Rank inequalities for the Heegaard Floer homology of Seifert homology spheres
aa r X i v : . [ m a t h . G T ] O c t RANK INEQUALITIES FOR THE HEEGAARD FLOER HOMOLOGY OFSEIFERT HOMOLOGY SPHERES
C¸ A ˘GRI KARAKURT AND TYE LIDMAN
Abstract.
We establish three rank inequalities for the reduced flavor of Heegaard Floer homologyof Seifert fibered integral homology spheres. Combining these inequalities with the known classifi-cations of non-zero degree maps between Seifert fibered spaces, we prove that a map f : Y ′ → Y between Seifert homology spheres yields the inequality | deg f | rank HF red ( Y ) ≤ rank HF red ( Y ′ ).These inequalities are also applied in conjunction with an algorithm of N´emethi to give a methodto solve the botany problem for the Heegaard Floer homology of these manifolds. Introduction
In the past several years, a great deal of progress has been made in the combinatorial descrip-tion of the Heegaard Floer invariants of various objects in low-dimensional topology. Many suchalgorithms come from the use of special Heegaard diagrams representing the objects in question,beginning with the advent of nice diagrams for closed 3-manifolds in [24]. This idea was used forknots and links in S [9, 10], which was then extended in [11] to give a completely combinato-rial description of the Heegaard Floer homology of closed three-manifolds and the Ozsv´ath-Szab´oinvariants of closed, smooth 4-manifolds with b +2 ≥ U map on the plus version of Heegaard Floer homology.Ozsv´ath and Szab´o’s result was extended by N´emethi in [12] which resulted in a fast algorithmcalculating the full Heegaard Floer homology of any 3-manifold which bounds a so–called almostrational plumbing. In the special case where the 3-manifold is a Seifert fibered integer homologysphere (or for short Seifert homology sphere), Can and the first author reformulated N´emethi’s algo-rithm in terms of a semigroup which is generated by a combination of Seifert invariants [2]. Despitethis progress, no closed formula is known for the Heegaard Floer homology of Seifert homologyspheres in terms of their Seifert invariants.The main purpose of the present article is to develop some combinatorial tools to compare theHeegaard Floer homologies of two given Seifert homology spheres without actually calculating themexplicitly. We shall prove three rank inequalities using these tools. Two of these inequalities arisefrom some geometric instances, namely the existence of certain kinds of maps between the spaces,but we never directly use these maps in our argument. Presumably one can prove more general rankinequalities (or possibly slightly different versions) by incorporating these maps with a compatibleversion of Floer homology. This strategy has been carried out for certain types of covering maps[7, 8]. We start by stating the claimed inequalities. Henceforth Σ( p , . . . , p l ) denotes the Seifert homol-ogy sphere corresponding to a given l -tuple of pairwise relatively prime positive integers ( p , . . . , p l )with p i ≥ i = 1 , . . . , l . Recall that the reduced version of Heegaard Floer homology HF red (Σ( p , . . . , p l )) is a finitely generated abelian group [20], and hence it has a well-defined rank.We also recall that S is the only Seifert homology sphere with l ≤ HF red ( S ) = 0.1.1. Rank inequalities.
The first rank inequality concerns a particular type of branched coverbetween Seifert homology spheres. Let Y = Σ( p , . . . , p l ). Fix n ∈ Z relatively prime to p , . . . , p l − and let Y ′ = Σ( p , . . . , p l − , np l ). The manifold Y ′ is the n -fold cyclic branched cover of Y branchedalong the singular fiber of order p l . Theorem 1.1. (Rank inequality for branched covers along singular fibers) We have (1.1) n [rank HF red (Σ( p , . . . , p l ))] ≤ rank HF red (Σ( p , . . . , p l − , np l )) . There also exists a certain degree one map f : Σ( p , . . . , p l ) → Σ( p , . . . , p k , p k +1 · · · p l ) calleda vertical pinch. See Section 8 for a geometric description. Our second inequality shows thatHeegaard Floer homology is sensitive to the existence of vertical pinches. Theorem 1.2. (Rank inequality for vertical pinches) Let l ≥ and fix ≤ k ≤ l − . Then (1.2) rank HF red (Σ( p , . . . , p k , p k +1 · · · p l )) ≤ rank HF red (Σ( p , . . . , p l )) . Remark . Recall that the rank of the Instanton Floer homology of a Seifert homology sphereequals its Casson invariant [3]. The splice-additivity of the Casson invariant, [4, 15], implies theanalogue of Theorem 1.2 for Instanton Floer homology.Consider the following partial order on the set of l -tuples of integers. We will write ( p , . . . , p l ) ≤ ( q , . . . , q l ), if there exists a permutation σ of the set { , . . . , l } such that p i ≤ q σ ( i ) for all i =1 , . . . , l . This relation naturally induces a partial order on the set of Seifert homology spheres with l singular fibers. The following result states that the rank of the reduced Heegaard Floer homologyis monotone under this partial order. Theorem 1.4. (Partial order rank inequality) Let Y = Σ( p , . . . , p l ) and Y ′ = Σ( q , . . . , q l ) beSeifert homology spheres such that ( p , . . . , p l ) ≤ ( q , . . . , q l ) . Then, we have (1.3) rank HF red ( Y ) ≤ rank HF red ( Y ′ ) . The botany problem for Seifert homology spheres.
Rank inequalities are useful whenone studies global problems about Heegaard Floer homology. Recall that the plus flavor of HeegaardFloer homology of any integral homology sphere is a Z -graded Z [ U ]-module. Question 1.5.
Let M be a Z -graded Z [ U ] -module. Is there a Seifert homology sphere realizing M as its Heegaard Floer homology? If there is at least one, what are all the Seifert homology sphereswhose Heegaard Floer homology is isomorphic to M ? Of course one can ask many different versions of this question. For example instead of Seiferthomology spheres, one can take any family of 3-manifolds. We focused on Seifert homology spheresbecause their Heegaard Floer homology can be calculated easily using an algorithm [12]. Neverthe-less the algorithm itself is not sufficient for problems involving infinite families. As an applicationof our rank inequalities we prove the following result.
ANK INEQUALITIES FOR THE HEEGAARD FLOER HOMOLOGY OF SEIFERT HOMOLOGY SPHERES 3
Theorem 1.6.
Suppose rank HF red (Σ( p , . . . , p l )) = n ≥ , then (1) 6 ≤ l ! < max { n, } , (2) max { p , . . . , p l } < n + 7 . Given n ≥
1, there are only finitely many tuples of pairwise relatively prime integers ( p , . . . , p l )satisfying the conditions in the above theorem. Hence one can solve the botany problem by calcu-lating the Heegaard Floer homology of a finite set of Seifert homology spheres. To illustrate this,we list all Seifert homology spheres with rank at most 12 in Table 1. Remark . Previously it was shown in [2] that there can be only finitely many Seifert homologyspheres with fixed Heegaard Floer homology (as a graded Z [ U ]-module), without explicitly givingbounds on the number of singular fibers and their multiplicities.1.3. Non-zero degree maps between Seifert homology spheres.Question 1.8.
Given -manifolds Y and Y ′ , does there exist a map f : Y ′ → Y with deg( f ) = 0 ? We will not attempt to answer this question here, but our work provides evidence that HeegaardFloer homology could be useful in this direction. Indeed, in order to show that such a map does notexist, one should look for some obstructions. Various topological quantities obstruct the existenceof a non-zero degree map. The rank of the first singular homology (i.e. the first betti number)is such a quantity, as one can easily check with elementary algebraic topology. Then a naturalquestion is whether the rank of the Heegaard Floer homology gives a similar obstruction. One ofthe goals of this paper is to study the behavior under non-zero degree maps of the Heegaard Floerhomology of Seifert homology spheres. It turns out that such maps are well-understood [5, 22, 23].By combining these results with our rank inequalities we prove the following theorem.
Theorem 1.9.
Let f : Y ′ → Y be a map between Seifert homology spheres. Then (1.4) | deg( f ) | rank HF red ( Y ) ≤ rank HF red ( Y ′ ) . Note that Theorem 1.9 is trivial if deg( f ) = 0.The organization is as follows: In Section 2, we review N´emethi’s method for calculating theHeegaard Floer homology of Seifert homology spheres. In Section 3, we build the theory of abstractdelta sequences and their morphisms to develop the combinatorial machinery to prove the three rankinequalities. Inequalities (1.1), (1.3), and (1.2) are proven in Section 4, Section 5, and Section 6respectively. We discuss the applications to the botany problem and nonzero degree maps inSection 7 and Section 8 respectively. In the last section we address some further directions andopen problems. Acknowledgments
We would like to thank Ian Agol for responding to our question on MathOverflow, leading to aproof of Proposition 8.4. In the course of this work, the first author was supported by a SimonsFellowship and the National Science Foundation FRG Grant DMS-1065178. The second authorwas partially supported by the National Science Foundation RTG Grant DMS-0636643.2.
