# Rational Quantum Cohomology of Steenrod Uniruled Manifolds

Semon Rezchikov

(October 2021)

###### Abstract

We show that if a semipositive symplectic manifold $M^{2n}$ is _Steenrod
uniruled_ , in the sense that the quantum Steenrod power of the point class
does not agree with its classical Steenrod power for infinitely many primes,
then the (rational) quantum product on $M$ is deformed. This bridges the gap
between the recent advances towards the Chance-McDuff conjecture utilizing
quantum Steenrod operations, and the natural formulation of the Chance-McDuff
conjecture in terms of rational Gromov-Witten theory.

##  1 Introduction

This paper shows that recent results towards the Chance-McDuff conjecture,
which use equivariant Floer theory to establish results about invariants of
semipositive symplectic manifolds taking values in fields of positive
characteristic, imply the original formulation of the Chance-McDuff conjecture
in terms of rational Gromov-Witten invariants.

Let $(M,\omega)$ be a semipositive symplectic manifold, and let $\phi:M\to M$
be a Hamiltonian diffeomorphism with finitely many periodic points. A natural
way to produce such a $\phi$ is via the flow of a Hamiltonian
$H:M\to\mathbb{R}$ which is the moment map of an $S^{1}$-action on $M$; in
this case, the periodic points for an irrational rotation are just the fixed
points, which are the critical points of the Hamiltonian. McDuff [McD09]
proved Hamiltonian $S^{1}$-manifolds are _strongly uniruled_ in the sense that
there is a nonzero Gromov-Witten $\langle pt,a_{2},a_{3}\rangle_{A}$ with
$A\neq 0$.

Let $\mathbb{F}$ be a field, and let the Novikov ring $\Lambda_{\mathbb{F}}$
consist of the series

$\sum_{A}c_{A}q^{A}\text{ such that }c_{A}\in\mathbb{F},A\in im(\pi_{2}(M)\to
H_{2}(M,\mathbb{Z})),$ (1)

where for each term either $A=0$ or $\int_{\omega}A>0$ and such that for any
real $C$, $\\#\\{c_{A}\neq 0|\int_{\omega}A<C\\}<\infty$. We associate the
grading $|q^{A}|=2c_{1}(A)$, although of course $\Lambda_{\mathbb{F}}$ is only
a filtered ring. Then McDuff’s result implies that the small quantum
cohomology $QH^{*}(M,\Lambda_{\mathbb{Q}})$ is _not_ isomorphic to
$H^{*}(M,\Lambda_{\mathbb{Q}})$ as a ring.

Since a Hamiltonian $H$ generating an $S^{1}$-action must be a perfect Morse
function, the Floer chain complex $CF^{*}(H,\Lambda_{\mathbb{F}})$ of $H$ has
trivial differential, and is thus isomorphic to
$H^{*}(M,\Lambda_{\mathbb{F}})$ as a graded abelian group. This property is
axiomatized via the notion of a _pseudorotation_.

######  Definition 1.

Let $\text{char }\mathbb{F}=0$ if $M$ is not semipositive, and otherwise let
$\mathbb{F}$ be an arbitrary field. We say that a Hamiltonian diffeomorphism
$\phi:M\to M$ is a _pseudorotation_ (see e.g. [She20]) if the set of periodic
points of $\phi$ is finite and coincides with the set of fixed points, if
there is identity of dimensions

$\sum_{x\in
M:\phi(x)=x}\dim_{\Lambda_{F}}HF_{loc}(x,\phi)=\dim_{\Lambda_{F}}H^{*}(M,\Lambda_{F})$

where $HF_{loc}$ is the _local Floer homology_ of a fixed point of $\phi$
([Flo89], [Gin10]), and ever term of the sum above is strictly positive.

A basic expectation about the symplectic topology of manifolds admitting
Hamiltonian diffeomorphisms with finitely many periodic points is the
following

######  Conjecture (Chance-McDuff).

Suppose $\phi$ has finitely many periodic points. Then
$QH^{*}(M,\Lambda_{\mathbb{Q}})\neq H^{*}(M,\Lambda_{\mathbb{Q}})$ as a ring.

Recent partial progress towards this conjecture been restricted to the case
when $\phi$ is a pseudorotation. Using one method [CGG19a], given additional
index-theoretic conditions on the periodic points of $\phi$, one can utilize
combinatorial arguments about index divisibility and iteration of the quantum
product in local Floer cohomology to conclude the existence of nontrivial
rational Gromov-Witten invariatns of $M$. Alternatively, several works have
given conceptual arguments using _equivariant localization in Floer homology_
[Sei15] [SZ22] together with _quantum Steenrod operations_ [Fuk97], [Wil20] to
argue the quantum Steenrod operations cannot agree with their classical
counterparts.

In particular, [She20] and subsequently [She19], [CGG19] establish that if $M$
is a monotone symplectic manifold admitting an
$\mathbb{F}_{2}$-pseudorotation, then $M$ is _$\mathbb{F}_{2}$ -uniruled_, a
notion introduced in [She20]. We review this notion in Section 2; being
$\mathbb{F}$-uniruled implies that for any compatible almost complex structure
on $M$, there is a pseudoholomorphic sphere passing through every point of
$M$. Moreover, an upcoming paper [SSW] will establish that if $M$ is a
semipositive symplectic manifold admitting an $\mathbb{F}_{p}$-pseudorotation
then $M$ is $\mathbb{F}_{p}$-uniruled.

The main theorem of this paper is that such results using equivariant Floer
homology are sufficient to establish the Chance-McDuff conjecture formulated
using rational Gromov-Witten theory.

######  Theorem 1.

Let $(M,\omega)$ be a symplectic manifold of dimension $2n$ which is either
monotone or has minimal Chern number $N>1$, and is $\mathbb{F}_{p}$-uniruled
for infinitely many primes $p$. Then the small rational quantum product on $M$
is deformed, i.e $QH^{*}(M,\Lambda_{\mathbb{Q}})\neq
H^{*}(M,\Lambda_{\mathbb{Q}})$ as a ring. If $M$ is not monotone and has
$N=1$, then the same conclusion holds when $M$ is $\mathbb{F}_{2}$-uniruled.

######  Remark 1.

The proof only uses a single prime $p$, and explicit descriptions of valid $p$
may be obtained from the class $[\omega]$ and the cohomological invariants of
$M$. See Lemma 3 for the main obstruction, which has to do with making sure
one can reduce $[\omega]$ mod $p$.

