( M ; F p ) , the operation Q Σ b is a covariantly constant endomor-phism (of degree p | b | ), meaning that it satisfies (1.7) . Lemma 1.2 still applies (the presence of the additional θ -variable makes no difference). Hence,the classical part (1.11), together with the quantum connection, determine Q Σ b modulo I diff . Remark 1.5.
Covariant constancy also means that Q Σ b is related to the fundamental solutionof the quantum differential equation (see e.g. [11] ). To explain this, let’s temporarily switchcoefficients to Q , and write ˜Λ for the associated Novikov ring. The fundamental solution is atrivialization of the quantum connection, (1.21) ∇ ˜Ψ = 0 , whose constant (in the q variables) term is the identity endomorphism. ˜Ψ is multivalued (has log( q A ) terms), and is also a series in t − . It is uniquely determined by those conditions, andone can write down an explicit formula in terms of Gromov-Witten invariants with gravitationaldescendants. Given β ∈ H ∗ ( M ; Z ) , write (1.22) ˜Ξ β ( γ ) = ˜Ψ( β ˜Ψ − ( γ )) . By construction, this is a covariantly constant endomorphism, whose constant term is cup-productwith β . It is single-valued; more precisely, (1.23) ˜Ξ β ∈ End ( H ∗ ( M ; ˜Λ))[[ t − ]] . For simplicity, suppose that H ∗ ( M ; Z ) is torsion-free. One can look at the denominators in ˜Ξ β ,order by order in the covariant constancy equation. The upshot is that factors of /p appear forthe first time in terms q A , A ∈ pH sphere ( M ; Z ) . As a consequence, ˜Ξ β has a well-defined partialreduction mod p , which we denote by (1.24) Ξ β ∈ End ( H ∗ ( M ; Λ /I diff ))[[ t − ]] , and which only depends on β ∈ H ∗ ( M ; F p ) . Let’s extend (1.24) linearly to β ∈ H ∗ ( M ; F p )[ t, θ ] ,in which case Ξ β can have both positive and negative powers of t . The case we are interested inis β = St ( b ) . Because of the uniqueness property from Lemma 1.2, we then have (1.25) Ξ St ( b ) = Q Σ b modulo I diff . Example 1.6.
Consider M = S , with the standard basis { , h } of cohomology. Take p > (thecase p = 2 is straightforward, but requires slightly different notation). Using Theorem 1.4, onecan compute that Q Σ h = − t p − Σ , where (1.26) Σ = (cid:18) σ σ σ σ (cid:19) , σ = − (cid:80) ( p − / k =1 (2 k − k !) ( k − q k t − k ,σ = − (cid:80) ( p +1) / k =2 (2 k − k − k − k ! q k t − k ,σ = (cid:80) ( p − / k =0 (2 k )!( k !) q k t − k ,σ = − σ . In particular, (1.27)
QSt ( h ) = − t p − σ − t p − σ h. uantum Steenrod 5 Note that after multiplying with t p − , all the powers of t in (1.26) become nonnegative. Moreprecisely, (1.28) − t p − Σ = (cid:18) q ( p +1) / q ( p − / (cid:19) + ( terms involving t ) , in agreement with (1.16) and the fact that the p -th quantum power of h is q ( p − / h . Example 1.7.
Let M be a cubic surface in C P (this is C P blown up at points, with itsmonotone symplectic form). Take p = 2 , and let h ∈ H ( M ; F ) be the Poincar´e dual of a point.Then (1.29) QSt ( h ) = St ( h ) = t h. This is interesting because of its implications for Hamiltonian dymanics: by the criterion from [2, 17] , it means that M cannot admit a quasi-dilation. We refer to Section 6a for furtherdiscussion. The proof of Theorem 1.4 goes roughly as follows. We introduce another operation, dependingon a ∈ H ( M ; Z ) as well as b ∈ H ∗ ( M ; F p ),(1.30) Q Π a,b : H ∗ ( M ; F p ) −→ ( H ∗ ( M ; Λ)[[ t, θ ]]) ∗ + | a |− p | b | . Geometrically, this is obtained from (1.15) by equipping the underlying Riemann surface with anadditional marked point, which can move around (we insert an incidence constraint dual to a atthat point). A localisation-type argument yields(1.31) t Q Π a,b ( c ) = Q Σ b ( a ∗ c ) − a ∗ Q Σ b ( c ) . We also have an analogue of the divisor equation:(1.32) Q Π a,b ( c ) = ∂ a Q Σ b ( c ) . Theorem 1.4 follows immediately by combining (1.31) and (1.32).
Remark 1.8.
Even though we have no immediate need for it here, it is worth while noting that Q Π a,b can be defined more generally for a ∈ H ∗ ( M ; F p ) , and still satisfies (1.31) , with suitableadded Koszul signs (see Remark 4.11). Remark 1.9.
The argument above is closely related to the Cartan relation for quantum Steenrodsquares. Namely, let’s set a = QSt ( b ) , b = b , c = 1 in (1.31) . Then, using (1.20) one sees that (1.33) t Q Π QSt ( b ) ,b (1) = ( − | b | | b | Q Σ b ( QSt ( b )) − QSt ( b ) ∗ QSt ( b )= ( − | b | | b | ( p ( p − / Q Σ b ∗ b (1) − QSt ( b ) ∗ QSt ( b )= ( − | b | | b | p ( p − / QSt ( b ∗ b ) − QSt ( b ) ∗ QSt ( b ) . In view of that, it is not surprising that in applications, computations based on covariant constancyclosely resemble those from [18] , where the Cartan relation was the main tool.Acknowledgments.
Both authors were partially supported by a Simons Investigator award fromthe Simons Foundation. Additional support for the first author was provided by the SimonsCollaboration for Homological Mirror Symmetry, and by NSF grant DMS-1904997. The secondauthor was additionally supported by a Heilbronn Research Fellowship.
PAUL SEIDEL, NICHOLAS WILKINS ∆ τ ∆ ∆ ∆ Figure 2.1.
The first cells from (2.3), (2.4).2.
A bit of equivariant (co)homology
This section introduces some of the algebra and topology underlying our construction. Eventhough this is elementary, it is helpful as a guiding model for the later discussion.
Write(2.1) S ∞ = { w = ( w , w , w , . . . ) ∈ C ∞ : w k = 0 for k (cid:29) , (cid:107) w (cid:107) = | w | + | w | + · · · = 1 } . Fix a prime p , and consider the Z /p -action on S ∞ generated by(2.2) τ ( w , w , . . . ) = ( ζw , ζw , . . . ) , ζ = e πi/p . Take the following subsets:∆ k = { w ∈ S ∞ : w k ≥ , w k +1 = w k +2 = · · · = 0 } , (2.3) ∆ k +1 = { w ∈ S ∞ : e − iθ w k ≥ θ ∈ [0 , π/p ] , w k +1 = w k +2 = · · · = 0 } . (2.4)Each of them is homeomorphic to a disc, of the dimension indicated by the subscript. Moreprecisely, ∆ k is a submanifold with boundary,(2.5) ∂ ∆ k = { w k = w k +1 = · · · = 0 } ∼ = S k − , and ∆ k +1 a submanifold with two boundary faces, whose intersection forms a corner stratum,(2.6) ∂ ∆ k +1 = { w k ≥ , w k +1 = w k +2 = · · · = 0 } ∪ { e − πi/p w k ≥ , w k +1 = w k +2 = · · · = 0 } . The subsets (2.3), (2.4) and their images under the Z /p -action form an equivariant (and regular)cell decomposition of S ∞ . The tangent space of ∆ k at the point where w k = 1 (and where allthe other coordinates are therefore zero) can be identified with C k by projecting to the first k coordinates; we use the resulting orientation. The tangent space of ∆ k +1 at the same point canbe similarly identified with C k × i R ; we use the orientation coming from the complex orientationof C k , followed by the positive vertical orientation of i R . For those orientations, the differentialin the cellular chain complex is ∂ ∆ k = ∆ k − + τ ∆ k − + · · · + τ p − ∆ k − , (2.7) ∂ ∆ k +1 = τ ∆ k − ∆ k . (2.8)Here and below, the convention is to ignore terms with negative subscripts.We adopt the quotient S ∞ / ( Z /p ) as our model for the classifying space B ( Z /p ). If we use F p -coefficients, the ∆ i become cycles on the quotient, and their homology classes form a basis for uantum Steenrod 7 H eq ∗ ( point ; F p ) = H ∗ ( S ∞ / ( Z /p ); F p ). (Moreover, from (2.7) one sees that the Bockstein sends∆ k to ∆ k − .) Consider the diagonal embedding δ on S ∞ / ( Z /p ), and the induced map(2.9) δ ∗ : H ∗ ( S ∞ / ( Z /p ); F p ) −→ ( H ∗ ( S ∞ / ( Z /p ); F p )) ⊗ . Lemma 2.1.