Calculating the Heegaard Floer homology of Seifert homology spheres
Graded roots are certain infinite trees that naturally encode Heegaard Floer homology [12].These objects can be described by sequences as follows. Let τ be a given sequence of integers whichis either finite or non-decreasing after a finite index N . For every n ∈ N , let R n be the infinitegraph with vertex set Z ∩ [ τ ( n ) , ∞ ) and the edge set { [ k, k + 1] : k ∈ Z ∩ [ τ ( n ) , ∞ ) } . We identifyall common vertices and edges in R n and R n +1 for each n ∈ N to get an infinite tree Γ τ . To eachvertex v of Γ τ , we can assign a grading χ τ ( v ) which is the unique integer corresponding to v in any C¸ A ˘GRI KARAKURT AND TYE LIDMAN R n to which v belongs. The pair (Γ τ , χ τ ) is called a graded root . Most of the time, we drop thegrading function χ τ from our notation for brevity. Clearly many different sequences can give thesame graded graded root. For example Γ τ does not depend on the values τ ( n ) for n > N . In fact,Γ τ is completely determined by the subsequence of local maximum and local minimum values of τ . See Figure 1 for an example of a graded root given by the sequence τ = ( − , − , − , , − , . . . )where τ is increasing after the first five terms. − − τ R R R R R Figure 1.
Graded root for τ = ( − , − , − , , − , ր ).To any graded root Γ τ , we associate a Z -graded Z [ U ]-module as follows: Let H (Γ τ ) be the free Z -module on the vertex set of Γ τ . We require that the degree of the generator corresponding toeach vertex v has degree 2 χ τ ( v ). We define a degree − U of H (Γ τ ) by sending eachvertex v to the sum of the vertices w where w is connected to v by an edge and χ τ ( w ) < χ τ ( v ), orto zero if no such vertices exist. The group H (Γ τ ) is not finitely generated, but we can build twofinitely generated groups by exploiting the U -action. Define H red (Γ τ ) := Coker( U n ) , for large n, b H (Γ τ ) := Ker( U ) ⊕ Coker( U )[ − . It can be checked from the definition of a graded root that the first group is well defined. Thesymbol [ −
1] in the second equation indicates that we lower the degree of each homogeneous elementof Coker( U ) by one.In his seminal work [12], N´emethi • Constructed an abstract graded root Γ Y for every rational homology sphere Y which boundsa special type of plumbed 4-manifold X , called an almost rational plumbing, starting withthe intersection form of X . • Showed that the Heegaard Floer homology group HF + ( − Y ) is isomorphic to H (Γ Y ) witha fixed degree shift. • Gave an algorithm for calculating a sequence τ Y which generates Γ Y , in a way similar toLaufer’s method for finding Artin’s fundamental cycle. • Explicitly calculated τ Y for all Seifert rational homology spheres (with base orbifold S ) interms of their Seifert invariants.We now review N´emethi’s formulation of τ Y for Seifert homology spheres. For every positiveinteger l , let ( p , p , . . . , p l ) be an l -tuple of pairwise relatively prime positive integers with p i ≥ i = 1 , . . . , l . Denote by Y = Σ( p , . . . , p l ) the Seifert fibered space with base orbifold S andnormalized Seifert invariants ( e , ( p ′ , p ) , . . . , ( p ′ l , p l )) where e , p ′ , p ′ , . . . , p ′ l are defined by(2.1) e p p · · · p l + p ′ p · · · p l + p p ′ · · · p l + · · · + p p · · · p ′ l = − , with 0 ≤ p ′ i ≤ p i −
1, for all i = 1 , . . . , l . Such manifolds Y are precisely the Seifert homologyspheres. The numbers p i are called the multiplicities of the singular fibers of Y . Permutations ofmultiplicities do not change the Seifert homology sphere. ANK INEQUALITIES FOR THE HEEGAARD FLOER HOMOLOGY OF SEIFERT HOMOLOGY SPHERES 5
Let ∆ Y : N → Z denote the function(2.2) ∆ Y ( n ) = 1 + | e | n − l X i =1 (cid:24) np ′ i p i (cid:25) , where ⌈ x ⌉ is smallest integer greater than x . Let τ Y be the unique solution of the difference equation(2.3) τ Y ( n + 1) − τ Y ( n ) = ∆ Y ( n ) , with τ Y (0) = 0 . It can be checked that τ Y is non-decreasing after a finite index so it defines a graded root Γ τ Y . Theorem 2.1 (Nemethi, Section 11.13 of [12]) . We have the following isomorphisms of Z [ U ] -modules up to an overall degree shift. (1) HF + ( − Y ) ∼ = H (Γ τ Y ) , (2) HF red ( − Y ) ∼ = H red (Γ τ Y ) , (3) d HF ( − Y ) ∼ = b H (Γ τ Y ) . This theorem is sufficient for calculating the Heegaard Floer homology of a fixed Seifert homologysphere. On the other hand, one needs to develop a better understanding of the term ∆ Y in Equation(2.2), in order to prove theorems regarding the Heegaard Floer homology of infinite families of Seiferthomology spheres. The following result serves that purpose. Recall that Y = Σ( p , . . . , p l ). Denoteby G Y the numerical semigroup generated by p p . . . p l p i for i = 1 , , . . . , l . Define the constant N Y = p p · · · p l ( l − − l X i =1 p i ! . Theorem 2.2 (Can-Karakurt, Theorem 4.1 of [2]) . (1) N Y is a positive integer, unless l ≤ or l = 3 with { p , p , p } = { , , } . (2) ∆ Y ( n ) ≥ , for all n > N Y . (3) ∆ Y ( n ) = − ∆ Y ( N Y − n ) , for all ≤ n ≤ N Y . (4) ∆ Y ( n ) ∈ Z ∩ [ − l + 2 , l − , for all n with ≤ n ≤ N Y . (5) For ≤ n ≤ N Y , one has ∆ Y ( n ) ≥ if and only if n ∈ G Y . (6) If n ∈ G Y is written in the form n = p p · · · p l l X i =1 x i p i ! then ∆ Y ( n ) = 1 + l X i =1 (cid:22) x i p i (cid:23) . Abstract delta sequences and their morphisms
The discussion in the previous section provides an explicit method for the calculation of thegraded roots and hence Heegaard Floer homology of Seifert homology spheres. On the other hand,it is a challenging question to give a closed formula for these objects. Rather than attemptingto find such a closed formula, we will develop some techniques to compare ranks of HeegaardFloer homology of two given Seifert homology spheres. That is why we shall define abstract deltasequences and study their morphisms in this section. These objects essentially reduce the rankcomparison problems to the existence of certain maps between some combinatorial objects.3.1.
Basic definitions and notation.Definition 3.1.
A delta sequence is a pair ( X, ∆) where (1) X is a well–ordered finite set, (2) ∆ : X → Z \ { } with ∆( z ) > , where z is the minimum of X . C¸ A ˘GRI KARAKURT AND TYE LIDMAN
We shall denote a delta sequence ( X, ∆) simply by ∆ if the set X is clear from the context. Anydelta sequence ∆ naturally induces a graded root Γ ∆ as follows. Write the ordered set as X = { z , z , . . . , z k − } , with z < z < · · · < z k − . Then define a function τ ∆ : { , , . . . , k } → Z using the recurrence relation τ ∆ ( n + 1) − τ ∆ ( n ) = ∆( z n ) , for n = 0 , . . . , k −
1, together with the initial condition τ ∆ (0) = 0 . The graded root Γ ∆ is the one induced by the function τ ∆ as explained in Section 2. Conversely,every graded root comes from an abstract delta sequence. Of course many different delta sequencesmay induce the same graded root. Let X + denote the well-ordered set X ∪{ z + } where z + > z for all z ∈ X . We find it convenient to think of the domain of τ ∆ as being X + rather than { , , . . . , | X |} .Hence sometimes we abuse notation and write τ ∆ ( z ) for z ∈ X + but actually mean τ ∆ ( n ( z )) where n : X + → { , . . . , | X |} is the order preserving enumeration of X + . With this convention, we have(3.1) τ ∆ ( z ) = X w ∈ Xw
0, for n = 1 , . . . , k −
2. If∆( z k − ) <
0, we also add one to c ∆ . Proposition 3.2.