The proof uses certain structural properties of the quantum Steenrod
operations elaborated in [Wil20], [Wil], [Sei19], and [SW21], which review in
Section 2, along with more basic facts about quantum cohomology. Section 3
shows that if $QH^{*}(M,\Lambda_{\mathbb{Q}})=H^{*}(M,\Lambda_{\mathbb{Q}})$
as rings, then in the total quantum Steenrod power of the fundamental class,
all coefficients of $q^{A}$ must be zero unless $c_{1}(A)<1$. Section 4
reviews the moduli spaces defining rational and equivariant Gromov-Witten
invariants. Section 5 shows that if $c_{1}(A)\leq 0$, then the
pseudoholomorphic spheres in the class $A$ cannot pass through a generic point
of $M$. Combining these propositions, we conclude the proof of the main
theorem below.

###### Proof of Theorem 1.

Suppose that $QH^{*}(M,\Lambda_{\mathbb{Q}})=H^{*}(M,\Lambda_{\mathbb{Q}})$.
By Proposition 1 the there is a prime $p$ such that the non-classical terms in
$Q\Sigma_{[M]}(1)$ are $f_{A}q^{A}$ for $c_{1}(A)<1$; but by Proposition 5,
$f_{A}=0$. Thus $M$ is not $\mathbb{F}_{p}$ uniruled for this $p$. We have
proven the contrapositive. ∎

Acknowledgements. I thank Egor Shelukhin for his interest in this work, for
pointing out an error in an initial version of the argument presented in this
paper, and for his comments on this manuscript. I also thank Denis Auroux for
providing feedback regarding dimension-counting arguments in the style of
Section 5. Additional acknowledgements to be added later.

##  2 Review of Quantum Steenrod Operations

On a semipositive sympletic manifold $M$, the $3$-point GW invariants [MS12]
are multilinear maps

$\langle\cdot,\cdot,\cdot\rangle_{A}:H^{*}(M,\mathbb{Z})^{\otimes
3}\to\mathbb{Z};$ (2)

by multilinearity they factor through $(H^{*}(M,\mathbb{Z})/Tors)^{\otimes
3}$. We can then define

$\langle\cdot,\cdot,\cdot\rangle_{A}:H^{*}(M,\mathbb{F})^{\otimes
3}\to\mathbb{F};$ (3)

by setting $\langle x,y,z\rangle_{A}=0$ whenever one of $(x,y,z)$ does not
land in
$Im(H^{*}(M,\mathbb{Z})/Tors\xrightarrow{\otimes\mathbb{F}}H^{*}(M,\mathbb{F}))$,
and otherwise by lifting $(x,y,z)$ to integral cohomology and applying the
homomorphism $\mathbb{Z}\to\mathbb{F}$ to the resulting Gromov-Witten
invariant.

These invariants give maps

$\begin{gathered}*_{A}:H^{*}(M,\mathbb{F})^{\otimes 2}\to
H^{*-2c_{1}(A)}(M,\mathbb{F})\\\ \int_{M}x*_{A}yz=\langle
x,y,z\rangle_{A}\end{gathered}$ (4)

which when extended $\Lambda_{\mathbb{F}}$-linearly and combined into

$x*y=\sum_{A}(x*_{A}y)q^{A}$ (5)

give the $\Lambda_{\mathbb{F}}$-module $H^{*}(M,\Lambda_{\mathbb{F}})$ the
structure of a commutative $\Lambda_{\mathbb{F}}$-algebra – the _small quantum
cohomology_ $QH^{*}(M,\Lambda_{\mathbb{F}})$ of $M$.

The discussion above establishes the following

######  Lemma 1.

If $\langle x,y,z\rangle_{A}=0$ for all $A\neq 0$ and $\mathbb{F}=\mathbb{Q}$,
then this holds for any other field $\mathbb{F}$.

###### Proof.

If one of $\\{x,y,z\\}$ will does not lift to integral cohomology then the
invariant is zero by definition. If they lift, then the invariant is zero any
of the lifts are torsion classes. If none of the lifts are torsion classes,
then the invariant agrees with the rational Gromov-Witten invariant associated
to the image of the lifts in rational cohomology. ∎

One defines the small quantum connection on $H^{*}(M,\Lambda)[[t]]$ via

$\nabla_{a}\gamma=t\partial_{a}\gamma+a*\gamma.$ (6)

where $*$ is extended $t$-linearly and $|t|=2$. Here $a\in
H^{2}(M,\mathbb{Z})$ and $\partial_{a}q^{A}=(a\cdot A)q^{A}$.

The Quantum Steenrod Operations are certain maps depending on a prime $p$

$Q\Sigma_{b}:H^{*}(M,\mathbb{F}_{p})\to(H^{*}(M,\Lambda)[[t,\theta]])^{*+p|b|}$
(7)

where $b\in H^{*}(M,\mathbb{F}_{p})$ and $|t|=2$, $|\theta|=1$, which
$q$-deform the usual operations in the sense that their reduction modulo the
maximal ideal of $\Lambda_{\mathbb{F}}$ agrees with the operation

$H^{*}(M,\mathbb{F}_{p})\ni x\mapsto St_{p}(b)x$ (8)

where $St_{p}$ is the total Steenrod power

$St_{p}:H^{|b|}(M,\mathbb{F}_{p})\to(H^{*}(M,\mathbb{F}_{p})[[t,\theta]])^{p|b|}.$
(9)

We use the conventions of [Sei19] for $St_{p}$; up to explicit signs and
nonzero constants, the coefficients of the terms of $St_{p}$ are the usual
Steenrod powers $P^{i}$ or their Bocksteins $\beta P^{i}$ (or alternatively
the Steenrod squares $Sq^{i}$ if $p=2$) defined as in [Hat00].

######  Definition 2 ([She20]).

We say that $M$ is $\mathbb{F}_{p}$-uniruled if

$Q\Sigma_{[M]}(1)\neq St_{p}([M])(1)=\pm[M]t^{(p-1)n}.$ (10)

where $[M]$ is the cohomology class Poincare-dual to a point (the “point
class”). See [Sei19, Eq. 10.9] for a formula for the sign.

This implies that for every compatible almost complex structure on $M$ there
is is a pseudoholomorphic sphere through every point of $M$; see Remark 4.