In homology with F p -coefficients, (2.10) δ ∗ ∆ i = (cid:88) i + i = i ∆ i ⊗ ∆ i if i is odd or p = 2 , (cid:88) i + i = ii k even ∆ i ⊗ ∆ i if i is even and p > .Proof. For p = 2, this is clear: from the relation between diagonal map and cup product, and thering structure on the cohomology of R P ∞ = S ∞ / ( Z / δ ∗ ∆ i must have nonzerocomponents in all groups H i ⊗ H i , and each of those is a copy of F .For p >
2, the same argument shows that exactly the terms in (2.10) must occur, but possiblywith some nonzero F p -coefficients, which have to be determined by looking a little more carefully.Choose generators θ ∈ H ( S ∞ / ( Z /p ); F p ) and t ∈ H ( S ∞ / ( Z /p ); F p ) so that(2.11) (cid:104) θ, ∆ (cid:105) = 1 , (cid:104) t, ∆ (cid:105) = − . Because ∆ was defined using the complex orientation, this means that t is the pullback ofthe (mod p ) Chern class of the tautological line bundle S ∞ → C P ∞ under the quotient map S ∞ / ( Z /p ) → S ∞ /S = C P ∞ . Looking at the orientations of the higher-dimensional cells yields(2.12) (cid:104) t k θ, ∆ k +1 (cid:105) = (cid:104) t k , ∆ k (cid:105) = ( − k . For k = k + k , we have (cid:104) t k ⊗ t k , δ ∗ ∆ k (cid:105) = (cid:104) δ ∗ ( t k ⊗ t k ) , ∆ k (cid:105) = (cid:104) t k , ∆ k (cid:105) , (2.13) (cid:104) t k θ ⊗ t k , δ ∗ ∆ k +1 (cid:105) = (cid:104) δ ∗ ( t k θ ⊗ t k ) , ∆ k +1 (cid:105) = (cid:104) t k θ, ∆ k +1 (cid:105) , (2.14)and that implies that the coefficients in (2.10) are all 1, as desired. (cid:3) What does this mean on the cochain level? For each k , take a smooth triangulation of S k − / ( Z /p ).Pull that back (taking preimages of the simplices) to a triangulation of ∂ ∆ k , and then extendthat to a triangulation of ∆ k . The outcome is an explicit smooth simplicial chain in S ∞ / ( Z /p ),denoted by ˜∆ k , which becomes a simplicial cycle when the coefficients are reduced modulo p ,and which represents the homology class of ∆ k in H ∗ ( S ∞ / ( Z /p ); F p ). A version of the sameprocess produces corresponding simplicial chains ˜∆ k − . With that in mind, let’s look at therelations underlying (2.10):(2.15) δ ˜∆ i ∼ (cid:88) i + i = i ˜∆ i × ˜∆ i if i is odd or p = 2, (cid:88) i + i = ii k even ˜∆ i × ˜∆ i if i is even and p > PAUL SEIDEL, NICHOLAS WILKINS L B Q P σL Figure 2.2.
The cells from (2.16)–(2.18).On the right hand side, one decomposes the products into simplices. After that, the relationmeans that there is a simplicial chain whose boundary (mod p ) equals the difference between thetwo sides. That chain can again be chosen to be smooth. One could in principle try to spell allof this out using explicit chains, but that is not necessary for our purpose. Consider the two-sphere S = ¯ C = C ∪ {∞} , again with a Z /p -action σ ( v ) = ζv , and thesubsets P = { v = 0 } , Q = { v = ∞} , (2.16) L = { v ≥ } ∪ { v = ∞} , (2.17) B = { e − iθ v ≥ θ ∈ [0 , π/p ] } ∪ { v = ∞} . (2.18)We use the real orientation of L , and the complex orientation of B . Let’s denote the associatedcellular chain complex simply by C ∗ ( S ). Its differential is ∂P = ∂Q = 0 , (2.19) ∂L = Q − P , (2.20) ∂B = L − σL . (2.21)Now look at S ∞ × Z /p S , which means identifying(2.22) ( w, σv ) ∼ ( τ w, v ) . This inherits a cell decomposition. The associated differential, which we denote by ∂ eq , is ∂ eq (∆ k × P ) = 0 , ∂ eq (∆ k +1 × P ) = 0 , (2.23) ∂ eq (∆ k × Q ) = 0 , ∂ eq (∆ k +1 × Q ) = 0 , (2.24) ∂ eq (∆ k × σ j L ) = ∆ k × ( Q − P ) + ∆ k − × ( L + σL + · · · + σ p − L ) , (2.25) ∂ eq (∆ k +1 × σ j L ) = − ∆ k +1 × ( Q − P ) + ∆ k × ( σ j +1 L − σ j L ) , (2.26) ∂ eq (∆ k × σ j B ) = − ∆ k × ( σ j +1 L − σ j L ) + ∆ k − × ( B + · · · + σ p − B ) , (2.27) ∂ eq (∆ k +1 × σ j B ) = ∆ k +1 × ( σ j +1 L − σ j L ) + ∆ k × ( σ j +1 B − σ j B ) . (2.28) uantum Steenrod 9 Lemma 2.2.
Take coefficients in F p . In the cellular complex of S ∞ × Z /p S , the following homologyrelationships hold: ∆ k × ( Q − P ) ∼ ∆ k − × ( B + σB + · · · + σ p − B ) , (2.29) ∆ k +1 × ( Q − P ) ∼ ∆ k − × ( B + σB + · · · + σ p − B ) . (2.30) Proof. (2.29) is obtained by subtracting (2.25) from the following, which comes from (2.28):(2.31) ∂ eq (cid:0) ∆ k +1 × ( σB + 2 σ B + · · · + ( p − σ p − B ) (cid:1) = − ∆ k +1 × ( L + · · · + σ p − L ) − ∆ k × ( B + · · · + σ p − B ) . The second relation (2.30) is a combination of (2.26), (2.27). (cid:3)
To fit this into the general framework of equivariant homology, note that as an application of thelocalisation theorem, the map induced by inclusion of the fixed point set,(2.32) H eq ∗ ( point ; F p ) ⊗ P ⊕ H eq ∗ ( point ; F p ) ⊗ Q −→ H eq ∗ ( S ; F p ) = H ∗ ( S ∞ × Z /p S ; F p )must be an isomorphism in sufficiently high degrees. Using the computations above, one can seehow that works out concretely: (2.32) is surjective, and it fails to be injective only in degrees 0and 1, where the kernel is generated by ∆ ⊗ ( Q − P ) and ∆ ⊗ ( Q − P ), respectively.More generally, take any (homologically graded) chain complex, carrying a ( Z /p )-action. Itsequivariant homology is defined by taking the tensor product with the previously consideredcellular complex of S ∞ , and then passing to coinvariants for the combined action in the samesense as in (2.22). The resulting equivariant differential is ∂ eq (∆ k ⊗ ξ ) = ∆ k − ⊗ ( ξ + σξ + · · · + σ p − ξ ) + ∆ k ⊗ ∂ξ, (2.33) ∂ eq (∆ k +1 ⊗ ξ ) = − ∆ k +1 ⊗ ∂ξ + ∆ k ⊗ ( σξ − ξ ) . (2.34)Here, ξ is an element of the original chain complex, and σ is the automorphism which generatesits ( Z /p )-action. These formulae generalize the ones we’ve previously written down for C ∗ ( S ). Dually to our previous construction, one can start with a cohomologically graded complex C with a ( Z /p )-action, and define an equivariant complex(2.35) C eq = C [[ t, θ ]]where the formal variables are as in (1.9), with differential d eq ( xt k ) = dx t k + ( − | x | ( σx − x ) t k θ, (2.36) d eq ( xt k θ ) = dx t k θ + ( − | x | ( x + σx + · · · + σ p − x ) t k +1 . (2.37)Write H ∗ eq ( C ) = H ∗ ( C eq ) for the resulting cohomology. Lemma 2.3. On C eq , the operations t and σt are homotopic.Proof. The desired homotopy is h ( xt k ) = 0, h ( xt k θ ) = ( − | x | xt k +1 . (cid:3) From now on, we work with F p -coefficients. In that case, the equivariant complex (2.35) car-ries a degree 1 endomorphism ˜ θ , which one can informally think of as a corrected version ofmultiplication with θ (acting on the left):˜ θ ( xt k ) = ( − | x | xt k θ, (2.38) ˜ θ ( xt k θ ) = ( − | x | ( σx + 2 σ x + · · · + ( p − σ p − x ) t k +1 . (2.39)The second part (2.39) contains the kind of expression we’ve seen previously in (2.31). It ishelpful to keep in mind that modulo p , id + σ + σ + · · · + σ p − = ( σ − id ) p − = σ ( σ − id ) p − = · · · , (2.40) σ + 2 σ + · · · + ( p − σ p − = − σ ( σ − id ) p − . (2.41) Lemma 2.4.
Up to homotopy, ˜ θ is multiplication by t if p = 2 , and for p > .Proof. In terms of (2.41), ˜ θ is the action of − σ ( σ − id ) p − t on the equivariant complex. But theaction of ( σ − id ) t is nullhomotopic by Lemma 2.3, and that implies the desired statement. (cid:3) A classical application of equivariant cohomology (basic to the definition of Steenrod operations)is to start with a general cochain complex C (without any ( Z /p )-action), and consider its p -foldtensor product C ⊗ p with the action that cyclically permutes the tensor factors. The equivariantcomplex ( C p ) eq is a homotopy invariant of C . We recall the following: Lemma 2.5.
Taking a cocycle x ∈ C to x ⊗ p ∈ ( C ⊗ p ) eq yields a map (2.42) H ∗ ( C ) −→ H p ∗ eq ( C ⊗ p ) , which becomes additive after multiplying by t .Proof. Since x ⊗ p is a ( Z /p )-invariant cocycle in C ⊗ p (note that the Koszul signs here are alwaystrivial), it is also a d eq -cocycle.The next step is to show that if we have two cohomologous cocycles, x − x = dz , then x ⊗ p and x ⊗ p are cohomologous in ( C ⊗ p ) eq . It is enough to consider the case where C is three-dimensional,with basis ( x , x , z ); the general case then follows by mapping this C into any desired complex.Take a one-dimensional complex D with a single generator y , and the map C → D which takesboth x k to y (and maps z to zero). This is clearly a quasi-isomorphism, and therefore induces aquasi-isomorphism ( C ⊗ p ) eq → ( D ⊗ p ) eq . Under that quasi-isomorphism, both x ⊗ p and x ⊗ p go to y ⊗ p . Therefore, they must be cohomologous in ( C ⊗ p ) eq .The additivity statement can be proved by an explicit formula: if we take(2.43) ( x + x ) ⊗ p − x ⊗ p − x ⊗ p and expand it out, we get 2 p − Z /p )-orbits. Take one repre-sentative for each orbit, add them up, and multiply the outcome by θ . This yields a cochain in( C ⊗ p ) eq whose boundary is t times (2.43), up to sign. (cid:3) uantum Steenrod 11 Finally, we return to the example of S . Take the cellular chain complex and reverse its grading,to make it cohomological. Then, on C −∗ ( S ) eq we have d eq ( P t k ) = 0 , d eq ( P t k θ ) = 0 , (2.44) d eq ( P t k ) = 0 , d eq ( P t k θ ) = 0 , (2.45) d eq ( σ j L t k ) = ( Q − P ) t k − ( σ j +1 L − σ j L ) t k θ, (2.46) d eq ( σ j L t k θ ) = ( Q − P ) t k θ − ( L + · · · + σ p − L ) t k +1 , (2.47) d eq ( σ j B t k ) = − ( σ j +1 L − σ j L ) t k + ( σ j +1 B − σ j B ) t k θ, (2.48) d eq ( σ j B t k θ ) = − ( σ j +1 L − σ j L ) t k θ + ( B + · · · + σ p − B ) t k +1 . (2.49)With F p -coefficients, we have the following analogue of Lemma 2.2, proved in the same way: Lemma 2.6.