We have (1) rank( b H (Γ ∆ )) = 2 c ∆ + 1 , (2) rank( H red (Γ ∆ )) = κ ∆ + min z ∈ X τ ∆ ( z ) .Proof. For part (1), observe that the number of univalent vertices of Γ ∆ is exactly c ∆ + 1. Fromthe description of the U action, it is now clear that rank(ker U ) = c ∆ + 1 and rank(coker U ) = c ∆ .Part (2) is [12, Corollary 3.7]. (cid:3) Operations on delta sequences.
In this subsection, we discuss some methods to generatenew delta sequences out of a given one. Henceforth for a given delta sequence ( X, ∆), we shallreserve the symbols τ, Γ , S, Q, κ , and c to denote the objects introduced in Section 3.1. Whenevera delta sequence admits a decoration, corresponding objects pick up the same decoration. Also thesubscript ∆ will be dropped for brevity. For example, if ( X , ∆ ) is a delta sequence, then S denotes the set of elements in X for which ∆ is positive. Definition 3.3.
Let ( X, ∆) be a delta sequence. A delta subsequence of ( X, ∆) is a delta sequence ( X , ∆ ) where X ⊂ X and ∆ = ∆ | X . ANK INEQUALITIES FOR THE HEEGAARD FLOER HOMOLOGY OF SEIFERT HOMOLOGY SPHERES 7
Proposition 3.4. If ( X , ∆ ) is a delta subsequence of ( X, ∆) then rank( b H (Γ )) ≤ rank( b H (Γ)) . Proof.
Clearly c ≤ c . By Proposition 3.2, we are done. (cid:3) For the rank of H red (Γ ), we shall prove a stronger inequality. To this end we introduce comple-mentary subsequences. Suppose ( X , ∆ ) is a delta subsequence of ( X, ∆). From this we constructa new delta sequence by letting X := X \ X , and ∆ := ∆ | X . The pair ( X , ∆ ) may not satisfyProperty (2) of Definition 3.1, but we can modify it to become a delta sequence as follows: Let x be the minimum of S . Remove all y ∈ Q with y ≤ x from X . By abuse of notation wedenote the resulting delta sequence by the same symbol, ( X , ∆ ). This delta sequence is calledthe complementary delta subsequence of ( X , ∆ ) in ( X, ∆). Proposition 3.5.
Let ( X, ∆) be a delta sequence. Let ( X , ∆ ) be a delta subsequence and ( X , ∆ ) its complementary subsequence. Then rank( H red (Γ )) + rank( H red (Γ )) ≤ rank( H red (Γ)) . Proof.
We use τ , τ , and τ for τ ∆ , τ ∆ , and τ ∆ respectively. First note that S = S ∪ S , Q ⊇ Q ∪ Q , and κ = κ + κ − X y ∈ Q \ ( Q ∪ Q ) ∆( y ) . Next we claim that(3.2) min τ ≥ min τ + min τ + X y ∈ Q \ ( Q ∪ Q ) ∆( y ) . Let us first see why this inequality finishes the proof. Rearranging the terms and adding κ + κ to both sides, we get min τ + κ ≥ min τ + min τ + κ + κ . By Proposition 3.2 we are done.To see why Inequality (3.2) holds, let w ∈ X + be an element where τ attains its minimum.Then by Equation (3.1) τ ( w ) = X z ∈ X z Refinements and merges do not change H red and b H .Proof. This follows easily from the definitions. (cid:3) Isomorphisms and embeddings. We will define various kinds of maps between delta se-quences and study their properties. Definition 3.7. A morphism between delta sequences ( X , ∆ ) and ( X , ∆ ) is a map φ : X → X with φ ( S ) ⊆ S and φ ( Q ) ⊆ Q . A morphism φ is called an isomorphism if it is an orderpreserving bijection satisfying ∆ ( φ ( z )) = ∆ ( z ) for all z ∈ X . Clearly isomorphic delta sequences induce the same graded root. Consequently we have the follow-ing. Proposition 3.8. If ( X , ∆ ) and ( X , ∆ ) are isomorphic delta sequences then rank( H red (Γ )) =rank( H red (Γ )) and rank( b H (Γ )) = rank( b H (Γ )) . In general, isomorphisms are difficult to construct. Most of the time we are content with isomor-phisms up to refinements. We say that two delta sequences ( X , ∆ ) and ( X , ∆ ) are equivalent if they are isomorphic after a sequence of refinements of each. Definition 3.9. An embedding of ( X , ∆ ) into ( X , ∆ ) is a morphism φ : ( X , ∆ ) → ( X , ∆ ) satisfying (1) For every x ∈ S and y ∈ Q , we have x < y if and only if φ ( x ) < φ ( y ) . (2) For every z ∈ φ ( X ) , | ∆ ( z ) | ≥ X w ∈ φ − ( z ) | ∆ ( w ) | . Property (1) of Definition 3.9 says that the order of elements of S relative to elements of Q ispreserved under an embedding. Note also that unlike what the name might suggest, an embeddingof delta sequences need not be injective. The following result says we can achieve injectivity afterwe do appropriate modifications to the domain and target delta sequences. Theorem 3.10. If there is an embedding φ : ( X , ∆ ) → ( X , ∆ ) then there are refinements ( X ′ , ∆ ′ ) and ( X ′ , ∆ ′ ) of ( X , ∆ ) and ( X , ∆ ) respectively such that ( X ′ , ∆ ′ ) is isomorphic to adelta subsequence of ( X ′ , ∆ ′ ) . In other words, if ∆ embeds into ∆ , then ∆ is equivalent to adelta subsequence of ∆ .Proof. We first make φ injective using refinements on ( X , ∆ ). Let z ∈ φ ( X ) such that | φ − ( z ) | ≥ 2. Write φ − ( z ) = { w , . . . , w t } . We refine ∆ at z to t + 1 elements z , . . . , z t +1 such that∆ ( z n ) = ∆ ( w n ) for all n = 1 , . . . , t . This is possible by Property (2) of Definition 3.9. Weextend φ to this refinement by sending w n to z n for all n = 1 , . . . , t . Repeating this process forevery z ∈ φ ( X ) with | φ − ( z ) | ≥ 2, we get a refined delta sequence ( X ′ , ∆ ′ ), and an injectiveembedding φ ′ : ( X , ∆ ) → ( X ′ , ∆ ′ ). If necessary, we can refine ( X ′ , ∆ ′ ) further to achieve that∆ ′ ( φ ′ ( w )) = ∆ ( w ) for all w ∈ X ; this is again guaranteed by Property (2) of Definition 3.9.The morphism φ ′ may not be order preserving, so we need to modify it to have this property.Since φ ′ is an embedding, the relative positions of elements of S do not change with respect tothe elements of Q . On the other hand, φ ′ can rearrange some consecutive elements in S (andrespectively in Q ). We can permute the order of these elements by a sequence of refinements ANK INEQUALITIES FOR THE HEEGAARD FLOER HOMOLOGY OF SEIFERT HOMOLOGY SPHERES 9 and merges and re-defining φ ′ accordingly. To see that this is possible, suppose x, ˜ x ∈ S with x < ˜ x but φ ′ (˜ x ) < φ ′ ( x ). Suppose also that there is no w ∈ X with x < w < ˜ x . Merge x and ˜ x together and subsequently refine the resulting element into x ′ , ˜ x ′ with x ′ < ˜ x ′ to obtain anew delta sequence ( X ′ , ∆ ′ ) so that ∆ ′ ( x ′ ) = ∆ (˜ x ) and ∆ ′ (˜ x ′ ) = ∆ ( x ). Define a morphism φ ′′ : ( X ′ , ∆ ′ ) → ( X ′ , ∆ ′ ) so that φ ′′ ( x ′ ) = φ ′ (˜ x ), φ ′′ (˜ x ′ ) = φ ′ ( x ), and φ ′′ agrees with φ ′ otherwise.Hence φ ′′ is still an injective embedding and now it preserves the order of the elements x ′ and˜ x ′ . Repeating this process for every pair in S (and respectively Q ) where the order preservingfails, eventually we make φ ′′ order preserving, since transpositions generate the whole permutationgroup. We use the same symbols φ ′′ : ( X ′ , ∆ ′ ) → ( X ′ , ∆ ′ ) to denote the resulting morphism aftermaking all necessary adjustments.Now φ ′′ becomes an order preserving injective morphism with ∆ ′ ( φ ′′ ( w )) = ∆ ′ ( w ) for all x ∈ X ′ .Hence it is an isomorphism onto its image. (cid:3) In light of Theorem 3.10, given an embedding of ∆ into ∆, we can find a delta subsequenceof ∆ equivalent to ∆ . By abuse of terminology, we will refer to the corresponding