The classical and quantum Cartan relations are the respective identities

$St_{p}(a)St_{p}(b)=(-1)^{c(a,b)}St_{p}(ab),\;Q\Sigma_{b}Q\Sigma_{c}=(-1)^{c(a,b)}Q\Sigma_{b*c},\text{
where }c(a,b)=|a||b|\frac{p(p-1)}{2},$ (11)

where we extend all of the above maps to endomorphisms of
$H^{*}(M,\Lambda)[[t,\theta]]$ by $t,\lambda,q^{A}$-linearity, and we extend
the lower argument of $Q\Sigma$ to take $\Lambda_{\mathbb{F}}$-linear sums by
requiring that

$Q\Sigma_{b}=\sum_{A}q^{pA}Q\Sigma_{b_{A}}\text{ for }b=\sum_{A}b_{A}a^{A}.$
(12)

Recently, Seidel and Wilkins [SW21] have established that _the quantum
Steenrod operations are covariantly constant with respect to the quantum
connection_ , namely that we have an identity

$Q\Sigma_{b}\nabla_{a}=\nabla_{a}Q\Sigma_{b}.$ (13)

Let $I_{diff,p}\subset\Lambda_{\mathbb{F}_{p}}$ be the ideal generated by
those elements $q^{A}$ such that $\partial_{a}q^{A}$ for all $a\in
H^{2}(M,\mathbb{Z})$. Equation (13) immediately implies

######  Lemma 2.

Suppose that $\langle x,y,z\rangle_{A}=0$ for all $A\neq 0$. Then
$Q\Sigma_{b}c=St(b)c\mod I_{diff}$.

###### Proof.

The assumption implies that $\nabla_{a}=t\partial_{a}$. Since $t$ is not a
zero-divisor (13) implies that for any $x\in H^{*}(M,\mathbb{F}_{p})$,

$\partial_{a}Q\Sigma_{b}(x)=Q\Sigma_{b}(\partial_{a}x)=0.$ (14)

∎

##  3 Steenrod Coefficients must be Calabi-Yau

In this section, we prove

######  Proposition 1.

Let $(M,\omega)$ be semipositive, and assume that the small quantum cohomology
of $M$ is classical. Then, if $M$ is monotone or has minimal chern number
$N>1$, then there are infinitely many primes $p$ such that if $q^{A}$ has a
nonzero-coefficient in $Q\Sigma_{[M]}(1)$ for this prime, then $c_{1}(A)\leq
0$. If $M$ is not monotone and has $N=1$, then this conclusion holds for
$p=2$.

We recall that the semipositivity condition (also refered to as _weak
monotonicity_ [HS-Novikov]) means [MS12, Exercise 6.4.3] that either

  1. 1.

$M$ is monotone: $c_{1}(M)(A)=\kappa[\omega](A)$ for some $\kappa>0$, for
every $A\in\pi_{2}(M)$;

  2. 2.

$M$ is spherically Calabi-Yau, i.e. $c_{1}(M)(A)=0$ for every
$A\in\pi_{2}(M)$; or

  3. 3.

$M$ has minimal Chern number $N\geq n-2$.

The proof will require several cases, and in the worst case ($n=3$, $N=1$) we
will use an analytical result proven in the final section. Before we proceed,
we discuss how to make $[\omega]$ into an integral class.

######  Lemma 3.

If $(M^{2n},\omega)$ is a semipositive symplectic manifold then there is a
constant $P>0$ such that for all $p>P$, $\omega$ can be deformed through
semipositive symplectic forms into a symplectic form $\tilde{\omega}$ such
that $[\tilde{\omega}]$ is integral and such that $[\tilde{\omega}]^{n}\neq
0\mod p$.

###### Proof.

Suppose $M$ is monotone or spherically Calabi-Yau. Write
$H^{2}_{S}(M)=Im(\pi_{2}(M)\to H_{2}(M,\mathbb{Z})/Tor)$. Write
$H^{2}_{S}(M)=Hom(H^{2}_{S}(M),\mathbb{Z})$, and
$H^{2}_{S}(M)^{\perp}=\ker(Hom(H_{2}(M,\mathbb{Z})/Tors,\mathbb{Z})\to
H^{2}_{S}(M)$. Then $H^{2}(M,\mathbb{R})=H^{2}_{S}(M)\otimes\mathbb{R}\oplus
H^{2}_{S}(M)^{\perp}\otimes\mathbb{R}$. Under this decomposition write
$[\omega]=[\omega]_{S}+[\omega]_{S}^{\perp}$.

If $M$ is monotone or spherically Calabi-Yau, since $c_{1}(M)$ is integral we
can let $\tilde{\omega}_{1}$ to be a rescaling of $\omega$ so that
$[\tilde{\omega}_{1}]_{S}$ an integral class. In general, the set of
cohomology classes of symplectic forms deformation equivalent to $\omega$
projects to a subset of $H^{2}(M,\mathbb{R})$ which contains an open subset
$U$ of $[\omega]$. The image of $H^{2}_{S}(M)^{\perp}$ in
$H^{2}_{S}(M)^{\perp}\otimes\mathbb{R}$ is dense; thus, we can perturb
$\tilde{\omega}_{1}$ among semi-positive symplectic forms with cohomology
classes in
$U\cap([\tilde{\omega}_{1}]_{S}+H^{2}_{S}(M)^{\perp}\otimes\mathbb{R})$ to a
symplectic form $\tilde{\omega}$ such that
$[\tilde{\omega}_{1}]_{S}=[\tilde{\omega}_{S}]$, while
$[\tilde{\omega}]^{\perp}_{S}$ is rational. Thus $[\tilde{\omega}]$ is
rational. Rescaling $\tilde{\omega}$ by an integer we can assume that
$\tilde{\omega}$ is integral. On the other hand, if $N\geq n-2$ then then any
other symplectic form on $M$ is semipositive; therefore, we can also deform
$\omega$ to a rational symplectic form through semipositive symplectic forms
by a small perturbation, and subsequently rescale to get an integral
symplectic form.

Write $[\tilde{\omega}]^{n}=K[M]$; since $\tilde{\omega}$ is symplectic,
$K\neq 0$. Then $P=K$ will do. ∎

######  Remark 2.

Continuing on Remark 1, the factors of $p$ are allowed are a problem in the
approximation of $\omega$ by rational classes, and in concrete cases, one may
follow the reasoning of the lemma above to derive explicit primes $p$ for
which the conclusion of the lemma is valid. In the remainder of the proof we
will have no further conditions on $p$, unless $N=1$ and $n=3$, where we will
change our strategy and take $p=2$.

######  Remark 3.

The rational Gromov-Witten invariants of a semipositive symplectic manifold
are independent of symplectic form up to deformation through semipositive
symplectic forms [MS12]. This holds because one can use almost complex
structures _tamed_ by $\omega$ to define the Gromov-Witten moduli spaces, and
because the set of almost complex structures tamed by $\omega$ is open in the
set of almost complex structures and correspondingly the set of symplectic
forms taming a fixed almost complex structure is open in the space of
symplectic forms. The quantum Steenrod powers as defined in [SW21] require one
to choose $\omega$-compatible almost complex structures; but for the Gromov
compactness results used in the paper, $\omega$-tame almost complex structures
would suffice. Thus, one can show that the deformations of Lemma 3 preserve
the property of the underlying symplectic manifold being Steenrod uniruled for
any given prime $p$.

###### Proof of Proposition 1.

If $c_{1}(M)=0$ the conclusion follows.