The following cohomology relations hold in C −∗ ( S ) eq : ( P − Q ) t k ∼ ( B + σB + · · · + σ p − B ) t k +1 , (2.50) ( P − Q ) t k θ ∼ ( B + σB + · · · + σ p − B ) t k +1 θ. (2.51) 3. Basic moduli spaces
This section introduces the relevant moduli spaces of pseudo-holomorphic curves, in their mostbasic form. This means that we look at a version of the small quantum product, and of one ofits properties, the divisor equation. Like the previous section, this should be considered as a toymodel which introduces some ideas that will recur in more complicated form later on.
Let M n be a weakly monotone closed symplectic manifold. Choose a Morse function f andmetric g , so that the associated gradient flow is Morse-Smale. Our terminology for stable andunstable manifolds is that dim( W s ( x )) = | x | is the Morse index, whereas dim( W u ( x )) = 2 n − | x | . Assumption 3.1.
We fix some compatible almost complex structure J with the following prop-erties (generically satisfied). (i) Consider simple J -holomorphic chains of length l ≥ . Such a chain consists of simple(non-multiply-covered) J -holomorphic maps, no two of which are reparametrizations ofeach other, (3.1) u , . . . , u l : C P = C ∪ {∞} −→ M,u k ( ∞ ) = u k +1 (0) for k = 1 , . . . , l − .Then, the moduli space of such simple chains is regular. (ii) On the space of simple J -holomorphic chains, the evaluation map ( u , . . . , u l ) (cid:55)→ u (0) istransverse to the stable and unstable manifolds of our Morse function.For l = 1 , (i) is just regularity of simple J -holomorphic spheres (which, because of the weakmonotonicity condition, implies the absence of spheres with negative Chern number). Our main moduli space uses a specific ( p +2)-marked sphere as the domain. We introduce specificnotation for it: taking ζ / = e πi/p ,(3.2) C = C P ,z C, = 0 , z C, = ζ / , z C, = ζ / , . . . , z C,p = ζ (2 p − / = ζ − / , z C, ∞ = ∞ . An inhomogeneous term is a J -complex anti-linear vector bundle map ν C : T C → T M , whereboth bundles involved have been pulled back to C × M . The associated inhomogeneous Cauchy-Riemann equation is(3.3) u : C −→ M, ( ¯ ∂ J u ) z = ν C,z,u ( z )) . Given critical points x , . . . , x p , x ∞ of f , we consider solutions of (3.3) with incidence conditionsat the (un)stable manifolds:(3.4) u ( z C, ) ∈ W u ( x ) , . . . , u ( z C,p ) ∈ W u ( x p ) , u ( z C, ∞ ) ∈ W s ( x ∞ ) . It is maybe better to think of this as having gradient half-flowlines(3.5) y , . . . , y p : ( −∞ , −→ M,y (cid:48) k = ∇ f ( y k ) ,y k (0) = u ( z C,k ) , lim s →−∞ y k ( s ) = x k and y ∞ : [0 , ∞ ) −→ M,y (cid:48)∞ = ∇ f ( y ∞ ) ,y ∞ (0) = u ( z C, ∞ ) , lim s →∞ y ∞ ( s ) = x ∞ . Assumption 3.2.
We impose the following requirements (generically satisfied) on our choice ofinhomogeneous term: (i)
The moduli space of solutions of (3.3) , (3.4) is regular. (ii) Take an element in the same space, with a simple J -holomorphic bubble attached at anarbitrary point. This means that we have a pair ( u, u ) with u as in (3.3) , (3.4) , a point z ∈ C , and a simple J -holomorphic u : C P → M with u ( z ) = u (0) . We want thismoduli space to be regular as well. (iii) Consider solutions with a simple holomorphic chain attached at each of a subset of the ( p + 2) marked points, and incidence constraints transferred accordingly. For simplicity,let’s spell out what this means only in the case of a single chain, attached at z C, ∞ . Inthat case, we have a solution of (3.3) , and a simple holomorphic chain ( u , . . . , u l ) , withthe conditions (3.6) u ( z C, ) ∈ W u ( x ) , . . . , u ( z C,p ) ∈ W u ( x p ) ,u ( z C, ∞ ) = u (0) , u l ( ∞ ) ∈ W s ( x ∞ ) . We require that the resulting moduli space should be regular. In the general case wherethere are several marked points with a chain attached to each, we transfer the adjacencycondition involving (un)stable manifolds to the end of the respective chain.
A few comments may be appropriate. In (ii), the bubble may be attached at one of the markedpoints. Let’s say that this point is z C, ∞ , in which case we have(3.7) u ( z C, ∞ ) = u (0) ∈ W s ( x ∞ ) . uantum Steenrod 13 Assumption 3.1(ii), for l = 1, says that the subspace of maps u satisfying u (0) ∈ W s ( x ∞ ) isregular. What we want to achieve is that the evaluation map on that subspace is transverse to u (cid:55)→ u ( z C, ∞ ). This is clearly satisfied for generic ν C . In the same way, genericity of (iii) dependson Assumption 3.1(ii), but this time for arbitrary l .Given A ∈ H ( M ; Z ), let M A ( C, x , . . . , x p , x ∞ ) be the space of solutions of (3.3), (3.4) such that u represents A . Given our regularity requirement, this is a manifold of dimension(3.8) dim M A ( C, x , . . . , x p , x ∞ ) = 2 c ( A ) + | x ∞ | − | x | − · · · − | x p | . We denote by ¯ M A ( C, x , . . . , x p , x ∞ ) the standard compactification. On the pseudo-holomorphicmap side, this involves the stable map compactification, and on the Morse-theoretic side oneallows the flow lines to break. Lemma 3.3. (i) If the dimension (3.8) is , we have (3.9) M A ( C, x , . . . , x p , x ∞ ) = ¯ M A ( C, x , . . . , x p , x ∞ ) , which means that the moduli space is a finite set.(ii) If the dimension is , the compactification is a manifold with boundary, with the interiorbeing the space M A ( · · · ) ; the boundary points involve no bubbling, and only once-broken gradientflow lines.Sketch of proof. This is standard [10, Section 6.7], but we still want to give a quick reminder ofthe arguments involved. Points in ¯ M A ( · · · ) are represented by maps whose domain is a nodalRiemann surface. That Riemann surface has a distinguished principal component, identifiedwith C , which carries a map that satisfies (3.3); while the other components carry J -holomorphicmaps. The domain also has ( p + 2) (smooth) marked points. Those points do not have to lie onthe principal component, but if they don’t, they lie on a bubble tree that is attached to principalcomponent at the relevant point z C,k .The first case is that, after collapsing ghost components (non-principal components carrying aconstant map), all the ( p + 2) marked points come to lie on the principal component. This meansthat the map on the principal component still satisfies an incidence condition as in (3.4), exceptfor the possible presence of broken Morse flow lines. If there are no non-principal components, thebroken flow lines are the only way in which this differs from a point in M A ( · · · ) itself, and thatsituation is unproblematic. On the other hand, assuming there are non-principal components,pick one which carries a non-constant J -holomorphic map, and whose image intersects that of theprincipal component. After replacing that by its underlying simple map, we are in the situationfrom Assumption 3.1(ii), except again for broken Morse flow lines; this outcome is of codimension ≥
2, hence cannot happen due to our regularity requirements.The other undesirable possibility in the stable map compactification is that one or several markedpoints lie on non-principal components which either carry a nontrivial J -holomorphic map, orare separated from the principal component by other non-principal components that carry sucha map. We collapse ghost components; then replace multiply-covered maps by the underlyingsimple ones; and finally remove redundant components, which are all of those we can get rid of ∞ those maps have the same imageconstant map (ghost)keep this0 12 3 ∞ replace by underlying simple map Figure 3.1.
Simplification process from the proof of Lemma 3.3, for p = 3. Thestable map at the top (with 7 components, and where the principal componentis shaded) yields a solution of (3.3) with a length 1 simple chain attached.while keeping the marked points and retaining connectedness of the image (see Figure 3.1). Theoutcome is as in (iii), again with possibly broken Morse flow lines, and the same codimensionargument applies. (cid:3) Given some coefficient field F , we denote by F x the one-dimensional vector space generated byorientations of W s ( x ), where the sum of the two orientations is set to zero. The Morse complexis(3.10) CM k ( f ) = (cid:77) | x | = k F x . A choice of orientations of W s ( x ) , . . . , W s ( x p ) , W s ( x ∞ ) determines an orientation of the modulispace M A ( C, x , . . . , x p , x ∞ ). In particular, every point in a zero-dimensional moduli space givesrise to a preferred isomorphism (an abstract version of a ± F x ⊗ · · · ⊗ F x p ∼ = F x ∞ .One adds up those contributions to get a map(3.11) m A ( C, x , . . . , x p , x ∞ ) : F x ⊗ · · · ⊗ F x p −→ F x ∞ , and those maps are the coefficients of a chain map(3.12) S A : CM ∗ ( f ) ⊗ p +1 −→ CM ∗− c ( A ) ( f ) . Up to chain homotopy, this map is independent of the choice of almost complex structure andinhomogeneous term, by a parametrized version of our previous argument. Of course, the outcomeis not in any sense surprising:
Lemma 3.4.
Up to chain homotopy, S A ( x , x , . . . , x p ) is the A -contribution to the ( p + 1) -foldquantum product x ∗ x ∗ · · · ∗ x p . uantum Steenrod 15
12 3 ∞ ∞∞∇ f ∇ f Figure 3.2.
A schematic picture of the proof of Lemma 3.4, with p = 3. Sketch of proof.