complemen-tary subsequence as the complementary delta subsequence of ∆ . Note that the complementarysubsequence implicitly depends on the choice of embedding. Corollary 3.11. Suppose ∆ embeds into ∆ and let ∆ denote the complementary delta subse-quence. Then (1) rank( b H (Γ )) ≤ rank( b H (Γ)) , (2) rank( H red (Γ )) + rank( H red (Γ )) ≤ rank( H red (Γ)) .Proof. The first inequality (second inequality respectively) follows from combining Proposition 3.4(respectively Proposition 3.5) with Proposition 3.6, Proposition 3.8, and Theorem 3.10. (cid:3) Corollary 3.12. If ( X , ∆ ) , . . . , ( X p , ∆ p ) can be disjointly embedded into ( X, ∆) then p X j =1 rank( H red (Γ j )) ≤ rank( H red (Γ)) . Proof. The disjointness ensures that ( X , ∆ ) , . . . , ( X p − , ∆ p − ) embed into the complementarysubsequence of ( X p , ∆ p ). Now use Corollary 3.11 and do induction on p . (cid:3) Right-veering maps and immersions. We now relax the condition on the preservation oforderings of our morphisms. Though we can still prove rank inequalities for H red under these newkinds of morphisms, we need to give up the rank inequality for b H . Definition 3.13. A right-veering morphism between delta sequences ( X , ∆ ) and ( X , ∆ ) is abijective morphism φ : ( X , ∆ ) → ( X , ∆ ) such that (1) Both of the maps φ | S and φ | Q are order preserving, (2) For all x ∈ S , y ∈ Q with x < y , we have φ ( x ) < φ ( y ) , (3) ∆ ( z ) = ∆ ( φ ( z )) for all z ∈ X . Note that a right-veering morphism φ is almost the same thing as an isomorphism except thatone may have elements x ∈ S , y ∈ Q with y ≤ x and φ ( x ) ≤ φ ( y ). In other words φ “moves” theelements of Q to the “right” of elements of S . Proposition 3.14. If there is a right-veering morphism φ : ( X , ∆ ) → ( X , ∆ ) then rank( H red (Γ )) ≤ rank( H red (Γ )) . Proof. First observe that κ = κ by Property (3) of Definition 3.13 and the fact that φ is a bijectivemorphism. Hence by Proposition 3.2, it suffices to show that min τ ≤ min τ . To do this we shallshow that given any z ∈ X +2 there exists z ∈ X +1 such that τ ( z ) ≤ τ ( z ). Let z ∈ X +2 be given. If z is the maximal element z +2 ∈ X +2 then we choose z to be the maximal element z +1 ∈ X +1 , andwe have τ ( z ) = τ ( z ) by Property (3) and the fact that φ is a bijection. Otherwise z ∈ X , andwe choose z := φ − ( z ).Using Properties (1) and (2) in Definition 3.13, we see that for all x ∈ S , if x ≤ z then φ ( x ) ≤ z . Therefore,(3.3) X x ∈ S x A morphism φ : ( X , ∆ ) → ( X , ∆ ) is called a semi-immersion if for every x ∈ S and y ∈ Q with x < y , we have φ ( x ) < φ ( y ) . A semi-immersion is called an immersion , ifit satisfies | ∆ ( z ) | ≥ X x ∈ φ − ( z ) | ∆ ( x ) | for all z ∈ φ ( X ) . Theorem 3.16. Suppose there exists an immersion from ( X , ∆ ) to ( X , ∆ ) . Then the followingexist. • A refinement ( X ′ , ∆ ′ ) of ( X , ∆ ) . • A refinement ( X ′ , ∆ ′ ) of ( X , ∆ ) . • A delta subsequence ( e X , e ∆ ) of ( X ′ , ∆ ′ ) . • A right-veering morphism e φ : ( X ′ , ∆ ′ ) → ( e X , e ∆ ) .Proof. Similar to the proof of Theorem 3.10. (cid:3) Corollary 3.17. If ( X , ∆ ) can be immersed into ( X , ∆ ) then rank( H red (Γ )) ≤ rank( H red (Γ )) . Proof. This follows from Proposition 3.5, Proposition 3.14, and Theorem 3.16. (cid:3) Well-behaved semi-immersions. Let φ : ( X , ∆ ) → ( X , ∆ ) be a one-to-one semi-immersion. For z ∈ X , the defect of z is the number d φ ( z ) := | ∆ ( z ) | − | ∆ ( φ ( z )) | . An element z ∈ X is called • a bad point if it has positive defect, • a good point if it has negative defect, ANK INEQUALITIES FOR THE HEEGAARD FLOER HOMOLOGY OF SEIFERT HOMOLOGY SPHERES 11 • a neutral point if it has zero defect.We shall call the images of bad (respectively good, neutral) points, bad (respectively good, neutral ) values .Of course a one-to-one semi-immersion φ is an immersion if it has no bad points. We shallinvestigate how much we can relax this condition to still obtain rank inequalities for H red underone-to-one semi-immersions. Denote the set of good points and the set of bad points by G ( φ ) and B ( φ ) respectively. Definition 3.18. A control function is an injection θ : B ( φ ) → G ( φ ) satisfying: (1) z ∈ S if and only if θ ( z ) ∈ S , (2) θ ( z ) < z if z ∈ S ∩ B ( φ ) , and θ ( z ) > z if z ∈ Q ∩ B ( φ ) , (3) | d φ ( z ) | ≤ | d φ ( θ ( z )) | for all z ∈ B ( φ ) . We will use control functions to “fix” defects of bad points with merges and refinements usingcorresponding good points. Call a one-to-one semi-immersion well-behaved if it admits a controlfunction. Theorem 3.19. Let φ : ( X , ∆ ) → ( X , ∆ ) be delta sequences. Suppose there is a well-behavedone-to-one semi-immersion φ : ( X , ∆ ) → ( X , ∆ ) . Then there exist refinements ( X ′ , ∆ ′ ) and ( X ′ , ∆ ′ ) of ( X , ∆ ) and ( X , ∆ ) respectively, and a one-to-one immersion φ ′ : ( X ′ , ∆ ′ ) → ( X ′ , ∆ ′ ) .Proof. Let θ : B ( φ ) → G ( φ ) be a control function. We define a refinement ( X ′ , ∆ ′ ) of ( X , ∆ ) ateach b ∈ B ( φ ) by replacing it with a consecutive pair b , b with the additional requirement that∆ ′ ( b ) = ∆ ( φ ( b )). Note that this automatically ensures | ∆ ′ ( b ) | = | d φ ( b ) | . Similarly define arefinement ( X ′ , ∆ ′ ) of ( X , ∆ ) at each g = φ ( θ ( b )) for b ∈ B ( φ ), by replacing g with a consecutivepair g , g with the requirement that ∆ ′ ( g ) = ∆ ( θ ( b )). Again this implies | ∆ ′ ( g ) | = | d φ ( θ ( b )) | .Now define a map φ ′ : ( X ′ , ∆ ′ ) → ( X ′ , ∆ ′ ) by(3.5) φ ′ ( z ) = φ ( z ) if z ∈ X \ ( B ( φ ) ∪ θ ( B ( φ ))) ,φ ( b ) if z = b for some b ∈ B ( φ ) ,g if z = θ ( b ) for some b ∈ B ( φ ) and g = φ ( θ ( b )) ,g if z = b for some b ∈ B ( φ ) and g = φ ( θ ( b )) . We check that φ ′ satisfies all the claimed properties. Clearly φ ′ is injective if φ is injective. Property(1) of control functions ensures that φ ′ is a morphism. Property (2) of being a control functionimplies φ ′ is a semi-immersion. It is left to show that φ ′ has no bad points. Clearly no elementof X \ ( B ( φ ) ∪ θ ( B ( φ ))) is bad. Suppose b ∈ B ( φ ), we will show none of b , b and θ ( b ) are bad.Equation (3.5) shows that under φ ′ , b and θ ( b ) are neutral. Finally by Property (3) of controlfunctions, we have | ∆ ′ ( b ) | = | d φ ( b ) | ≤ | d φ ( θ ( b )) | = | ∆ ′ ( g ) | . Hence b is not a bad point. (cid:3) Corollary 3.20. If there is a well-behaved one-to-one semi immersion φ : ( X , ∆ ) → ( X , ∆ ) then rank( H red (Γ )) ≤ rank( H red (Γ )) . Proof. This follows from Proposition 3.8, Corollary 3.17, and Theorem 3.19. (cid:3) Delta sequences of Seifert homology spheres. It is our goal to reduce the comparisonof the Heegaard Floer homology of Seifert homology spheres to the comparison of the associateddelta sequences. In order to use the techniques developed in this section we need to see that thegraded root of a Seifert homology sphere is naturally induced by an abstract delta sequence.Let Y = Σ( p , . . . , p l ) be a Seifert homology sphere, and ∆ Y its delta function. We shall useTheorem 2.2 and the notation introduced