Otherwise we will prove the contrapositive; thus, assume that $\langle
x,y,z\rangle_{A}=0$ for all $A\neq 0$. Then quantum multiplication agrees with
the classical cup product.

First, suppose $\omega$ is integral. Note that (reducing $[\omega]$ mod $p$),
$St_{p}([\omega])=G_{p}([\omega])$ for $G_{p}(z)\in\mathbb{F}_{p}[z]$ an
explicit polynomial. Indeed, there is a map $f:M\to\mathbb{C}P^{\infty}$ such
that $f^{*}c_{1}=[\omega]$, where
$H^{*}(\mathbb{C}P^{\infty},\mathbb{F}_{p})=\mathbb{F}_{p}[c_{1}]$. The claim
follows because the Steenrod operations are natural.

We will repeatedly utilize the following

######  Lemma 4.

Let $a\in H^{2}(M,\mathbb{F})$; then if $p>n$ we can write

$Q\Sigma_{a}(b)=St_{p}(a)b+\sum_{i}f_{i}a^{A_{i}}+\sum_{j}g_{j}q^{B_{j}}$ (15)

where $c_{1}(A_{i})=0$, $c_{1}(B_{j})=p$, $A_{i}\neq 0$, and
$f_{i},g_{j}\in(t,\theta)H^{*}(M,\mathbb{F})[[t,\theta]]$. If $b=1$ or $N>1$
then $g_{j}=0$ for each $j$.

###### Proof.

We have that $Q\Sigma_{a}b=St(a)b+f$ for $f\in I_{diff,p}$ of degree $2p$.
Write $f=\sum_{i}f_{i}q^{A_{i}}+\ldots$ for $\partial_{a}q^{A_{i}}=(a\cdot
A_{i})a^{A_{i}}=0$, $A_{i}\neq 0$, and $f_{i}\in
H^{*}(M,\mathbb{F})[[t,\theta]]$. Write $A_{i}=pB_{i}+T_{i}$ where $T_{i}$ is
a torsion class, and suppose $f_{i}\neq 0$.

We see that
$|f_{i}q^{A_{i}}|=|f_{i}|+2c_{1}(pB_{i}+T_{i})=|f_{i}|+2pc_{1}(B_{i})=|b|+2p$.
Dividing by $2p$ and using that $p>n$ we see that $c_{1}(B_{i})\leq 1$. So if
$N>1$ we have that $c_{1}(A_{i})\leq 0$.

By Remark 5 we actually have that if $c_{1}(A_{i})\geq 0$ for any $i$. So we
have established the theorem when $N>1$.

Moreover, writing $f_{i}\in f^{0}_{i}+(t,\theta)$ we have that $f^{0}_{i}=0$
since

$\sum_{i}f^{0}_{i}a^{A_{i}}=a^{*n}*b-a^{n}b=0$ (16)

since again, the quantum product is classical. By degree considerations this
proves the last sentence when $b=1$.

∎

In the case where $M$ is monotone or when $N>1$, we apply Lemma 3, and replace
$\omega$ with $\tilde{\omega}$ as in the lemma. By Remark 3, this preserves
the rational Gromov-Witten invariants of $M$, as well as the property of being
$\mathbb{F}_{p}$-Steenrod uniruled. We then have that
$[\omega]^{*n}=[\omega]^{n}=K[M]$. Note that
$Q\Sigma_{[M]}(1)=(1/K)Q\Sigma_{[\omega]^{*n}}(1)$, which makes sense since
$0\neq K\in\mathbb{F}_{p}$ is invertible.

We have by Lemma 4 that

$Q\Sigma_{[\omega]}(1)=St_{p}([\omega])+\sum_{i}f_{i}q^{A_{i}}=G_{p}([\omega])+\sum_{i}f_{i}q^{A_{i}},\;c_{1}(A_{i})=0,A_{i}\neq
0.$ (17)

This is the base case of an induction on $k$ establishing that

$Q\Sigma_{[\omega]^{k}}(1)=Q\Sigma_{[\omega]}^{k}(1)=St([\omega]^{k})+\sum_{i}f_{i}q^{A_{i}},\;c_{i}(A_{i})=0,A_{i}\neq
0.$ (18)

In the case that $M$ is monotone we further have that $f_{i}=0$ in the base
case and the inductive hypothesis.

Now, for any $a\in H^{2}(M,\mathbb{Z})$, there is an operation [SW21]

$\Pi_{a,b}:H^{*}(M,\mathbb{F}_{p})\to(H^{*}(M)[[t,\theta]])^{*+|a|-2+p|b|}$
(19)

satisfying

$t\Pi_{a,b}(c)=Q\Sigma_{b}(a*c)-a*Q\Sigma_{b}(c).$ (20)

Moreover, it also satisfies

$\Pi_{a,b}(c)=\partial_{a}Q\Sigma_{b}(c)=0.$ (21)

Since $t$ is not a zero-divisor we conclude that
$Q\Sigma_{b}(\omega^{k})=\omega^{k}Q\Sigma_{b}(1)$ for any $k\geq 0$.

Thus assuming (18) we have that

$Q\Sigma_{[\omega]}Q\Sigma_{[\omega]}^{k}(1)=Q\Sigma_{[\omega]}(G_{p}({[\omega]})^{k})+\sum_{i}q^{A_{i}}Q\Sigma_{[\omega]}(f_{i})=G_{p}({[\omega]})^{k}Q\Sigma_{{[\omega]}}(1)+\sum_{i}q^{A_{i}}Q\Sigma_{[\omega]}(f_{i}).$
(22)

In the case that $N>1$ we see that $g_{i}=0$ in Lemma 4; applying that lemma
to the terms $Q\Sigma_{[\omega]}(f_{i})$ in formula above we see that the
inductive hypothesis is preserved. On the other hand, if we are in the
monotone case the $f_{i}$ were zero and thus applying the formula from the
base case we see that the inductive hypothesis has been preserved as well.
This establishes the induction and for $k=n$ proves the theorem when $M$ is
monotone or when $N>1$.

The remaining case is $N=1$ and $M$ is not monotone, which implies that $2\leq
n\leq 3$.

Suppose $n=2$. Then by Poincare duality we can find a pair of classes $a,b\in
H^{2}(M,\mathbb{F}_{p})$ such that $ab=[M]$. Choosing $p>n=2$, we use quantum
Cartan to write $Q\Sigma_{[M]}(1)=Q\Sigma_{b}Q\Sigma_{a}(1)$ and conclude the
proposition by applying Lemma 4 twice.

The final and trickest case is if $n=3$. In this case we will specialize to
$p=2$. By Poincare duality we can find a pair of classes $a\in
H^{4}(M,\mathbb{Z})$, $b\in H^{2}(M,\mathbb{Z})$ such that $ab=[M]$
integrally; and thus the same holds for the mod-$2$ reductions of these
classes.