This is a familiar argument, which involves degenerating C to a nodal curve eachof whose components has three marked points, one option being that drawn in Figure 3.2(i); eachcomponent will again carry a Cauchy-Riemann equation with an inhomogeneous term. In ourMorse-theoretic context, there is an additional step, familiar from the proof that the PSS map isan isomorphism [12]. Namely, one adds a length parameter, and inserts a finite length flow lineof our Morse function at each node. As the length goes to infinity, each of the flow lines we haveinserted breaks, see Figure 3.2(ii); and that limit gives rise to the Morse homology version of theiterated quantum product. The parametrized moduli space (consisting of, first, the parameterused to degenerate C ; and then in the second step, using the finite edge-length as a parameter)then yields a chain homotopy between those two operations. (cid:3) Remark 3.5.
Our use of inhomogeneous terms means that, in principle, S A can be nonzero evenfor classes A ∈ H ( M ; Z ) which do not give rise to monomials in Λ (because (cid:82) A ω M is eithernegative, or it’s zero but A (cid:54) = 0 ). However, by choosing the inhomogeneous term small, one canrule out that undesired behaviour for any specific A . Since the outcome is independent of thechoice up to chain homotopy, the resulting cohomological level structure is indeed defined over Λ . Fix an oriented codimension 2 submanifold Ω ⊂ M . When choosing an almost complexstructure, there are additional restrictions: Assumption 3.6.
In the situation of Assumption 3.1, we additionally require that the evaluationmap on the space of simple J -holomorphic chains should be transverse to Ω . We equip the Riemann surface (3.2) with “an additional marked point which can move freely”(and which will carry an Ω-incidence constraint). Formally, this means that we consider a family of genus zero surfaces with sections(3.13) C −→ S,z C , , . . . , z C ,p , z C , ∞ , z C , ∗ : S −→ C where the parameter space S is again a copy of C P , and such that the following holds:(i) The critical values of (3.13) are precisely the marked points from (3.2). If v is a regularvalue, the fibre C v is canonically identified with C ; that identification takes the points z C r , , . . . , z C r ,p , z C r , ∞ arising from (3.13) to their counterparts in (3.2), and the remainingpoint z C v , ∗ to v .(ii) If v is a singular value, C v = C v, + ∪ C v, − is a nodal surface with two components. The firstcomponent C v, + is again identified with C , and the second component C v, − is a rationalcurve attached to the first one at v . The first component carries all the marked pointsthat C does, with the exception of the one which is equal to v ; and the second componentcarries the two remaining marked points, considered to be distinct and also different fromthe node (so, the second component has three special points, which identifies it up tounique isomorphism).Explicitly, (3.13) is constructed by starting with the trivial family C × S → S , and then blowingup the points ( v, v ), where v is one of the marked points in (3.2). One takes the proper transformsof the constant sections and of the diagonal section, which yield the z C ’s from (3.13).Denote by C sing ⊂ C the set of ( p + 2) nodes, and by C reg its complement. We write T ( C reg /S ) forthe fibrewise tangent bundle, which is a complex line bundle on C reg . A fibrewise inhomogeneousterm on C is a complex anti-linear map ν C : T ( C reg /S ) → T M , where both bundles involved havebeen pulled back to C reg × M , and with the property that ν C is zero outside a compact subset(meaning, in a neighbourhood of C sing × M ⊂ C × M ). Suppose that we have chosen such a term.One can then consider the moduli space of pairs ( v, u ), where(3.14) v ∈ S, u : C v −→ M, ( ¯ ∂ J u ) z = ν C v ,z,u ( z ) . In the case where C v has a node, the second equation is imposed separately on each of itscomponents (with the assumption that both preimages of the node must be mapped to thesame point, so as to constitute an actual map on C v ). This makes sense since, near each of thepreimages of the node, the equation reduces to the ordinary J -holomorphic curve equation. Theincidence conditions are(3.15) u ( z C v , ) ∈ W u ( x ) , . . . , u ( z C v ,p ) ∈ W u ( x p ) , u ( z C v , ∞ ) ∈ W s ( x ∞ ) , u ( z C v , ∗ ) ∈ Ω . Assumption 3.7.
We impose the following requirements: (i)
The space of all solutions of (3.14) , (3.15) should be regular. This should be understoodas two distinct conditions: on the open set of regular values v , regularity holds in theparametrized sense; and for each singular value v , it holds in the ordinary unparametrizedsense. uantum Steenrod 17 (ii) Take an element ( v, u ) in the same space, with a simple J -holomorphic bubble attached atan arbitrary point, in the same sense as in Assumption 3.2(ii) (the attaching point canbe a marked point, or even the node if v is singular). Then, that moduli space should beregular as well. As in (i), this should be interpreted as two different conditions, dependingwhether v is regular or not. (iii) Consider solutions for regular v , which have a simple holomorphic chain attached at asubset of the ( p +3) marked points, and where the incidence constraint has been transferredto the end of that chain, as in Assumption 3.2(iii). Then, the resulting moduli spaceshould again be regular. (iv) Take a singular v , We look at a situation similar to (iii), but where additionally, theremay be a simple holomorphic chain separating the two components of C v . Let’s spell outwhat that means (ignoring the possible existence of chains at the marked points). Write z ± ∈ C v, ± for the preimages of the node. In the definition of the moduli space (3.14) ,the C v, ± carry maps u ± which necessarily satisfy u − ( z − ) = u + ( z + ) . However, in ourlimiting situation, we instead have a simple chain ( u , . . . , u l ) such that (3.16) u − ( z − ) = u (0) , u + ( z + ) = u l ( ∞ ) . Again, we require that the resulting space should be regular.
As before, given A ∈ H ( M ; Z ), we write M A ( C , x , . . . , x ∞ , Ω) for the space of solutions of(3.14), (3.15) representing A . The added parameter v ∈ S compensates exactly for the evaluationconstraint at z C , ∗ , so that we get the same expected dimension as before,(3.17) dim M A ( C , x , . . . , x p , x ∞ , Ω) = 2 c ( A ) + | x ∞ | − | x | − · · · − | x p | . Concerning the analogue of the stable map compactification, we have a version of Lemma 3.3(with essentially the same proof):
Lemma 3.8. (i) If the dimension (3.17) is , we have a finite set (3.18) M A ( C , x , . . . , x p , x ∞ , Ω) = ¯ M A ( C , x , . . . , x p , x ∞ , Ω) . (ii) If the dimension is , the compactification is a manifold with boundary, with the boundarypoints only involving once-broken gradient flow lines.In both cases (i) and (ii), the moduli space and its compactification contain only points where v is a regular value. We define m A ( C , x , . . . , x p , x ∞ , Ω) to be the signed count of points in the zero-dimensional modulispaces. As before, one can assemble these into a chain map(3.19) P A, Ω : CM ∗ ( f ) ⊗ p +1 −→ CM ∗− c ( A ) ( f ) . Up to chain homotopy, this is independent of the choices of J and ν C , and also depends only on[Ω] ∈ H ( M ; Z ). The remaining topic in this section is the analogue of the divisor axiom. As one wouldexpect, this is not particularly difficult, but requires a bit of technical discussion around forgetting a marked point. For the submanifold Ω, we want to assume that it is transverse to the stableand unstable manifolds of the Morse function.
Lemma 3.9.
In the situation of Lemma 3.3, the following holds generically: any map u in azero-dimensional space M A ( C, x , . . . , x p , x ∞ ) intersects Ω transversally, and moreover, all thoseintersections happen away from the marked points. The same is true within the smaller space ofthose ν C which vanish close to the marked points. This is standard (transversality of evaluation maps). The only wrinkle specific to our case isthat the intersections avoid the marked points: but if they didn’t, we would have an incidenceconstraint with Ω ∩ W s ( x ) or Ω ∩ W u ( x ), and those can be ruled out for dimension reasons. Proposition 3.10.