there. In particular G Y denotes the numerical semigroupgenerated by p · · · p l p i for i = 1 , . . . , l. The graded root and hence the Heegaard Floer homology of Y is completely determined by thevalues of ∆ Y on the interval [0 , N Y ] because it becomes non-negative afterward. On the interval[0 , N Y ], ∆ Y takes all its positive values on the set S Y := G Y ∩ [0 , N Y ], and it takes all its negativevalues on the set Q Y := { N Y − x : x ∈ S Y } . Note that S Y ∩ Q Y = ∅ . Let X Y := S Y ∪ Q Y . Proposition 3.21. For any Seifert homology sphere Y which is not S or Σ(2 , , , the pair ( X Y , ∆ Y | X Y ) is an abstract delta sequence in the sense of Definition 3.1. The graded root inducedby this abstract delta sequence is the same as the graded root of Y .Proof. It is clear that Property (1) of Definition 3.1 is satisfied. Property (2) holds, since ∆ Y (0) = 1.Therefore, ( X Y , ∆ Y | X Y ) is an abstract delta sequence. For the second claim, we simply observethat the zeros of ∆ Y and the values of ∆ Y after N Y do not affect the graded root. (cid:3) In particular, we have that H red (Γ ∆ Y ) (respectively b H (Γ ∆ Y )) is isomorphic to HF red ( − Y ) (re-spectively d HF ( − Y )) by Theorem 2.1. The duality of Heegaard Floer homology under orienta-tion reversal [19, Proposition 2.5] implies rank H red (Γ ∆ Y ) = rank HF red ( Y ) and rank b H (Γ ∆ Y ) =rank d HF ( Y ).Next we discuss a practical method to generate semi-immersions between delta sequences ofSeifert homology spheres from maps between their semigroups. Definition 3.22. Suppose Y and Y ′ are Seifert homology spheres. If φ : G Y → G Y ′ is a one-to-onefunction such that (1) φ ( x ) ≥ x , (2) φ ( x ) − x ≤ N Y ′ − N Y ,for every x ∈ G Y , then φ is called rigid . Lemma 3.23. Let φ : G Y → G Y ′ be rigid. Then, φ naturally determines a one-to-one semi-immersion on the associated abstract delta sequences by restricting φ to S Y and by extending φ to Q Y by φ ( N Y − x ) = N Y ′ − φ ( x ) .Proof. Let us first verify that φ defines a morphism from ∆ Y to ∆ Y ′ . Observe that Properties (1)and (2) in Definition 3.22 force that N Y ≤ N Y ′ . If x ∈ S Y , then x < N Y . Use Property (2) inDefinition 3.22 to see that φ ( x ) ≤ N Y ′ x − N Y < N Y ′ + N Y ≤ N Y ′ . Hence φ ( S Y ) ⊆ S Y ′ . Furthermore, φ ( N Y − x ) = N Y ′ − φ ( x ) by definition. Since φ ( x ) ≥ 0, we have φ ( N Y − x ) > 0. Therefore φ ( Q Y ) ⊆ Q Y ′ .The last thing to check is that if x < N Y − x for x , x ∈ S Y then φ ( x ) < N Y ′ − φ ( x ). Againit follows from Property (2) in Definition 3.22 that φ ( x ) − x + φ ( x ) − x ≤ N Y ′ − N Y . ANK INEQUALITIES FOR THE HEEGAARD FLOER HOMOLOGY OF SEIFERT HOMOLOGY SPHERES 13 Hence φ ( x ) + φ ( x ) ≤ N Y ′ − ( N Y − x − x ) < N Y ′ , where the last inequality follows from x < N Y − x . (cid:3) Guide. The following diagram shows where various types of morphisms are defined and wherethey are going to be used in our argument.Isomorphisms: Definition 3.7Rank equality by Proposition 3.8Right-veerings: Definition 3.13Rank inequality by Proposition 3.14Embeddings: Definition 3.9Rank inequality by Corollary 3.11Disjoint EmbeddingsRank inequality by Corollary 3.12Immersions: Definition 3.15Rank inequality by Corollary 3.17Well-behaved semi-immersions: Section 3.5Rank inequality by Corollary 3.20Rank inequality forbranched covers:Theorem 1.1, Section 4Partial orderrank inequality:Theorem 1.4, Section 5Semi-immersions:Definition 3.15No rank inequality Rank inequality forvertical pinches:Theorem 1.2, Section 6 ........................................................................................................................................................................................................................................................................................... .................................................................................................................................................................................................................................................................................................................. ................................................................................................................................................................................................................... ....................................................................................................................................................................................................................................................................................................................... ..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... .......................................................................................................................................................................................................................................... ....................................................................................................................................................................................................................................................................................................................................................................................... .......................................................................................................................................................................................................................................................................................................................................... .............................................................................................................................................................................................................. ............ Proof of the Rank inequality for Branched Covers We start with a remark about the content of the proof. Remark . For S or Σ(2 , , HF red = 0. Hence Theorems 1.1 and 1.4 trivially holdif one of the Seifert homology spheres being considered is either S or Σ(2 , , S orΣ(2 , , 5) cannot arise in the statement of Theorem 1.2. Proof of Theorem 1.1. Let Y = Σ( p , . . . , p l ) and Y ′ = Σ( p , . . . , p l − , np l ). Consider the associatedabstract delta sequences ( X Y , ∆ Y ) and ( X Y ′ , ∆ Y ′ ) as described in Section 3.6. Our aim is to show that n rank H red (Γ ∆ Y ) ≤ rank H red (Γ ∆ Y ′ ), and appeal to Theorem 2.1. By Corollary 3.12, it sufficesto find n disjoint embeddings of ∆ Y into ∆ Y ′ . For 0 ≤ k ≤ n − 1, define φ k : X Y → X Y ′ , z nz + kp · · · p l − . We first need to see that each φ k is an embedding. In particular we must show that each φ k is amorphism. It is clear that φ k takes elements of G Y to G Y ′ . Let z ∈ X Y . We observe that φ k ( z ) ≤ φ n − ( z ) = nz + ( n − p · · · p l − ≤ nN Y + ( n − p · · · p l − = N Y ′ This implies that φ k ( S Y ) ⊂ S Y ′ . Now, if y = N Y − x ∈ Q Y , where x ∈ S Y , we have φ k ( y ) = ny + kp · · · p l − = nN Y + kp · · · p l − − nx (4.1) = N Y ′ − ( n − p · · · p l − + kp · · · p l − − nx (4.2) = N Y ′ − ( n − − k ) p · · · p l − − nx. (4.3)Since 0 ≤ k ≤ n − 1, we have that ( n − − k ) p · · · p l − + nx = φ n − − k ( x ) ∈ S Y ′ . Therefore, φ k ( y ) ∈ Q Y ′ . Thus, each φ k is a morphism between the delta sequences ∆ Y and ∆ Y ′ . Furthermore,it is clear that each φ k is order preserving (it is of the form ax + b with a > φ k are embeddings, it remains to check that | ∆ Y ′ ( φ k ( z )) | ≥ | ∆ Y ( z ) | for all z ∈ X Y , since the φ k are injective. The following claim establishes this. Claim . For each z ∈ X Y , ∆ Y ′ ( φ k ( z )) = ∆ Y ( z ). Proof of Claim 4.2. Recall X Y = S Y ∪ Q Y . We begin with the case of x ∈ S Y . Express x by x = p · · · p l l X i =1 a i p i . By Theorem 2.2 part (6), ∆ Y ′ ( φ k ( x )) = ∆ Y ′ ( nx + kp · · · p l − )= l − X i =1 (cid:22) a i p i (cid:23) + (cid:22) na l + knp l (cid:23) + 1 . Since 0 ≤ k < n , it is straightforward to check that (cid:22) na l + knp l (cid:23) = (cid:22) a l p l (cid:23) . Therefore,(4.4) ∆ Y ′ ( φ k ( x )) = l X i =1 (cid:22) a i p i (cid:23) + 1 = ∆ Y ( x ) . On the other hand, if y = N Y − x ∈ Q Y , then by Equations (4.1)-(4.4) and Theorem 2.2 part (3),∆ Y ′ ( φ k ( y )) = ∆ Y ′ ( N Y ′ − ( n − − k ) p · · · p l − − nx )= − ∆ Y ′ ( nx + ( n − − k ) p · · · p l − )= − ∆ Y ( x )= ∆ Y ( y ) . (cid:3) ANK INEQUALITIES FOR THE HEEGAARD FLOER HOMOLOGY OF SEIFERT HOMOLOGY SPHERES 15 It remains to see that for i = j , the images of φ i and φ j are disjoint. Suppose that φ i ( z ) = φ j ( z )for some z , z ∈ X Y . In this case, either both z , z ∈ S Y or z , z ∈ Q Y . First, if z , z ∈ S Y , then φ i ( z ) ≡ ip · · · p l − (mod n ) and φ j ( z ) ≡ jp · · · p l − (mod n ) . Therefore, ip · · · p l − ≡ jp · · · p l − (mod n ) . Therefore, i ≡ j (mod n ) since gcd( p k , n ) = 1 for all 1 ≤ k ≤ l − 1. However, since 0 ≤ i, j ≤ n − i = j . A similar argument applies to the case of z , z ∈ Q Y . (cid:3) The proof of Theorem 1.1 also implies a weaker rank inequality for d HF . Proposition 4.3. We have rank d HF (Σ( p , . . . , p l )) ≤ rank d HF (Σ( p , . . . , p l − , np l )) . Proof. Let Y = Σ( p , . . . , p l ) and Y ′ = Σ( p , . . . , p l − , np l ) as before. In the proof of Theorem 1.1we constructed an embedding of ∆ Y into ∆ Y ′ . Hence Corollary 3.11 and Theorem 2.1 finish theproof. (cid:3) Proof of the Partial Order Inequality Our purpose is to prove Theorem 1.4. First we prove a lemma about our numerical semigroups.Let p , . . . , p l be pairwise relatively prime positive integers with p i ≥ i = 1 , . . . , l . Let G be the numerical semigroup generated by p · · · p l p i for i = 1 , . . . , l . Lemma 5.1. Every element n of G can be uniquely written as (5.1) n = p · · · p l k + l X i =1 x i p i ! , for some integers k ≥ and ≤ x i < p i , for all i = 1 , . . . , l .Proof. First we show that the form (5.1) exists for every element of G . Let n ∈ G , and write it asa linear combination of the generators n = p · · · p l l X i =1 a i p i . Then apply the division algorithm to each term a i p i to get non-negative integers k i and x i such that a i = k i p i + x i , with 0 ≤ x i < p i . Let k := l X i =1 k i . Then n is in the form (5.1).Next we show that the form (5.1) is unique. Suppose(5.2) p · · · p l k + l X i =1 x i p i ! = p · · · p l k + l X i =1 y i p i ! . Taking reductions of both sides modulo p i , we see that x i ≡ y i (mod p i ) for all i = 1 , . . . , l. Since both x i and y i are in [0 , p i ), this forces x i = y i for all i = 1 , . . . , l . After this observation wesee that Equation (5.2) forces k = k . (cid:3) Definition 5.2. We shall call the unique expression (5.1) in Lemma 5.1, the normal form of theelement n of the numerical semigroup G .Proof of Theorem 1.4. Let Y = Σ( p , . . . , p l ) and Y ′ = Σ( q , . . . , q l ) with p i ≤ q i for all i , andconsider the associated abstract delta sequences ( X Y , ∆ Y ) and ( X Y ′ , ∆ Y ′ ) as described in Section3.6. By Corollary 3.17, it suffices to construct an immersion of ∆ Y into ∆ Y ′ . Let n ∈ G Y and writeit in the normal form(5.3) n = p · · · p l k + l X i =1 x i p i ! , for some integers k ≥ ≤ x i < p i , for all i = 1 , . . . , l . Then n ′ := q · · · q l k + l X i =1 x i q i ! is the normal form of an element n ′ of G Y ′ , since p i ≤ q i for all i . Define φ : G Y → G Y ′ by φ p · · · p l k + l X i =1 x i p i !! = q · · · q l k + l X i =1 x i q i ! . We claim that φ ( S Y ) ⊆ S Y ′ . To see this, suppose n ≤ N Y and n ∈ G Y with normal form (5.3).Note that k + l X i =1 x i p i ≤ l − − l X i =1 p i . Then φ ( n ) q · · · q l = k + l X i =1 x i q i ≤ k + l X i =1 x i p i ≤ l − − l X i =1 p i ≤ l − − l X i =1 q i = N Y ′ q · · · q l , which implies φ ( n ) ≤ N Y ′ . Hence we have a map φ : S Y → S Y ′ . Next we extend this to a morphism φ : ( X Y , ∆ Y ) → ( X Y ′ , ∆ Y ′ ) by requiring that φ ( N Y − x ) = N Y ′ − φ ( x ), for all x ∈ S Y .We claim that φ is the required immersion. Let us first show that φ satisfies(5.4) φ ( x ) + φ (˜ x ) ≤ φ ( x + ˜ x ) for all x, ˜ x ∈ G Y . Write x and ˜ x in normal form x = p · · · p l k + l X i =1 x i p i ! , ˜ x = p · · · p l k + l X i =1 ˜ x i p i ! . Observe that both φ ( x ) + φ (˜ x ) and φ ( x + ˜ x ) belong to G Y ′ . A comparison of their normal formsshows the following: • φ ( x ) + φ (˜ x ) = φ ( x + ˜ x ) if x i + ˜ x i < p i or x i + ˜ x i ≥ q i for all i = 1 , . . . , l . • φ ( x ) + φ (˜ x ) < φ ( x + ˜ x ) if for some i = 1 . . . l , we have p i ≤ x i + ˜ x i < q i .Having shown that φ satisfies Inequality (5.4), we will easily prove that φ is a semi-immersion.Suppose x ≤ y for some x ∈ S Y and y ∈ Q Y . Write y = N Y − ˜ x for ˜ x ∈ S Y , then φ ( x ) + φ (˜ x ) ≤ φ ( x + ˜ x ) ≤ N Y ′ , which implies φ ( x ) ≤ N Y ′ − φ (˜ x ) = φ ( N Y − ˜ x ) = φ ( y ) . Finally we verify that φ is an immersion. By the uniqueness of normal forms, φ is one-to-one. Bypart (6) of Theorem 2.2, ∆ Y ( x ) = ∆ Y ′ ( φ ( x )), for all x ∈ S Y . By symmetry (part (3) of Theorem2.2), we conclude ∆ Y ( y ) = ∆ Y ′ ( φ ( y )), for all y ∈ Q Y as well. Hence φ is an immersion. (cid:3) ANK INEQUALITIES FOR THE HEEGAARD FLOER HOMOLOGY OF SEIFERT HOMOLOGY SPHERES 17 Proof of the Rank Inequality for Pinches Description. The goal of this section is to prove Theorem 1.2. It is clear that in order to proveTheorem 1.2, it suffices to prove rank HF red (Σ( p , . . . , p l , qr )) ≤ rank HF red (Σ( p , . . . , p l , q, r )). Wewill therefore focus on this inequality for the rest of the section. In order to obtain the latter, byCorollary 3.20 it suffices to construct a well-behaved one-to-one semi-immersion from ∆ Y to ∆ Y ′ ,where Y = Σ( p , . . . , p l , qr ) and Y ′ = Σ( p , . . . , p l , q, r ).6.2. Numerical semigroups on two elements. We are going to define two auxiliary (infinite)delta sequences and a well-behaved semi-immersion between them. This data will be used laterin the next subsection in order to define the required well-behaved one-to-one semi-immersion of∆ Y into ∆ Y ′ . Fix relatively prime integers q, r ≥ S q,r := { aq + br | a, b ≥ } . Recall that the Frobenius number of S q,r is qr − q − r [1,Theorem 1.2]. In other words qr − q − r S q,r , and every integer n ≥ ( q − r − 1) is in thesemigroup S q,r . We point out that since q and r are relatively prime, ( q − r − is an integer. Lemma 6.1. We have | N r S q,r | = | [0 , qr − q − r ] ∩ S q,r | = ( q − r − .Proof. This follows from Sylvester’s Theorem which says that precisely half of the integers between1 and ( q − r − 1) belong to S q,r . See for example [1, Therem 1.3] for the proof. (cid:3) Lemma 6.2. Suppose x ∈ [0 , qr − q − r ] . Then x ∈ S q,r if and only if qr − q − r − x / ∈ S q,r Proof. We already know qr − q − r / ∈ S q,r , so qr − q − r − x is not in the semigroup if x ∈ S q,r .Lemma 6.1 proves that the converse is also true. (cid:3) Lemma 6.3. There exists a unique bijective function ψ : N → S q,r such that: (1) for x , x ∈ N , x < x implies ψ ( x ) < ψ ( x ) , (2) if x ≥ ( q − r − , then ψ ( x ) = x + ( q − r − .Proof. First, define ψ (0) = 0. Next, we define ψ ( i ) to be the i th non-zero element in S q,r , where theorder structure on S q,r is just the induced one from N . This is clearly the unique order preservingbijection. We now just want to show that part (2) holds. As discussed, for all z > qr − q − r , wehave z ∈ S q,r . Therefore, if ψ ( x ) = z , where z > qr − q − r , then ψ ( x + k ) = ψ ( x ) + k for all k ≥ ψ ( ( q − r − ) = ( q − r − ( q − r − elements of [0 , ( q − r − S q,r by Lemma 6.1. (cid:3) We can define delta functions, ∆ qr on N by ∆ qr ( x ) = 1 + ⌊ xqr ⌋ and ∆ q,r on S q,r by ∆ q,r ( aq + br ) =1 + ⌊ ar ⌋ + ⌊ bq ⌋ . We regard ( N , ∆ qr ) and ( S q,r , ∆ q,r ) as abstract delta sequences, even though bothsets are infinite and the delta functions attain only positive values. The map ψ is a semi-immersionwhich is also one-to-one. Therefore, we are still able to study good and bad points as defined inSection 3.5. As before, G ( ψ ) and B ( ψ ) denote the sets of good and bad points of ψ respectively.For any point x belonging to either one of these sets, d ψ ( x ) denotes its defect. Lemma 6.4. The one-to-one semi-immersion ψ : ( N , ∆ qr ) → ( S q,r , ∆ q,r ) is well-behaved. In fact ψ admits a control function θ q,r : B ( ψ ) → G ( ψ ) such that d ψ ( b ) = − d ψ ( θ q,r ( b )) = 1 for all b ∈ B ( ψ ) .Proof. Before constructing the control function we need to make a few preparations. First observethe following about the values of our delta functions. If k is a non-negative integer, and x is aninteger in the interval [ kqr, ( k + 1) qr ), then ∆ qr ( x ) = k + 1. If x is also in S q,r , then ∆ q,r ( x ) = k or k + 1. Moreover, ∆ q,r ( x ) = k + 1 if and only if x − kqr ∈ S q,r .Next we show that ψ shifts all its bad points by ( q − r − . To see this, suppose b ∈ B ( ψ ). Thenby definition ∆ qr ( b ) > ∆ q,r ( ψ ( b )) ≥ 1. So ∆ qr ( b ) is at least 2 which implies b ≥ qr . By Lemma 6.3part (2), we have ψ ( b ) = b + ( q − r − . Claim . We have [ kqr, ( k + 1) qr ) ∩ B ( ψ ) ⊆ [ kqr, kqr + ( q − r − ), for every integer k ≥ Proof of Claim 6.5. Suppose to the contrary that b ≥ kqr + ( q − r − and that b is a bad point of ψ . Then ψ ( b ) ≥ kqr + ( q − r − 1) by the discussion above. Since ψ ( b ) − kqr ≥ ( q − r − ψ ( b ) − kqr ∈ S q,r . This implies that ∆ q,r ( ψ ( b )) ≥ k + 1. Since b is bad, we must have that∆ qr ( b ) ≥ k + 2, implying b ≥ ( k + 1) qr . This contradicts b being in [ kqr, ( k + 1) qr ). (cid:3) We saw that every bad point b of ψ must be in [ kqr, kqr + ( q − r − ) for some k ≥ 1. Wetherefore have ψ ( b ) ∈ [ kqr, ( k + 1) qr ). These give ∆ qr ( b ) = k + 1 and ∆ q,r ( ψ ( b )) = k , so the defectof b is 1.We are now ready to construct our control function. Fix k ≥ 1, and consider b ∈ [ kqr, kqr + ( q − r − ) ∩ B ( ψ ). Define(6.1) θ q,r ( b ) := 2 kqr − b − . This defines a map θ q,r : B ( ψ ) → N . We will check that θ q,r satisfies all the properties of beinga control function given in Definition 3.18. We can see that θ q,r ( b ) ∈ [ kqr − ( q − r − , kqr ), so∆ qr ( θ q,r ( b )) = k and θ q,r ( b ) < b . It is straightforward to check that θ q,r is injective. The followingclaim completes the proof, showing that θ q,r ( b ) is good with the appropriate defect. Claim . We have ∆ q,r ( ψ ( θ q,r ( b ))) = k + 1. Hence the defect of θ q,r ( b ) is − Proof of Claim 6.6. Since ∆ q,r ( ψ ( b )) = k , we must have b + ( q − r − − kqr S q,r . Thus b + ( q − r − − kqr < ( q − r − α b = qr − q − r − (cid:18) b + ( q − r − − kqr (cid:19) ∈ S q,r . Note that by construction, α b ∈ [0 , ( q − r − ). Let’s consider β b given by β b = kqr + α b = (2 k + 1) qr − q − r − b − ( q − r − . We have that β b ∈ [ kqr, kqr + ( q − r − ). Since α b ∈ S q,r and β b ∈ [ kqr, kqr + ( q − r − ), we have∆ q,r ( β b ) = k + 1. Furthermore, since θ q,r ( b ) ≥ ( q − r − , Lemma 6.3 gives ψ ( θ q,r ( b )) = θ q,r ( b ) + ( q − r − kqr − b − q − r − β b , and thus ∆ q,r ( ψ ( θ q,r ( b ))) = ∆ q,r ( β b ) = k + 1 . (cid:3)(cid:3) Constructing the map. We complete the proof of Theorem 1.2 in two steps. The first isto find a rigid function (see Definition 3.22) from G Y to G Y ′ where Y = Σ( p , . . . , p l , qr ) and Y ′ = Σ( p , . . . , p l , q, r ), and G Y and G Y ′ are their associated semigroups given in Theorem 2.2.The second is to show that the semi-immersion induced by Lemma 3.23 is well-behaved. We wouldlike a standard form for studying the elements of G Y and G Y ′ . We will always assume that anelement x of G Y is expressed in the form x = p · · · p l " z + qr l X i =1 x i p i ! , ANK INEQUALITIES FOR THE HEEGAARD FLOER HOMOLOGY OF SEIFERT HOMOLOGY SPHERES 19 where 0 ≤ x i < p i , but we may have z be arbitrarily large. Note that this is different from thenormal form introduced in Section 5. Like the normal form, this expression is unique. We define afunction π : G Y → N by π ( x ) = z .Similarly for G Y ′ , we have a decomposition x ′ = p · · · p l " aq + br + qr l X i =1 x ′ i p i ! , where 0 ≤ x ′ i < p i , but a and b may be arbitrarily large. We define π ′ : G Y ′ → S q,r by π ′ ( x ′ ) = aq + br . Though the above decomposition is not unique, the map π ′ is well-defined.The values of ∆ Y and ∆ Y ′ can be computed using Theorem 2.2:(6.2) ∆ Y ( x ) = ∆ qr ( π ( x )) and ∆ Y ′ ( x ′ ) = ∆ q,r ( π ′ ( x ′ )) . Now define φ : G Y → G Y ′ by(6.3) φ p · · · p l " z + qr l X i =1 x i p i ! = p · · · p l " ψ ( z ) + qr l X i =1 x i p i ! , where ψ is defined as in Lemma 6.3. Note that since ψ ( z ) ∈ S q,r , the codomain of φ is G Y ′ . Lemma 6.7. The function φ is rigid.Proof. Recall from Lemma 6.3 that ψ is injective. This implies φ is also injective. Since ψ ( π ( x )) ≥ π ( x ) for all x ∈ N , we have φ ( x ) ≥ x for all x ∈ G Y . We also have that N Y ′ − N Y = p · · · p l " lqr − ( q + r ) − qr l X i =1 p i ! − " ( l − qr − − qr l X i =1 p i ! = p · · · p l ( qr − q − r + 1) = p · · · p l ( q − r − . Therefore by Lemma 6.3, φ ( x ) − x = p · · · p l ( ψ ( π ( x )) − π ( x )) ≤ p · · · p l ( q − r − N Y ′ − N Y . Thus, φ is rigid. (cid:3) Proof of Theorem 