We will decompose $Q\Sigma_{[M]}(1)=Q\Sigma_{b}Q\Sigma_{a}(1)$. Let us study
the terms in

$Q\Sigma_{a}(1)=St_{2}(a)+\sum_{i}f_{i}q^{A_{i}},A_{i}\neq 0.$ (23)

Then $St_{2}(a)=t^{2}Sq_{0}(a)+t\theta Sq_{1}(a)+Sq_{2}(a).$ Since $a$ is
integral, $Sq_{1}(a)=0$ since $Sq_{1}$ is the Bockstein. For the remaining
terms we have $|f_{i}q^{A_{i}}|=4$, so arguing as in Lemma 4 we write
$A_{i}=pB_{i}+T_{i}$ where $T_{i}$ is torsion, and conclude that
$|f_{i}|+4c_{1}(B_{i})=8$. We write
$f_{i}=\theta^{x_{i}}t^{y_{i}}f^{\prime}_{i}$ with $f^{\prime}_{i}\in
H^{*}(M,\mathbb{F}_{2})$. Then $|x_{i}|+|y_{i}|>0$, and in particular
$|f_{i}|>0$. Thus $c_{1}(B_{i})\leq 1$. If $c_{1}(B_{i})=1$ then
$|f^{\prime}_{i}|\leq 3$. We will establish by an analytical argument in
Section 5 that

######  Proposition 2.

In the situation described above, $f_{i}=0$ for each $i$.

Given this proposition we conclude by writing $Sq_{2}(a)=r[M]=rba$, and
utilizing (20), (21) to compute that

$Q\Sigma_{[M]}(1)=Q\Sigma_{b}((t^{2}+rb)a)=(t^{2}+rb)Q\Sigma_{b}a=(t^{2}+rb)\left(Sq_{2}(b)a+\sum_{i}f_{i}q^{2B_{i}+T_{i}}\right)$
(24)

where $B_{i},T_{i}\in H_{2}(M,\mathbb{Z})$ with $B_{i}$ torsion-free and
$T_{i}$ torsion. By degree considerations as above we conclude that
$c_{1}(B_{i})\leq 1$, and if $c_{1}(B_{i})=1$ then
$f_{i}=\theta^{x_{i}}t^{y^{i}}f^{\prime}_{i}$ with $f^{\prime}_{i}\in
H^{*}(M,\mathbb{F}_{2})$ and $|f^{\prime}_{i}|\leq 3$. We then invoke one more
proposition proven in Section 5:

######  Proposition 3.

In the situation above, if
$Q\Sigma_{[M]}(1)=\sum_{i}\theta^{x_{i}}t^{y_{i}}f^{\prime}_{i}q^{A_{i}}$ with
$c_{1}(A_{i})=2$ then $|f^{\prime}_{i}|>3$.

Thus $c_{1}(2B_{i}+T_{i})\leq 0$ and we have proven the proposition.

∎

##  4 Rational and Equivariant Gromov Witten Invariants

In this section we review the definitions of the moduli spaces giving rise to
the usual Gromov-Witten invariants as well as the quantum Steenrod powers.

Let $f$ be a Morse function with critical point set $Crit(f)$, and let $g$ be
a metric such that the gradient flow of $f$ is Morse-Smale. If $x$ is a
critical point of $f$ with Morse index $|x|$, then in our conventions the
stable manifold $W^{s}(x)$ has dimension $|x|$, while the unstable manifold
$W^{u}(x)$ has dimension $2n-|x|$. Write $\mathbb{F}_{x}$ for the
$1$-dimensional $\mathbb{F}$-vector space generated by the two orientations of
$W^{s}(x)$ modulo the relation that the sum of the orientations is zero. The
Morse complex is

$CM^{k}(f)=\bigoplus_{|x|=k}\mathbb{F}_{x}$ (25)

and it has a cohomological differential such that its cohomology computes
$H^{*}(M,\mathbb{F})$.

Write $\mathcal{J}(M)$ or for the set of compatible almost complex structures
on $M$ and write $\mathcal{J}(S^{2},M)$ for the set of smooth maps from
$S^{2}$ to $\mathcal{J}(M)$.

### Rational GW theory.

Given $A\in\operatorname{im}(\pi_{2}(M)\to H_{2}(M,\mathbb{Z}))$ and an
element $J_{z}\in\mathcal{J}(S^{2},M)$ with $z\in S^{2}$, we have the moduli
space

$\mathcal{M}_{0,3}(M,A,J_{z})=\\{u:S^{2}\to M;\bar{\partial}_{J_{z}}u=0\\}.$
(26)

This is equipped with maps

$ev_{p}:\mathcal{M}_{0,3}\to M\text{ for each }p\in S^{2}$ (27)

where we will choose $p\in\\{0,1,\infty\\}$. Write
$Aut(\mathbb{P}^{1},\\{S\\})$ for the group of automorphisms of
$\mathbb{P}^{1}$ which fix $S\subset\mathbb{P}^{1}$. If $J_{z}$ is independent
of $z$ (i.e., $J_{z}\in\mathcal{J}(M)$) then we also have moduli spaces
$\overline{\mathcal{M}}_{0,2}$ and $\overline{\mathcal{M}}_{0,1}$ defined by
quotienting $\overline{\mathcal{M}}_{0,3}$ by
$Aut(\mathbb{P}^{1},\\{\infty\\})$ and $Aut(\mathbb{P}^{1},,\\{\infty,1\\})$,
respectively.

Each of the above moduli spaces is contained in its Gromov-Kontsevich
compactification, and contains the subspace of simple maps:

$\mathcal{M}^{*}_{0,k}(M,A,J_{z})\subset\mathcal{M}_{0,k}(M,A,J_{z})\subset\overline{\mathcal{M}}_{0,k}(M,A,J_{z})$
(28)

The methods of [MS12] can be used to establish

######  Proposition 4.

There exist $J_{z}$ such that

$ev_{0}\times ev_{1}\times ev_{\infty}:\mathcal{M}^{*}_{0,3}(M,A,J_{z})\to
M\times M\times M$

is a pseudocycle [MS12] of dimension $2n+2c_{1}(A)$.

The rational Gromov Witten invariant $\langle a,b,c\rangle_{A}$ is defined by
defining the map on generators of $CM^{*}(f)$ via

$\langle x,y,z\rangle_{A}=\\#\mathcal{M}^{*}_{0,3}(M,A,J_{z})\times_{M\times
M\times M}W^{u}(x)\times W^{u}(y)\times W^{u}(z).$ (29)

The cardinality of the set on the right hand side is finite whenever
$|x|+|y|+|z|-2c_{1}(A)=2n$, and the points are counted with sign according to
the canonical orientation on the fiber product; otherwise the count is defined
to be zero.