Fix some A . For suitable choices made in the definitions, the maps (3.12) and (3.19) are related by P A, Ω = ( A · Ω) S A . (For arbitrary choices, the same relation willtherefore hold up to chain homotopy.)Proof. Even more explicitly, our statement says that one can arrange that(3.20) m A ( C , x , . . . , x p , x ∞ , Ω) = ( A · Ω) m A ( C, x , . . . , x p , x ∞ ) . We start with J as in Assumption 3.6, and a ν C as in Lemma 3.9. Because the inhomogeneousterm is zero near the marked points, it can be pulled back to give a fibrewise inhomogeneousterm ν C . To clarify, if C v is a singular fibre, then ν C v is supported on C v, + ∼ = C , and zero onthe other component C v, − . Let’s consider the structure of the resulting moduli spaces. Given anelement in the stable map compactification ¯ M A ( C , x , . . . , x p , x ∞ , Ω), one can forget the positionof the ∗ marked point, and then collapse unstable components (components which are not C , andwhich carry a constant J -holomorphic map and less than three special points). The outcome isa (continuous) map(3.21) ¯ M A ( C , x , . . . , x p , x ∞ , Ω) −→ ¯ M A ( C, x , . . . , x p , x ∞ ) . Now suppose that the dimension is zero. Then, the target in (3.21) is M A ( C, x , . . . , x p , x ∞ ), andconsists only of maps u : C → M whose intersection points with Ω are not marked points. Thepreimage of u under (3.21) is necessarily an element of M A ( C , x , . . . , x p , x ∞ , Ω), with v a regularvalue; such preimages correspond bijectively to points in u − (Ω), hence form a finite set, and(because of the transversality condition in Lemma 3.9) are regular points in the parametrizedmoduli space. Finally, the sign of their contribution to m A ( C , x , . . . , x p , x ∞ , Ω) is given bymultiplying the contribution of u to m A ( C, x , . . . , x p , x ∞ ) with the local intersection number(sign) of u and Ω at the relevant point.We have now shown that M A ( C , x , . . . , x p , x ∞ , Ω) = ¯ M A ( C , x , . . . , x p , x ∞ , Ω) is regular, andthat counting points in it exactly yields the right hand side of (3.20). The ν C used for thispurpose may not satisfy Assumption 3.7, so this setting is not strictly speaking part of ourgeneral definition of m A ( C , x , . . . , x p , x ∞ , Ω). However, we can find a small perturbation of ν C which does satisfy Assumption 3.7, and points in the associated zero-dimensional moduli spaceswill correspond bijectively to those for the original ν C , because of the compactness and regularityof the original space. (cid:3) uantum Steenrod 19 Quantum Steenrod operations
This section concerns the operations (1.15) and (1.30). We first set up the various equivariantmoduli spaces, then define Q Σ b , and discuss its properties. Then we proceed to do the same for Q Π a,b , and go as far as establishing (1.32). We equip C = C P with the ( Z /p )-action generated by the same rotation as in Section2c, but here denoted by σ C . Fix a compatible almost complex structure J . An equivariantinhomogeneous term ν eq C is a smooth complex-antilinear map T C → T M , where both bundleshave been pulled back to S ∞ × Z /p C × M . More concretely, one can think of it as a family ν eq C,w ofinhomogeneous terms (in the standard sense) parametrized by w ∈ S ∞ , with the property that(4.1) ν eq C,τ ( w ) ,z,x = ν eq C,w,σ C ( z ) ,x ◦ Dσ z : TC z → TM x for ( w, z, x ) ∈ S ∞ × C × M .Consider the following parametrized moduli problem:(4.2) w ∈ S ∞ , u : C −→ M, ( ¯ ∂ J u ) z = ν eqC,w,z,u ( z )) . Note that this inherits a ( Z /p )-action, generated by(4.3) ( w, u ) (cid:55)−→ ( τ ( w ) , u ◦ σ C ) . Fix critical points x , . . . , x p , x ∞ , and impose the same incidence constraints as in (3.4) or equiv-alently (3.5). Moreover, we fix an integer i ≥ w to oneof the cells from (2.3), (2.4). More precisely, the condition is that(4.4) w ∈ ∆ i \ ∂ ∆ i ⊂ S ∞ . Take solutions of (4.2), (3.4), (4.4) that represent some class A ∈ H ( M ; Z ), and denote theresulting moduli space by M A (∆ i × C, x , . . . , x p , x ∞ ). The expected dimension increases by thenumber of parameters,(4.5) dim M A (∆ i × C, x , . . . , x p , x ∞ ) = i + 2 c ( A ) + | x ∞ | − | x | − · · · − | x p | . Note that while one could define such moduli spaces for more general cells τ j (∆ i ), that is re-dundant because of (4.3). To express that more precisely, write ( x ( j )1 , . . . , x ( j ) p ) for the p -tupleobtained by cyclically permuting ( x , . . . , x p ) j times (to the right, so x (1)1 = x p ). Then,(4.6) M A ( τ j (∆ i ) × C, x , . . . , x p , x ∞ ) ∼ = −→ M A (∆ i × C, x , x ( j )1 , . . . , x ( j ) p , x ∞ ) , ( w, u ) (cid:55)−→ ( τ − j ( w ) , u ◦ σ − jC ) . There is also a natural compactification, denoted by ¯ M A ( · · · ) as usual. This combines the(parametrized) stable map compactification, breaking of Morse flow lines, and instances wherethe parameter w reaches the boundary of ∆ i . Lemma 4.1.
For generic J and ν eq C , the following properties are satisfied.(i) If the dimension (4.5) is zero, we get a finite set (4.7) M A (∆ i × C, x , . . . , x p , x ∞ ) = ¯ M A (∆ i × C, x , . . . , x p , x ∞ ) . (ii) If the dimension is , the moduli space is regular, and its compactification is a manifoldwith boundary. Besides the usual boundary points arising from broken Morse flow lines, one hassolutions ( w, u ) where w ∈ ∂ ∆ i . Using (4.6) , the set of such boundary points can be identifiedwith a disjoint union (4.8) (cid:91) j M A (∆ i − × C, x , x ( j )1 , . . . , x ( j ) p , x ∞ ) over (cid:40) j = 0 , . . . , p − i even ,j = 0 , i odd. In (ii), note that the only points w ∈ ∂ ∆ i that occur lie in the interior of the cells of dimension( i − i have corners can be disregarded.The proof of Lemma 4.1 is simply a parametrized version of that of Lemma 3.3: one imposesAssumption 3.1 on J , and the parametrized analogue of Assumption 3.2 on ν eq C , where theparameter space is taken to be each ∆ i \ ∂ ∆ i . We will not discuss the argument further, andmove ahead to its implications.As usual, we count points in zero-dimensional moduli spaces, and collect those coefficients into asingle map(4.9) Σ A (∆ i , . . . ) : CM ∗ ( f ) ⊗ CM ∗ ( f ) ⊗ p −→ CM ∗− i − c ( A ) ( f ) . Lemma 4.1(ii), with the orientations of the ∆ i taken into account as in (2.7), (2.8), shows that, d being the Morse differential,(4.10) d Σ A (∆ i , x , . . . , x p ) − ( − i p (cid:88) j =0 ( − | x | + ··· + | x j − | Σ A (∆ i , x , . . . , dx j , . . . , x p )= (cid:88) j ( − ∗ Σ A (∆ i − , x , x ( j )1 , . . . , x ( j ) p ) i even,( − ∗ Σ A (∆ i − , x , x (1)1 , . . . , x (1) p ) − Σ A (∆ i − , x , x , . . . , x p ) i odd.Here, ( − ∗ is the Koszul sign associated with permuting ( x , . . . , x p ). Remark 4.2.
Our sign conventions for parametrized pseudo-holomorphic map equations are asfollows. Consider, just for the simplicity of notation, operations induced by a Cauchy-Riemannequation on the sphere, with one input and one output. If we have a family of such equationsdepending on a parameter space ∆ which is a manifold with boundary, then the resulting endo-morphism of CM ∗ ( f ) satisfies (4.11) dφ ∆ − ( − | ∆ | φ ∆ d = φ ∂ ∆ . Note that this differs from the convention in [13, Section 4c] ; one can translate betwen the twoby multiplying φ ∆ with ( − | ∆ | ( | ∆ |− . From now on, we will exclusively work with coefficients in F = F p . Lemma 4.3.
Suppose that b is a Morse cocycle. Then, for each i and A , (4.12) x (cid:55)−→ ( − | b | | x | Σ A (∆ i , x, b, . . . , b ) is a chain map (an endomorphism of the Morse complex) of degree p | b | − i − c ( A ) . uantum Steenrod 21 This is immediate, by specializing (4.10) to x = · · · = x p = b . We like to combine theseoperations into a series, which is a chain map(4.13) Σ A,b : CM ∗ ( f ) −→ ( CM ( f )[[ t, θ ]]) ∗ + p | b |− c ( A ) ,x (cid:55)−→ ( − | b | | x | (cid:88) k (cid:16) Σ A (∆ k , x, b, . . . , b ) + ( − | b | + | x | Σ A (∆ k +1 , x, b, . . . , b ) θ (cid:17) t k , One can also sum formally over all A and extend the outcome Λ-linearly,(4.14) Σ b = (cid:88) A q A Σ A,b : CM ∗ ( f ; Λ) −→ CM ∗ + p | b | ( f ; Λ) . Lemma 4.4.
Up to homotopy, (4.13) depends only on cohomology class [ b ] , and moreover, thatdependence is linear.Proof. Take CM ∗ ( f ) ⊗ p , with the Z /p -action given by cyclic permutation, and form the associatedequivariant complex as in (2.35). Consider the t -linear map(4.15) Σ eq A : CM ∗ ( f ) ⊗ ( CM ∗ ( f ) ⊗ p ) eq −→ ( CM ( f )[[ t, θ ]]) ∗− c ( A ) ,x ⊗ ( x ⊗ · · · ⊗ x p ) (cid:55)−→ (cid:88) k (cid:16) Σ A (∆ k , x , . . . , x p ) + ( − | x | + ··· + | x p | Σ A (∆ k +1 , x , . . . , x p ) θ (cid:17) t k ,x ⊗ ( x ⊗ · · · ⊗ x p ) θ (cid:55)−→ (cid:88) k (cid:16) Σ A (∆ k , x , . . . , x p ) θ − ( − | x | + ··· + | x p | (cid:88) j j ( − ∗ Σ A (∆ k +1 , x , x ( j )1 , . . . , x ( j ) p ) t (cid:17) t k , where ( − ∗ is again the Koszul sign. The equation (4.10) amounts to saying that (4.15) is achain map. As an elementary algebraic consequence, one has the following: if c is any cocycle in( CM ∗ ( f ) ⊗ p ) eq , then(4.16) x (cid:55)−→ ( − | c | | x | Σ eq A ( x ⊗ c )is an endomorphism of the chain complex CF ∗ ( f ) of degree | c | − c ( A ). The homotopy class ofthat endomorphism depends only on the cohomology class of c . Moreover, they are additive in c . Applying that construction to c = b ⊗ · · · ⊗ b yields precisely (4.13).From Lemma 2.5, we know that the cohomology class [ b ⊗· · ·⊗ b ] ∈ H ∗ eq ( CM ∗ ( f ) ⊗ p ) only dependson that of [ b ], which proves our first claim. By the same Lemma, if we use t ( b ⊗ · · · ⊗ b ) instead,the associated operation becomes linear in [ b ]. But that operation is just t times (4.13), so itfollows that (4.13) itself must be linear in [ b ]. (cid:3) Definition 4.5.
For b ∈ H ∗ ( M ; F p ) and A ∈ H ( M ; Z ) , we define the operation Q Σ A,b from (1.10) to be the cohomology level map induced by (4.13) . Correspondingly, (4.14) is the chainmap underlying Q Σ b . Here, we are implicitly using the fact that the chain level operations are independent of all choicesup to chain homotopy. The proof is standard, using moduli spaces with one extra parameter, andwill be omitted. Among the previously stated properties of Q Σ, (1.16) concerns the contribution of the cell ∆ , which is the operation from Section 3a, hence is exactly Lemma 3.4. The nexttwo Lemmas correspond to (1.11) and (1.18). Lemma 4.6.
For A = 0 , Q Σ A,b is the cup product with St ( b ) .Sketch of proof. It will be convenient for this purpose to allow a slightly larger set of choices inthe construction. Namely, we choose s -dependent vector fields(4.17) Z ,w,s , . . . , Z p,w,s ∈ C ∞ ( T M ) for w ∈ S ∞ , s ≤
0, with Z k,w,s = ∇ f if s (cid:28) Z ∞ ,w,s ∈ C ∞ ( T M ) for w ∈ S ∞ , s ≥
0, with Z ∞ ,w,s = ∇ f if s (cid:29) dy k /ds = Z w,k,s . The effect isthat in the incidence conditions (3.4), the (un)stable manifolds are replaced by perturbed versions.In particular, the transversality of those incidence conditions imposed on pseudo-holomorphiccurves can then be achieved by choosing (4.17) generically. We impose an additional symmetrycondition, which ensures that (4.6) still holds:(4.18) Z k +1 ,w,s = Z k,τ ( w ) ,s for k = 1 , . . . , p − A = 0, this means that we can take the inhomogeneous term to be zero through-out, so that all maps u are constant (and of course regular). The resulting moduli spaces arepurely Morse-theoretical, see Figure 4.1(i) for a schematic representation. Without violating thesymmetry property (4.6), we can deform our moduli spaces as indicated in Figure 4.1(ii). Thisseparates the coincidence condition at the endpoints of the half-flow lines into two parts, joinedby a finite length flow line of some other auxiliary s -dependent vector field. More precisely, weuse the length as an additional parameter, and all vector fields involved may depend on that.One can arrange that as the length goes to ∞ , the limit consists of split solutions as in Fig-ure 4.1(iii), where the vector fields on the bottom part are independent on w ∈ S ∞ . It is nowstraightforward to see that this limit is the combination of the Morse-theoretic cup product andthe Morse-theoretic version of the Steenrod operation [1, 3]. (cid:3) Lemma 4.7. Q Σ A,b (1) agrees with the A -contribution to the quantum Steenrod operation Q St ( b ) ,as defined in [7, 18] (for p = 2 ) or [14] (all p ).Sketch of proof. Morse-theoretically, 1 is represented by the sum of local minima of the Morsefunction. Hence, the associated incidence condition (3.4) requires u (0) to lie in an open dense set,and is generically satisfied on every zero-dimensional moduli space. In other words, Σ A,b (1) canbe computed by forgetting the zero-th marked point and its incidence condition. The outcome isexactly the definition of the quantum Steenrod operation, generalizing the p = 2 case from [18]in a straightforward way; compared to the slightly more abstract formulation in [14, Section 9],the only difference is that we stick to a specific cell decomposition of B Z /p = S ∞ / ( Z /p ). (cid:3) The final piece of our discussion of Q Σ operations concerns (1.20). We assume that theunderlying cochain level map Σ b has been extended to b ∈ CM ∗ ( f ) ⊗ Λ, as in (1.19). uantum Steenrod 23 ∞ (i) (ii)0 1 2 30 finite length trajectory1 2 30 (iii)intermediate critical pointonly this part depends on w ∈ S ∞ ∞∞ Figure 4.1.
A schematic picture of the proof of Lemma 4.6.
Proposition 4.8.
Fix Morse cocycles b and ˜ b , and write ˜ b ∗ b ∈ CM ∗ ( f ) ⊗ Λ for a cochainrepresentative of their quantum product. Then, there is a chain homotopy (4.19) Σ ˜ b ◦ Σ b (cid:39) ( − | b | | ˜ b | p ( p − Σ ˜ b ∗ b . Sketch of proof.
We introduce a family of Riemann surfaces with (2 p + 2) marked points, whichdepends on an additional parameter η ∈ (1 , ∞ ). Each of those surfaces C η is a copy of C , andthe marked points are z C η ,k = z C,k , k ∈ { , . . . , p, ∞} , from (3.2) together with(4.20) ˜ z C η , = ηz C, , . . . , ˜ z C η ,p = ηz C,p . There are natural degenerations at the end of our parameter space: as η →
1, each point ˜ z C η ,k collides with its counterpart z C η ,k , and one can see this as each pair bubbling off into an extracomponent of a nodal curve C . As η → ∞ , all the ˜ z C η ,k collide with z C η , ∞ , and one can see asdegeneration of C η into a nodal curve C ∞ with two components, each of which is modelled onthe original (3.2) (see Figure 4.2).We choose an equivariant inhomogeneous term ν eq C η on each of our curves, which is well-behavedunder the two degenerations (and is zero in a neighbourhood of the nodes; the details aresimilar to our previous definition of fibrewise inhomogeneous terms). Given critical points x , x , ˜ x , . . . , x p , ˜ x p , x ∞ of the Morse function f , and a cell ∆ i , we define a moduli space oftriples ( η, w, u ), where: η ∈ (1 , ∞ ), w is as in (4.4), and u : C η → M is a map, representing thegiven homology class A , which satisfies the η -parametrized version of (4.2), and the incidence ∞ ∞∞ η → ∞ η → Figure 4.2.
The family underlying the proof of Proposition 4.8, for p = 2.conditions (3.4) as well as(4.21) u (˜ z C η , ) ∈ W u (˜ x ) , . . . , u (˜ z C η ,p ) ∈ W u (˜ x p ) . To understand the algebraic relations which this parametrized moduli space provides, we haveto look at the contributions from limits with η = 1 or η = ∞ . The η = 1 contribution is givenby a suitable moduli space of maps on C , and is fairly easy to interpret. Namely, one followsthe proof of Lemma 3.4 and separates the components of C by finite length gradient trajectories(to preserve the Z /p -symmetry, all the lengths must be the same, so there is only one lengthparameter). As the length goes to infinity, the Morse flow lines split, and we end up with acomposition of quantum product (of x k and x (cid:48) k ) and a remaining component where we have thepreviously defined operation (4.9). We can apply the same strategy to the η = ∞ limit, insertinga finite length gradient flow line between the two pieces. As the length goes to infinity, we endup with two separate components carrying equations of the kind which underlies (4.9). However,the two equations are coupled because they carry the same parameter w ∈ S ∞ . In other words,the resulting moduli spaces end up being(4.22) (cid:91) M A (∆ i × C, x , . . . , x p , x ) × S ∞ M A (∆ i × C, x, ˜ x , . . . , ˜ x p , x ∞ ) , where the (disjoint) union is over A + A = A and all critical points x .In the same spirit as in (4.9), we denote the operations obtained from (4.22) by(4.23) Ξ A ( δ (∆ i ) , . . . ) : CM ∗ ( f ) ⊗ CM ∗ ( f ) ⊗ p −→ CM ∗− i − c ( A ) ( f ) . We also find it convenient to add up over all A , with the usual q A coefficients. Fix cocycles b and˜ b and insert them into (4.23) at the marked points labeled (1 , . . . , p ) and (˜1 , . . . , ˜ p ), respectively,with signs as in (4.13). This yields a chain map(4.24) Ξ ˜ b,b ( δ (∆ i ) , · ) : CM ∗ ( f ) −→ ( CM ( f ) ⊗ Λ) ∗ + p | b | + p | ˜ b | . uantum Steenrod 25 The outcome of the parametrized moduli space argument outlined above is a chain homotopy(4.25) Ξ ˜ b,b ( δ (∆ i ) , · ) (cid:39) Σ ˜ b ∗ b (∆ i , · ) . We will be somewhat brief about the final step, since that is a general issue involving equivariantcohomology, and not really specific to our situation. One can construct chain maps like (4.24)not just for δ (∆ i ), but for other F p -coefficient cycles in S ∞ / ( Z /p ) × S ∞ / ( Z /p ), such as ∆ i × ∆ i .In that case, there is a simple decomposition formula(4.26) Ξ ˜ b,b (∆ i × ∆ i , · ) = ( − | b | | ˜ b | p ( p − Ξ ˜ b (∆ i , Ξ b (∆ i , · ))where the Koszul sign arises from reordering (˜ b, b, ˜ b, b, . . . ) into (˜ b, . . . , ˜ b, b, . . . , b ). Finally, ho-mologous cycles give homotopic maps. One can use that, and the decomposition of δ (∆ i ) intoproduct cycles from Section 2b, to obtain a further homotopy(4.27) Ξ ˜ b,b ( δ (∆ i ) , · ) (cid:39) (cid:88) i + i = i Ξ ˜ b,b (∆ i × ∆ i , · ) if i is odd or p = 2, (cid:88) i + i = ii k even Σ ˜ b,b (∆ i × ∆ i , · ) if i is even and p > (cid:3) We now merge ideas from Sections 3b and 4a, by which we mean that we take modulispaces parametrized by cells in S ∞ / ( Z /p ), and add an additional freely moving marked pointto the domain. The starting point is, once more, the family (3.13). From its construction as ablowup of C × S → S , this inherits a (diagonal) ( Z /p )-action, which we denote by σ C .Fix an almost complex structure J . An equivariant fibrewise inhomogeneous term is a complexanti-linear map(4.28) ν eq C /S : T ( C reg /S ) −→ T M, where both bundles have been pulled back to S ∞ × Z /p C reg × M . When restricted to any S k − × Z /p C reg × M , it should vanish outside a compact subset (meaning, it’s zero in a neighbourhood of S k − × Z /p C sing × M ; the restriction to S k − follows our usual process of treating S ∞ as adirect limit of finite-dimensional manifolds). As before, one can think of it more explicitly asa family ν eq C /S,w of fibrewise inhomogeneous terms parametrized by w ∈ S ∞ , and satisfying a( Z /p )-equivariance property as in (4.1):(4.29) ν eq C /S,τ ( w ) ,z,x = ν eq C /S,w,σ C ( z ) ,x ◦ Dσ C : T ( C reg /S ) z → TM x . The associated moduli space consists of triples ( w, v, u ), where the parameters are ( w, v ) ∈ S ∞ × S , v being a regular value of (3.13), and u : C v → M is a solution of the inhomogeneous Cauchy-Riemann equation given by ν eq C v ,w . These inherit a ( Z /p )-action as in (4.3):(4.30) ( w, v, u ) (cid:55)−→ ( τ ( w ) , σ − ( v ) , u ◦ σ C ) . We impose the usual incidence conditions, given by the (un)stable manifolds of critical points x , . . . , x p , x ∞ , and by a codimension 2 submanifold Ω at the ∗ marked point. Finally, we restrictto the interior of cells (4.4). Denote the resulting moduli spaces by M A (∆ i × C , x , . . . , x p , x ∞ , Ω).Their expected dimension remains as in (4.5).