### Equivariant GW theory.

The terms the quantum Steenrod operation $Q\Sigma_{a}$ can also be described
in a similar way [SW21]. Let

$S^{\infty}=\\{(w_{0},w_{1},\ldots)|w_{i}\in\mathbb{C},w_{i}=0\text{ for
}i>>0\\}=\bigcup_{k\geq 0}S^{k}.$ (30)

There is an action of $\mathbb{Z}/p$ on $S^{\infty}$ where the generator
$\sigma$ acts by multiplication by $\zeta=e^{2\pi i/p}$. There are smooth
manifolds with corners $\Delta_{i}\subset S^{\infty}$ homeomorphic to disks;
the $\Delta_{i}$ together with their images under the $\mathbb{Z}/p$ action
give give $S^{\infty}$ the structure of a free $\mathbb{Z}/p$-CW complex.

There is an action of $\mathbb{Z}/p$ on $\mathbb{P}^{1}$ generated
multiplication by $\zeta$ and we write

$C=\mathbb{P}^{1},z_{C,0}=0,z_{C,1}=\zeta^{1/2},z_{C,2}=\zeta^{3/2},\ldots,z_{C,p}=\zeta^{-1/2},z_{C,\infty}=\infty.$
(31)

Given an element $J_{z}\in\mathcal{J}(S^{2},M)$, write $Hom(TS^{2},TM)$ for
the space of linear maps maps from $TS^{2}$ to $TM$ when each bundle is pulled
back to $S^{2}\times M$. Note that there is an action of $\sigma$ on
$\mathcal{J}(S^{2},M)$ and on $Hom(TS^{2},TM)$ by pullback.

The operation $Q\Sigma_{a}$ is defined by choosing perturbation data

$\begin{gathered}\\{J_{z,w}\\}_{w\in
S^{\infty}}\in\mathcal{J}(S^{2},M),\\{\eta_{z,w}\\}_{w\in S^{\infty}}\in
Hom(TS^{2},TM),\\\
J_{\sigma(z),w}=J_{z,\sigma(w)},\;\eta_{\sigma(z),w}=\eta_{z,\sigma(w)},\;\eta_{z,w}i=-J_{z}\eta_{z,w}.\end{gathered}$
(32)

which vary smoothly on each $\Delta_{i}$, and defining for each $A\in
H_{2}(M,\mathbb{Z})$, $x_{0},\ldots,x_{p},x_{\infty}\in Crit(f)$, the moduli
space

$\mathcal{M}_{A}(\Delta_{i}\times
C,x_{0},\ldots,x_{p},x_{\infty})=\left\\{\begin{array}[]{c|l}w\in\Delta_{i}\setminus\partial\Delta_{i},&\bar{\partial}_{J_{w,z}}u=\eta_{w,z},u_{*}[C]=A\\\
u:C\to M&u(z_{C,i})\in W^{u}(x_{i}),i=0,\ldots,p;u(z_{C,\infty})\in
W^{s}(x_{\infty})\end{array}\right\\}.$ (33)

For generic choices of perturbation data this moduli space has dimension

$i+2c_{1}(A)+|x_{\infty}|-\sum_{i=0}^{p}|x_{i}|.$ (34)

We then define

$\Sigma_{A}^{i}:CM^{*}(f)\otimes CM^{*}(f)^{\otimes p}\to
CM^{*-i-c_{1}(A)}(f)$ (35)

by using (33) to define this map’s coefficients. This descends to a map on
homology, and finally if $a,b\in H^{*}(M,\mathbb{F})$, we have

$Q\Sigma_{a}(b)=(-1)^{|a||b|}\sum_{A}q^{A}\sum_{k}\Sigma_{A}^{2k}(a,b,\ldots
b)t^{k}+(-1)^{|a|+|b|}\Sigma_{A}^{2k+1}(a,b,\ldots b)t^{k}\theta.$ (36)

##  5 Uniruling Cohomology Classes

In this section we first prove an equivariant symplectic incarnation of a well
known fact in algebraic geometry, that a projective variety cannot be uniruled
by spheres of low Chern number:

######  Proposition 5.

Write

$Q\Sigma_{[M]}(1)=f_{0}+\sum_{A\neq 0}f_{A}q^{A}.$ (37)

If $f_{A}\neq 0$ then $c_{1}(A)>1$.

###### Proof.

To define Morse cohomology let us use a Morse function with a unique minimum
$p_{-}$ and a unique maximum $p^{+}$. We choose a sequence of perturbation
data $J^{i}_{z,w},\eta^{i}_{z,w}$ as in (32) which limit to
$J^{\infty}_{z,w}=J^{\infty}\in\mathcal{J}(M),\eta^{\infty}_{z,w}=0$; If
$f_{A}$ is nonzero then the moduli spaces (33) defining the corresponding
count are nonempty for all $i$. Thus, the Gromov compactification of the
corresponding moduli space for the perturbation data $(J^{\infty}_{z,w},0)$ is
also nonempty. Let

$\mathcal{M}_{A}(C,x_{0},\ldots,x_{p};x_{\infty})=\left\\{\begin{array}[]{c|l}u:C\to
M&\bar{\partial}_{J^{\infty}}u=0,u_{*}[C]=A\\\ &u(z_{C,i})\in
W^{u}(x_{i}),i=0,\ldots,p;u(z_{C,\infty})\in
W^{s}(x_{\infty})\end{array}\right\\}$ (38)

and let $\overline{\mathcal{M}}_{A}(C,x_{0},\ldots,x_{p};x_{\infty})$ be its
Gromov compactification. Writing
$f_{A}=\theta^{f_{A}^{\theta}}t^{f_{A}^{t}}f^{\prime}_{A}$ with
$f^{\prime}_{A}\in H^{*}(M,\mathbb{F})$ we thus have that for some $x\in
Crit(f)$ whose coefficient for a chain level representative of
$f^{\prime}_{A}$ is nonzero, we have that

$\overline{\mathcal{M}_{A}(\Delta_{f_{A}^{\theta}+2f_{A}^{t}}\times
C,p_{-},p_{+},\ldots,p_{+},x)}=\Delta_{f_{A}^{\theta}+2f_{A}^{t}}\times\overline{\mathcal{M}}_{A}(C,p_{-},p_{+}\ldots,p_{+};x)$
(39)

is nonempty, where the left hand side is defined using using the perturbation
data $(J^{\infty}_{z,w},0)$. Thus in particular we have that the Gromov
compactification $\overline{\mathcal{M}}_{A}(C^{\prime};p_{+},\ldots,p_{+})$
of

$\mathcal{M}_{A}(C^{\prime};p_{+},\ldots,p_{+})=\left\\{\begin{array}[]{c|l}u:C\to
M&\bar{\partial}_{J^{\infty}}u=0,u_{*}[C]=A\\\
&u(z_{C,i})=p_{+},i=1,\ldots,p;\end{array}\right\\}$ (40)

(where $C^{\prime}$ of course refers to $C$ with $z_{C,0}$ and $z_{C,\infty}$
forgotten) is nonempty.