We omit the discussion of transversality and of the compactifications, which is simply a combina-tion of those in Sections 3b and 4a. The outcome of isolated-point-counting in our moduli spaceare maps(4.31) Π A (∆ i , . . . ) : CM ∗ ( f ) ⊗ CM ∗ ( f ) ⊗ p −→ CM ∗− i − c ( A ) ( f )which, due to the structure of the compactified one-dimensional moduli spaces, satisfy the sameequation as the Σ A (∆ i , . . . ), see (4.10). Specializing to coefficients in F p , and fixing a Morsecocycle b , one can therefore use (4.31) to define a chain(4.32) Π A,b : CM ∗ ( f ) −→ ( CM ( f )[[ t, θ ]]) ∗ + p | b |− c ( A ) exactly as in (4.12). Moreover, up to homotopy that map depends linearly on [ b ], as in Lemma4.4. Again up to homotopy, it is also independent of all choices, including that of Ω within itscohomology class a = [Ω] ∈ H ( M ; Z ). Definition 4.9.
For a ∈ H ( M ; Z ) , b ∈ H ∗ ( M ; F p ) and A ∈ H ( M ; Z ) , we define Q Π A,a,b to bethe cohomology level map induced by (4.32) . Adding up those maps with weights q A yields (1.30) . Proposition 4.10.
Fix some A and integer i . For suitable choices made in the definition, wehave Π A (∆ i , . . . ) = ( A · Ω)Σ A (∆ i , . . . ) . As a consequence, we have Q Π A,a,b = ( A · Ω) Q Σ A,b forall i and A , which is equivalent to (1.32) .Proof. The geometric part of this is exactly as in Proposition 3.10: for suitably correlated choicesof inhomogeneous terms, the underlying moduli spaces bear the same relationship. Since thatargument involves making a small perturbation, we can only apply it to finitely many modulispaces at once, and that explains the bound on i in the statement. As a consequence, we getequality of the i -th coefficient in Q Π A,a,b and ( A · Ω) Q Σ A,b . (cid:3) Remark 4.11.
Both in Section 3b and here, we have used an evaluation constraint at a codimen-sion two submanifold Ω ⊂ M , which limits Q Π a,b to a ∈ H ( M ; Z ) . One can replace that by apseudo-cycle of arbitrary dimension d (see e.g. [19] ) and then, the definition goes through withoutany significant changes for a ∈ H d ( M ; Z ) . In fact, one could even take a mod p pseudo-cycle.This consists of an oriented manifold with boundary N d , such that ∂N carries a free ( Z /p ) -action,and a map f : N → M such that f | ∂N is ( Z /p ) -invariant, with the following properties: the limitpoints of f are contained in the image of a map from a manifold of dimension ( d − , and thelimit points of f | ∂N are contained in the image of a map from a manifold of dimension ( d − .While we do not intend to develop the theory of mod p pseudo-cycles here, this should allow oneto define Q Π a,b for all a ∈ H d ( M ; F p ) . The proof of (1.31) given in the next section extends tosuch generalizations in a straightforward way, but of course, there is no analogue of (1.32) indegrees d > . Proof of Theorem 1.4
This section derives (1.31). Together with the previously established (1.32), that completes ourproof of Theorem 1.4. uantum Steenrod 27
We decompose the moduli spaces underlying Q Π a,b into pieces, where the position of theadditional marked point is constrained to lie in one of the cells from Section 2c. This means thatinstead of using ∆ i × S ⊂ S ∞ × S as parameter spaces, we look at the subspaces ∆ i × W , where(5.1) W ∈ { P , Q , σ j ( L ) , σ j ( B ) } . Within the framework of Section 4c, it is unproblematic to ensure that all the resulting modulispaces, denoted by M A (∆ i × C | W, x , . . . , x p , x ∞ , Ω), satisfy the usual regularity and compactnessproperties. Point-counting in them gives rise to maps(5.2) Π A (∆ i × W, . . . ) : CM ∗ ( f ) ⊗ CM ∗ ( f ) ⊗ p −→ CM ∗− i − c A ( A ) −| W | +2 ( f ) . As in (4.10), adjacencies between cells determine relations between the associated invariants. Inour case, these are governed by (2.7)–(2.8) and (2.19)–(2.21). Explicitly, the relations are(5.3) d Π A (∆ i × W, x , . . . , x p ) − ( − i + | W | p (cid:88) k =0 ( − | x | + ··· + | x k − | Π A (∆ i × W, x , . . . , dx k , . . . , x p )= (cid:88) j ( − ∗ Π A (∆ i − × σ j W, x , x ( j )1 , . . . , x ( j ) p ) i even,( − ∗ Π A (∆ i − × σW, x , x (1)1 , . . . , x (1) p ) − Π A (∆ i − × W, x , x , . . . , x p ) i odd+ (extra term depending on W ) . The last-mentioned term is zero if W ∈ { P , Q } , with the remaining cases being(extra term for W = σ j L )= ( − i (cid:0) Π A (∆ i × Q , x , x ( j )1 , . . . , x ( j ) p ) − Π A (∆ i × P , x , x ( j )1 , . . . , x ( j ) p ) (cid:1) , (5.4) (extra term for W = σ j B )= ( − i +1 (cid:0) Π A (∆ i × σ j +1 L , x , x (1)1 , . . . , x (1) p ) − Π A (∆ i × σ j L , x , x , . . . , x p ) (cid:1) . (5.5)As usual, we now specialize to coefficients in F = F p . The relations above immediately imply thefollowing: Lemma 5.1.
Fix a cocycle b ∈ CM ∗ ( f ) . Then, the t -linear map (5.6) Π eq A,b : C −∗ ( S ) eq ⊗ CM ∗ ( f ) −→ ( CM ( f )[[ t, θ ]]) ∗ + p | b |− c ( A )+2 ,W ⊗ x (cid:55)−→ ( − | b | ( | W | + | x | ) (cid:88) k (cid:16) Π A (∆ k × W, x, b, . . . , b )+ ( − | x | + | b | + | W | Π A (∆ k +1 × W, x, b, . . . , b ) θ (cid:17) t k ,W θ ⊗ x (cid:55)−→ ( − | b | ( | W | + | x | ) (cid:88) k (cid:16) ( − | x | Π A (∆ k × W, x, b, . . . , b ) θ − ( − | b | + | W | (cid:88) j j Π A (∆ k +1 × σ j W, x, b, . . . , b ) t (cid:17) t k , is a chain map. Following (4.15), one can think of (5.6) as a special case of a more general structure, which wouldbe a t -linear chain map(5.7) ( C −∗ ( S ) ⊗ CM ∗ ( f ) ⊗ CM ∗ ( f ) ⊗ p ) eq −→ ( CM ( f )[[ t, θ ]]) ∗− c ( A )+2 . Here, the group Z /p acts on C −∗ ( S ), as well as on CM ∗ ( f ) ⊗ p by cyclic permutations. As inthe previous situation, (5.7) would be useful in order to prove that (5.6) only depends on thecohomology class of b , and is additive. For our purposes, however, we can work around that,since all necessary computations can be done using a fixed cocycle b . At this point, everything we need can be extracted from an analysis of the chain map (5.6).
Lemma 5.2.
Suppose that we specialize (5.6) to using only B + σB + · · · + σ p − B ∈ C ( S ) eq .Then, the resulting chain map CM ∗ ( f ) → ( CM ( f )[[ t, θ ]]) ∗ + p | b |− c ( A ) is equal to Π A,b .Proof.
This is essentially by definition. We are considering the map(5.8) x (cid:55)−→ ( − | b | | x | (cid:88) j,k (cid:16) Π A (∆ k × σ j ( B ) , x, b, . . . , b )+ ( − | b | + | x | Π A (∆ k +1 × σ j ( B ) , x, b, . . . , b ) θ (cid:17) t k . The regularity of the spaces M A (∆ × C | W, x , . . . , x p , x ∞ , Ω) for cells W of dimension < M A (∆ × C , x , . . . , x p , x ∞ , Ω), none of the points arises from aparameter value v ∈ S which belongs to one of those cells. In other words, that space M A (∆ × C , x , . . . , x p , x ∞ , Ω) is the disjoint union of M A (∆ × C | σ j B, x , . . . , x p , x ∞ , Ω). (cid:3)
Lemma 5.3.
Suppose that we specialize (5.6) to using only P ∈ C ( S ) eq , and pass to co-homology. Then, the resulting map is equal to the following: take all possible decompositions A = A + A , and add up (5.9) H ∗ ( M ; F p ) ∗ A a −−−→ H ∗ +2 − c ( A ) ( M ; F p ) Q Σ b,A −−−−−→ ( H ( M ; F p )[[ t, θ ]]) ∗ + p | b | +2 − c ( A ) , where a = [Ω] ∈ H ( M ; Z ) .Proof. This time, the reason is geometric. Using P means that we are restricting to a particularfibre of (3.13), which is the nodal surface from Figure 5.1(i). Recall that each component ofthat surface carries an inhomogeneous term, which additionally depends on parameters in S ∞ .However, without violating regularity or other restrictions, one can arrange that the inhomoge-neous term on the component which is a three-pointed sphere ( C , − in the notation from Section3b) is independent of those parameters. After that, one inserts a finite length Morse flow linebetween the two components, as in Figure 5.1(ii). In the same way as in Lemma 3.4, the resulting(varying length) moduli space gives a chain homotopy between our operation and the chain mapunderlying the composition (5.9), in its Morse-theoretic incarnation. (cid:3) Lemma 5.4.
Suppose that we specialize (5.6) to using only Q ∈ C ( S ) eq , and pass to cohomologyThen, the resulting map is equal to the following: take all possible decompositions A = A + A ,and add up (5.10) H ∗ ( M ; F p ) Q Σ b,A −−−−−→ H ( M ; F p )[[ t, θ ]]) ∗ + p | b |− c ( A ) ∗ A a −−−→ ( H ( M ; F p )[[ t, θ ]]) ∗ + p | b | +2 − c ( A ) , uantum Steenrod 29
12 3 ∞∗ ∞∗ ∇ f Figure 5.1.
A schematic picture of the proof of Lemma 5.3, with p = 3. where a = [Ω] as before. The proof is the same as for Lemma 5.3. Note that the operations in (5.10) appear in the oppositeorder from (5.9). The reason is that over v = 0, the component C y, − is attached to C v, + at thepoint 0 ∈ C , which serves as input of the Σ operation; whereas for v = ∞ , it is attached at theoutput point ∞ ∈ C . Finally, we have the following, which establishes (1.31): Proposition 5.5. tQ Π a,b equals the difference between (5.9) and (5.10) .Proof. By Lemma 5.2, Π
A,b t is obtained by specializing (5.6) to ( B + · · · + σ p − B ) t . From(2.50) and (2.51), we see that this is chain homotopic to specializing the same map to ( P − Q ).Using Lemma 5.3 and 5.4 then yields the desired result. (cid:3) Computations
In this section, we explore the power of Theorem 1.4 as a computational tool.
Our first task is to work out the details of Example 1.6, where M is the two-sphere. Weuse the standard generator of H ( M ; Z ), and correspondingly write Λ as a power series ring inone variable q . The quantum connection is(6.1) ∇ = tq∂ q + (cid:18) q (cid:19) . Let’s temporarily use Q -coefficients, and allow inverses of t . If ξ satisfies(6.2) ( tq∂ q ) ξ = qξ, then the following endomorphism is covariantly constant with respect to (6.1):(6.3) Ξ = (cid:18) − ξ ( tq∂ q ξ ) − ( tq∂ q ξ ) ξ ξ ( tq∂ q ξ ) (cid:19) . It is straightforward to write down an explicit solution of (6.2):(6.4) ξ = ∞ (cid:88) k =0 k !) q k t − k . Pick a prime p >
2. Take (6.3) with (6.4), and truncate it by dropping all powers q p or higher.The remaining denominators are coprime to p , so we can reduce coefficients to F p . The outcome,using some elementary combinatorics to simplify the formulae, is the matrix Σ from (1.26). Byconstruction, this endomorphism is covariantly constant modulo q p ; and the constant term (in q ) of − t p − Σ matches the cup product with St ( h ) = − t p − h (see (1.13) for the sign convention).Therefore, − t p − Σ and Q Σ h must agree modulo q p . But for degree reasons, Q Σ h can’t haveterms of order q p or higher. The consequence is that Q Σ h = − t p − Σ, as previously stated.
Remark 6.1.
It is worthwhile spelling out the comparison with the fundamental solution of thequantum differential equation, mentioned in Remark 1.5. For S , the fundamental solution is [9,Section 28.2] (note the differences in notation and conventions: our t is their − (cid:126) ; our q is their e t ; our t is their H ) (6.5) Ψ = (cid:18) − tq∂ q η − tq∂ q ξη ξ (cid:19) , where ξ is as in (6.4) , and (6.6) η = ∞ (cid:88) k =0 k !) q k t − k − (cid:0) − log( q ) + 2 k (cid:88) j =1 j (cid:1) is a multivalued solution of the same equation (6.2) as ξ . By forming (1.23) with β = h , one getsexactly the matrix from (6.3) : (6.7) Ξ = Ψ (cid:18) (cid:19) Ψ − . Following ideas from [18], let’s look at the following situation:
Assumption 6.2.
The quantum ring structure on H ∗ ( M ; Λ) is generated by H ( M ; F p ) . This implies that H ∗ ( M ; F p ) is zero in odd degrees. It also implies that each class in H ( M ; F p )can be lifted to H ( M ; Z ), as one sees by looking at(6.8) · · · → H ( M ; Z ) → H ( M ; F p ) → H ( M ; Z ) p −→ H ( M ; Z ) → H ( M ; F p ) → · · · Lemma 6.3.
Suppose that Assumption 6.2 holds. Then, the quantum product and QSt ( b ) , for b ∈ H ( M ; F p ) , determine all the quantum Steenrod operations. uantum Steenrod 31 Proof.
Write the covariant constancy property as(6.9) QΣ b ( a ∗ c ) = t∂ a QΣ b ( c ) + a ∗ QΣ b ( c ) , a ∈ H ( M ; Z ) , b, c ∈ H ∗ ( M ; F p ) . This shows that Q Σ b ( c ) and the quantum product determine Q Σ b ( a ∗ c ). Therefore, if one knows QSt ( b ) = Q Σ b (1) and Assumption 6.3 holds, the entire operation Q Σ b can be computed fromthat.By (1.20),(6.10) QSt ( b ∗ c ) = Q Σ b ( Q St ( c )) . If we know
QSt ( b ) and QSt ( c ), for some b ∈ H ( M ; F p ) and c ∈ H ∗ ( M ; F p ), then our previousargument determines Q Σ b , and we can get QSt ( b ∗ c ) from that by (6.10). In view of Assumption6.3, this implies the desired result. (cid:3) Here is a concrete class of examples to which this strategy applies.
Proposition 6.4.
Suppose that M is a monotone symplectic manifold, satisfying Assumption6.2. Then the quantum Steenrod operations can be computed in terms of the quantum productand classical Steenrod operations.Proof. Take b ∈ H ( M ; F p ). Then QSt ( b ) has degree 2 p . The monomials in it that can havenonzero coefficients are t j q A , where j + c ( A ) ≤ p . The terms with j = 0 and c ( A ) = p are partof (1.16). The remaining terms are determined by covariant constancy, since any monomial q A that lies in I diff must necessarily have c ( A ) ≥ p . Having determined QSt ( b ), Lemma 6.3 doesthe rest. (cid:3) As a concrete illustration, let’s consider a cubic surface M ⊂ C P , which is a del Pezzo surface,and hence a monotone symplectic manifold. For simplicity, instead of the whole Novikov ring,we will work with a single Novikov variable q , which counts the Chern number of holomorphiccurves. Let’s first take coefficients in Z . Take h to be the first Chern class of M , and h to bethe Poincar´e dual of a point. Computations in [4, 5] show that(6.11) h ∗ h = 3 h + 9 q h + 180 q ,h ∗ h = 36 q h + 252 q . At one point we will use another class in H ( M ), the Poincar´e dual of a Lagrangian sphere,denoted by l . This satisfies(6.12) l ∗ l = − h + 4 q h + 12 q . Example 6.5.
Take the cubic surface with p = 2 (this computation is of the same kind as thosein [18] , only expressed in slightly different language). First of all, (6.13) QSt ( c ) = c ∗ c + tc for all c ∈ H ( M ; F ) .A priori, QSt ( c ) could also have a tq term, which would lie in H ( M ; F ) . This would comefrom classes with c ( A ) = 1 . To get a nonzero output in H ( M ; F ) , one would need to have a stable A -curve going through every point of M . But each A is represented by a unique embedded ( − -sphere, hence the term must vanish, leaving (6.13) .By combining (6.11) , (6.13) , and (6.9) , one gets (6.14) QSt ( h ) = h + ( q + t ) h ,Q Σ h ( h ) = tq∂ q QSt ( h ) + h ∗ QSt ( h ) = ( q + t ) h + q h ,Q Σ h ( h ∗ h ) = tq∂ q Q Σ h ( h ) + h ∗ Q Σ h ( h ) = ( tq + q ) h + q h . Using (6.10) , we get the result announced in Example 1.7: (6.15)
QSt ( h ) = QSt ( h ∗ h + qh ) = Q Σ h ( Q St ( h )) + q QSt ( h )= Q Σ h ( h ∗ h + th ) + q QSt ( h ) = t h . Example 6.6.
Let’s again look at the cubic surface, but now with p = 3 . Here, the fact that wework with a single Novikov variable q will limit the effectiveness of our computation, leading to anincomplete result. As explained in Proposition 6.4, we can use covariant constancy to determinethe quantum Steenrod operations on H ( M ; F ) . In the same way, one can compute Q Σ b ( c ) for b, c ∈ H ( M ; F ) except for the q t term, which lies in H ( M ; F ) . We will only describe theoutcome (code that carries out this computation is available at [15] ): (6.16) QSt ( h ) = − t h , QSt ( l ) = − t l , QΣ l ( l ) = − t h + ( term lying in H ( M ; F ) q t ) . From that one gets, using (6.12) , (6.17) QSt ( h ) = QSt ( l ∗ l − qh ) = Q Σ l ( QSt ( l )) − q QSt ( h )= t h + q t h + ( term lying in H ( M ; F ) q t ) . Note that, unlike the p = 2 case, QSt ( h ) contains a non-classical (quantum) term. We conclude our discussion with a higher-dimensional case: the intersection of two quadricsin C P , which is a monotone symplectic 6-manifold. Let’s first work with Z -coefficients. Theeven degree cohomology has a basis { , h , h , h } , where the subscript denotes the dimension.There is also odd degree cohomology, H ( M ; Z ) = Z , but that will play no role in our argument.We can identify the Novikov ring with Z [[ q ]], but since c ( M ) is twice the positive area generatorof H ( M ; Z ), the formal variable q has degree 4. The quantum product, as computed in [6],satisfies(6.18) h ∗ h = 4( h + q ) ,h ∗ h = h + 2 qh ,h ∗ h = 4 qh + 4 q ,h ∗ h = 2 qh + 3 q . Example 6.7.
Taking our intersection of quadrics, let’s set p = 2 . The classical Steenrodoperations are (6.19) Sq ( h k ) = t k/ h k . uantum Steenrod 33 For h , this is because h = 0 ∈ H ( M ; F ) , which one can read off from the classical term in (6.18) . For h , its Poincar´e dual of is represented by a line C P ⊂ M . The normal bundleof that line has first Chern class ; by the geometric description of Steenrod squares throughStiefel-Whitney classes, this implies vanishing of Sq ( h ) .Since the quantum product with h agrees with its classical counterpart, the cup product with anyelement of H ∗ ( M ; F ) is a covariantly constant endomorphism for the quantum connection. Fromthat, Theorem 1.4, and (6.19) , one gets (6.20) Q Σ h ( c ) = th c + (terms lying in H k ( M ; F ) with k < | c | − ), Q Σ h ( c ) = t h c + q c + (terms lying in H k ( M ; F ) with k < | c | ).Therefore, (6.21) QSt ( h ) = th , QSt ( h ) = t h + q , QSt ( h ) = Q Σ h ∗ h (1) = Q Σ h ( Q St ( h )) = Q Σ h ( t h + q ) = t h + q th . Example 6.8.
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