######  Remark 4.

This standard argument implies that for arbitrary $J^{\infty}$, there is a
$J^{\infty}$-holomorphic sphere through any $p^{+}\in M$ if $M$ is
$\mathbb{F}_{p}$-uniruled for some $p$. This is explained and first stated in
[She20, Remark 2].

We will argue that this moduli space must be empty if we choose $J^{\infty}$
generically.

Each component of the moduli space corresponds to a tree with a root vertex;
the root vertex can be identified with $C$, and each vertex corresponds to a
pseudoholomorphic sphere for $J^{\infty}$, Moreover, if we collapse all non-
root spheres then the points $z_{C,1},\ldots,z_{C,p}$ must map to
corresponding points on $C$. In particular these points lie in distinct tree
components.

We will further stratify
$\overline{\mathcal{M}}_{A}(C^{\prime};p_{+},\ldots,p_{+})$ by additionally
considering how many of the marked points $z_{C,1},\ldots,z_{C,p}$ coincide
upon replacing the root component with the $J^{\infty}$-holomorphic sphere
that it may cover. The evaluation maps to $M$ used to define the intersection
of the corresponding space of curves with the stable and unstable manifolds
factor through smooth manifolds of maps in which we replace each sphere in the
tree with the simple sphere that it covers; we will argue that the dimension
of this manifold is too low for each stratum of
$\overline{\mathcal{M}}_{A}(C^{\prime};p_{+},\ldots,p_{+})$ to be nonempty.

On a semi-positive symplectic manifold, the Chern number of any
$J^{\infty}$-holomorphic sphere is non-negative. Since the total Chern number
of the tree is at most $1$, at most one sphere is Chern number $1$ (and is
thus a simple sphere) while the remaining spheres are Chern number zero. In
particular if there is a nonempty stratum then $c_{1}(A)\geq 0$.

######  Remark 5.

The same method of argument shows that for any $a,b\in H^{*}(M,\mathbb{F})$,
if $Q\Sigma_{a}(b)=f_{0}+\sum_{A\neq 0}f_{A}q^{A}$ then $f_{A}\neq 0$ implies
$c_{1}(A)\geq 0$.

We note that adding spheres of Chern number zero to the tree always decreases
the dimension of the associated manifold. If the Chern number of the root
component is $1$ then it is simple; in this case it suffices to upper bound
the dimension of the configuration where the only sphere is the root
component, which has dimension $2n+2-p(2n)<0$ (unless $n=1$, in which case the
computation of [Wil20] for $M=S^{2}$ together with Remark 4 makes it possible
to verify the theorem directly).

If the root component is not simple then its Chern number is zero and it also
covers a sphere of zero Chern number, which be simple or constant. Whether it
is simple or constant turns out not to matter for the dimension computation.
Suppose the total Chern number is $1$. We reduce the case where the Chern
number of the root component is zero to the case where either the sphere of
Chern number $1$ is in a tree component that contains none of the $z_{C,i}$,
or that it is; and there are no other holomorphic spheres. We then divide the
cases into the number of distinct points there are after taking the images of
the $z_{C,i}$ by the covering map; the worst case is manifestly when all the
points are collapsed to one. So there are only two cases to check and they
both have negative expected dimension.

If the total Chern number is $0$ then the same reasoning applies but we have
one more case to check: namely, the case when there are no components except
the root component, which is a cover of a Chern number zero sphere on which
all the marked points $z_{C,i}$ have been sent to a single point. In this case
it is crucial that we have taken $J^{\infty}$ to be domain independent; the
expected dimension of this configuration is $2n-2n-4<0$, where the $4$ comes
from the automorphisms of the final curve that fix the single marked point. ∎

Next, we prove the final technical proposition claimed in Section 3:

###### Proof of Proposition 3.

We argue exactly as above. The argument using Gromov compactness shows that
the Gromov compactification
$\overline{\mathcal{M}}_{A}(C^{\prime\prime};p_{+},\ldots,p_{+};x)$ of

$\mathcal{M}_{A}(C^{\prime\prime};p_{+},\ldots,p_{+})=\left\\{\begin{array}[]{c|l}u:C\to
M&\bar{\partial}_{J^{\infty}}u=0,u_{*}[C]=A\\\
&u(z_{C,i})=p_{+},i=1,\ldots,p;u(z_{C,\infty})\in
W^{s}(x)\end{array}\right\\}$ (41)

(where $C^{\prime\prime}$ refers to $C^{\prime}$ with $z_{C,0}$ and
$z_{C,\infty}$ forgotten) is nonempty for some $x\in Crit(f)$ with nonzero
coefficient in a chain-level representative of $f^{\prime}_{i}$. Because
$\overline{W^{s}}(x)$ is a manifold with corners of dimension equal to that of
$W^{s}(x)$, for the purpose of the dimension computation we can assume that
the codimension constraint on $z_{C,\infty}$ is as if $z_{C,\infty}$ was
constrained to lie strictly in $W^{s}(x)$, since the corresponding moduli
spaces where $z_{C,\infty}$ maps to lower-dimensional strata of
$\overline{W^{s}}(x)$ are only easier to rule out. We note also that
$W^{s}(x)\cap\\{p_{+}\\}=\emptyset$.

If the root component is simple and has Chern number $2$ then it suffices to
check the case where there are no additional Chern number $0$ spheres
attached, and this case is ruled out by a dimension count. If the root
component is a point then either there are two Chern number $1$ spheres, in
which case we are in a moduli space of dimension strictly lower than that
where the root component has Chern number $1$; or there is one Chern number
two sphere, and we can reduce to the case where are no other spheres. The
Chern number $2$ sphere cannot be attached to an input since
$W^{s}(x)\cap\\{p_{+}\\}=\emptyset$; so it must be attached to the output. In
this case, since $W^{s}(x)\cap\\{p_{+}\\}=\emptyset$, even if the sphere is a
multiple cover, the covering map must still send the two marked points to
distinct points, and so the dimension computation shows that this
configuration cannot exist as $10-2-3-6<0$. If the root component has Chern
number $1$, then all other spheres are Chern number zero or simple, and for
the purpose of the dimension count we can assume that there is only one more
sphere of Chern number $1$ (which is then also simple); there are two cases
and each are ruled out by counting dimensions. Finally, if the root component
is a double cover then all other spheres have Chern number zero, so we can
assume there are no other spheres; as before the double cover can collapse the
two inputs but cannot collapse an input and an output, and the dimension count
$10-2-3-6<0$ excludes this configuration as well. ∎

Finally, we prove, the remaining proposition used in Section 3:

###### Proof of Proposition 2.

We have $p=2$ fixed. The computation is similar to the previous one: we will
choose $(J^{i}_{z,w},\eta^{i}_{z,w})$ converging to some
$(J^{\infty}_{z,w}=J^{\infty},0)$ and analyze the resulting strata of the
limit. However, to exclude one of the possible strata we will have to choose a
particular Morse function to calculate cohomology, and a more careful
constraint on $J^{\infty}$ and on the $J^{i}_{z,w}$ and $\eta^{i}_{z,w}$. We
will have $x\in Crit(f)$ have nonzero coefficient in a chain-level
representative of $f^{\prime}_{i}$, as before.

Specifically, by [Tho54, II.27], the Poincare dual homology class to $a$ can
be represented by an oriented $2$-dimensional closed submanifold $N$ of $M$;
by attaching a local handle to $N$ we can assume that the genus of $N$ is
positive. By the $h$-principle it can be represented by a symplectic
submanifold [EM02]. We choose the Morse function $f$ such that there is a
single critical point $y$ representing $a$ in integral cohomology, such that
the closure of $W^{u}(y)$ is exactly $N$ and such that
$N\cap\overline{W^{s}(x)}$ has dimension $1$: this can be done by choosing a
Morse function on $N$ with a single maximum, extending by a negative definite
quadratic form on a tubular neighborhood $U$ of $N$ with closure $\bar{U}$,
and then extending to a Morse function on all of $M$. We require that
$J^{\infty}|_{\bar{U}}$ is integrable, with $N$ a complex submanifold; the
projection from $U$ to $N$ holomorphic, making $\bar{U}$ into a closure of a
neighborhood of zero of a holomorphic vector bundle structure on the normal
bundle to $N$. We additionally fix the germ of a complex hypersurface $D$
containing $N$. Finally, we require that the $J^{i}_{z,w}$ agree with
$J^{\infty}$ on $\bar{U}$ for all $w$ and $i$, and that $\eta^{i}_{z,w}$ is
zero on $U$ for all $w$ and $i$. This constraint does not violate
transversality because _there are no $J^{\infty}$ holomorphic spheres
contained entirely in $\bar{U}$_. Indeed, if there there was such a sphere,
its projection to $N$ would give a holomorphic map to a surface of positive
genus, which would have to factor through a point of $N$; so the curve would
lie in a fiber of the projection to $N$, which would define a bounded
holomorphic map from a sphere to a $\mathbb{C}^{2}$, contradicting Liouville’s
theorem.

Finally, we require that $J^{\infty}$ is chosen such that the _moduli space of
simple holomorphic spheres with two marked points, one passing through $N$ and
one passing through a stratum of $\overline{W^{s}(x)}$, and having contact of
order $2$ to $D$_ is of expected dimension. The notion of spheres having
contract of positive order to a holomorpic divisor was introduced in [CM07],
[CM17], and an elaborate theory for the case where the divisor is locally
defined near a point was established in [MS21]. The transversality results
[CM07, Prop. 6.9] and [CM17, Prop 3.1] (see also [MS21, Lemma 2.2.1]) suffice
to show that this moduli space is of expected dimension for generic
$J^{\infty}$; the dimension of this moduli space is $2$ less than the
dimension of the corresponding moduli space of simple spheres with no order of
contact contraint at $D$.

As in the proof of Proposition 3 we will assume that the constraint on the
output is a codimension
$6-|f^{\prime}_{i}|=\operatorname{codim}W^{s}(x)=\operatorname{codim}\overline{W^{s}(x)}\geq
3$ constraint since $\overline{W^{s}(x)}$ is a manifold with corners
compactifying $W^{s}(x)$.

If the root component has Chern number $2$ then all other components have
Chern number zero, and so we can assume the only component is the root
componnet for the purpose of upper bounding the dimension. If the root
component is simple then the moduli space has dimension
$6+4-4-4-(6-|f^{\prime}_{i}|)<0$. If the root component is a two-fold cover
then the resulting curve is simple; as before we can reduce to the case where
there are no other spheres.

If the images of $z_{C,1}$ and $z_{C,2}$ are distinct the dimension is only
lower than in the previous case. If the images of $z_{C,1}$ and $z_{C,2}$
agree then we upper bound the dimension by $6+2-2-(6-|f^{\prime}_{i}|)-4)<0$.

If the Chern number of the root component is $1$ then it is simple. We can
reduce to the case where there is exactly one other sphere of Chern number
$1$, attahed to the root component either away from the marked points or at
one of the marked points; all three cases have negative expected dimension.

Suppose the root component is a constant map. We reduce to two cases: either
there is one additional sphere of Chern number $2$ and no other spheres, or
there there are two additional spheres attached at different points. If there
is a sphere of Chern number $2$ then either it is attached to an input, in
which case if it is simple then lies in a moduli space of dimension bounded by
$6+4-2-4-5$ (where we have used that one of the marked points on this sphere
lies in the _intersection_ of the stable manifold of the output with the
unstable manifold of the input). Alternatively, it is attached to the output;
but then by [CM07, Lemma 7.2b] this sphere must have contact of order $2$ to
the divisor $D$ containing $N$, and thus lies in expected dimension $2$ lower
than if it was only required to be contained in $N$; for a usual transverse
intersection the expected dimension (if this sphere is simple) is
$6+4-2-(6-|f^{\prime}_{i}|)-4\leq 1$, and so with the tangency constraint we
have a negative upper bound. If this sphere is a double-cover then it might
collapse the $N$ constraint and the $W^{s}(x)$ constraint, but then the unique
marked point lies in $N\cap W^{s}(x)$ which has dimension $1$ and thus the
expected dimension of this configuration is negative as well. This latter
argument also resolves the case where the Chern number $2$ sphere was attached
to an input and was a double cover.

Finally, suppose the Chern number of the root component is zero. Then we can
again reduce to the case where there is additionally just one Chern number $2$
sphere or $2$ Chern number $1$ spheres.

If the simplification of the root component sends $z_{C,1}$ and $z_{C,2}$ to
distinct points just the root component together with its intersections with
the two input cycles cannot exist for dimension reasons.

If the simplification of the root component sends $z_{C,1}$ and $z_{C,2}$ to
the same point then: in the case with one more sphere of Chern number $2$ we
attach to the input or the output, and get dimension at most $-1$ (these are
degenerations of previously studied configurations) and if we have two Chern
number $1$ spheres then we distribute them among input and output in the $3$
different ways possible and upper bound dimensions separately. ∎

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