1.2. As discussed, it suffices to showrank HF red ( Y ) ≤ rank HF red ( Y ′ )where Y = Σ( p , . . . , p l , qr ) and Y ′ = Σ( p , . . . , p l , q, r ). We have given a rigid map φ : G Y → G Y ′ in Equation (6.3). We study the induced one-to-one semi-immersion φ : ( X Y , ∆ Y ) → ( X Y ′ , ∆ Y ′ )(see Lemma 3.23). We want to see that φ is well-behaved. Hence we need to construct a controlfunction θ Y : B ( φ ) → G ( φ ) in the sense of Definition 3.18. First we need to do a few preparations.Throughout we shall assume that every element x ∈ S Y is written in the form(6.4) x = p · · · p l " π ( x ) + qr l X i =1 x i p i ! . Claim . Suppose x ∈ S Y . Then d φ ( x ) = d ψ ( π ( x )). In particular, x ∈ B ( φ ) if and only if π ( x ) ∈ B ( ψ ). Similarly, x ∈ G ( φ ) if and only if π ( x ) ∈ G ( ψ ). Proof of Claim 6.8. We simply calculate the defect of x using Equation (6.4) to see that d φ ( x ) = | ∆ Y ( x ) | − | ∆ Y ′ ( φ ( x )) | = ∆ qr ( π ( x )) − ∆ q,r ( ψ ( π ( x ))) = d ψ ( π ( x )) . (cid:3) Claim . Suppose y ∈ Q Y with y = N Y − x where x ∈ S Y . Then d φ ( y ) = d ψ ( π ( x )). In particular, y ∈ B ( φ ) if and only if π ( x ) ∈ B ( ψ ). Similarly, y ∈ G ( φ ) if and only if π ( x ) ∈ G ( ψ ). Proof of Claim 6.9. Again we calculate the defect of y using Equation (6.4) to see that d φ ( y ) = | ∆ Y ( y ) | − | ∆ Y ′ ( φ ( y )) | = ∆ qr ( π ( x )) − ∆ q,r ( ψ ( π ( x ))) = d ψ ( π ( x )) . (cid:3) We are now ready to define the control function θ Y . First define it on S Y ∩ B ( φ ). Suppose x ∈ S Y ∩ B ( φ ). Then we let θ Y ( x ) := p · · · p l " θ q,r ( π ( x )) + qr l X i =1 x i p i ! , where θ q,r is as defined in Equation (6.1). Note that Claim 6.8 implies that the term θ q,r ( π ( x ))above makes sense. Next define θ Y on Q Y ∩ B ( φ ). Suppose y ∈ Q Y ∩ B ( φ ) with y = N Y − x . Thenwe let θ Y ( y ) := N Y − p · · · p l " θ q,r ( π ( x )) + qr l X i =1 x i p i ! . Note that Claim 6.9 implies that the term θ q,r ( π ( x )) above makes sense.Having seen that the map θ Y is defined on B ( φ ), let us now show that it has the correctcodomain. We must show that 0 ≤ θ Y ( b ) ≤ N Y for all b ∈ B ( φ ). Indeed, since θ q,r is a controlfunction, 0 ≤ θ q,r ( π ( x )) < π ( x ) for all x ∈ S Y ∩ B ( φ ). This implies0 ≤ θ Y ( x ) < x < N Y for all x ∈ S Y ∩ B ( φ ) , (6.5) 0 < y < θ Y ( y ) ≤ N Y for all y ∈ Q Y ∩ B ( φ ) . (6.6)It is now clear that we have a map θ Y : B ( φ ) → X Y such that θ Y ( S Y ∩ B ( φ )) ⊆ S Y and θ Y ( Q Y ∩ B ( φ )) ⊆ Q Y .We start checking that θ Y satisfies the properties listed in Definition 3.18. Since θ q,r is a controlfunction, Claim 6.8 and Claim 6.9 respectively imply that θ Y ( S Y ∩ B ( φ )) ⊆ S Y ∩ G ( φ ) ,θ Y ( Q Y ∩ B ( φ )) ⊆ Q Y ∩ G ( φ ) . Hence θ : B ( φ ) → G ( φ ) satisfies Property (1) of Definition 3.18. Clearly θ Y is injective since θ q,r isinjective. Property (2) was shown in Inequalities (6.5) and (6.6) above. Finally Lemma 6.4 togetherwith Claim 6.8 and Claim 6.9 imply that | d φ ( b ) | = | d φ ( θ Y ( b )) | = 1 for all b ∈ B ( φ ). Hence Property(3) follows.Therefore, we have shown that the one-to-one semi-immersion φ admits a control function, andhence it is well-behaved. The theorem now follows from Corollary 3.20. (cid:3) ANK INEQUALITIES FOR THE HEEGAARD FLOER HOMOLOGY OF SEIFERT HOMOLOGY SPHERES 21 Botany In this section we will prove Theorem 1.6 and some other results related to the botany question. Proof of Theorem 1.6. Without loss of generality assume the multiplicities of the singular fibersare in the increasing order, 1 < p < p < . . . < p l . Therefore p j > j for all j = 1 , . . . , l . Now n = rank HF red (Σ( p , . . . , p l )) ≥ l ≤ 2, since S is the only Seifert homology sphere with at most 2 singular fibers and HF red ( S ) = 0. Part (1) is trivial when l = 3. Suppose l ≥ 4. We have ( p , p , p , p ) ≥ (2 , , , HF red (Σ(2 , , , n = rank HF red (Σ( p , . . . , p l )) ≥ rank HF red (Σ( p , . . . , p l − p l )) ≥ p l rank HF red (Σ( p , . . . , p l − ))... ≥ p l · · · p rank HF red (Σ( p , p , p , p )) ≥ l !4! rank HF red (Σ(2 , , , > l !2 , which establishes part (1) of the theorem.For part (2), suppose to the contrary that p l ≥ n + 7. One can directly verify that the rank of HF red (Σ(2 , , n + 7)) is n + 1. By repeatedly applying Theorem 1.2 and Theorem 1.1, we get n = rank HF red (Σ( p , . . . , p l )) ≥ p . . . p l − rank HF red (Σ( p l − , p l − , p l )) ≥ rank HF red (Σ(2 , , n + 7)) (by Theorem 1.4)= n + 1 , which is a contradiction. (cid:3) Recall that for a Spin c structure s on a rational homology sphere Y , the group HF + ( Y, s ) comesequipped with a Q -valued grading [16]. The correction term, d ( Y, s ), is defined to be the minimalgrading of an element of HF + ( Y, s ) which lies in the image of the obvious map from HF ∞ ( Y, s )to HF + ( Y, s ). In the case of a homology sphere, we omit the Spin c structure from the notation.Furthermore, in this case, the correction term is always an even integer. Corollary 7.1. For fixed n ∈ N , there are at most finitely many Seifert homology spheres Y with rank HF red ( Y ) ≤ n . Therefore, there are at most finitely many integers d that can occur as thecorrection term of a Seifert homology sphere Y with rank HF red ( Y ) = n .Proof. By Theorem 1.6, n easily provides an upper bound on the number of singular fibers, andtheir multiplicities for which rank HF red = n . (cid:3) Corollary 7.2. For a fixed integer n , there are at most finitely many Seifert homology spheres Y with λ ( Y ) = n , where λ denotes the Casson invariant.Proof. We apply [16, Theorem 1.3] relating HF red , the correction term, and the Casson invariant: λ ( Y ) + d ( Y )2 = χ ( HF red ( Y )) . After a possible change in orientation, Y bounds a positive-definite four-manifold, arising from aplumbing. It is shown in [17] that HF + ( Y ) is supported in even gradings. Therefore, χ ( HF red ( Y )) = rank HF red ( Y ). Furthermore, in this case, d ( Y ) ≤ 0. Therefore, λ ( Y ) ≥ rank HF red ( Y ). Corol-lary 7.1 now gives the proof. (cid:3) Botany of Seifert homology spheres with rank at most . Here we list all the Seiferthomology spheres whose rank of the reduced Heegaard Floer homology is at most 12. By Theo-rem 1.6, these Seifert homology spheres necessarily have 3 singular fibers excepting the 3–sphere.The same result says that the maximum multiplicity of these singular fibers is no more than 79. Welooked at all the triples of pairwise relatively prime integers ( p , p , p ) with 1 < p < p < p < Rank Triple n = 0 S , [2 , , n = 1 [2 , , , [2 , , n = 2 [3 , , , [2 , , , [3 , , , [2 , , , [2 , , , [2 , , n = 3 [2 , , , [2 , , , [2 , , , [2 , , , [2 , , , [3 , , n = 4 [3 , , , [2 , , , [3 , , , [2 , , , [2 , , , [2 , , n = 5 [4 , , , [3 , , , [3 , , , [2 , , , [2 , , , [2 , , , [2 , , n = 6 [2 , , , [2 , , , [2 , , , [4 , , , [3 , , , [3 , , , [2 , , , [3 , , , [2 , , , [2 , , n = 7 [2 , , , [3 , , , [3 , , , [3 , , , [2 , , , [2 , , , [2 , , n = 8 [2 , , , [3 , , , [2 , , , [2 , , , [2 , , , [2 , , , [4 , , , [2 , , , [3 , , , [2 , , , [5 , , , [3 , , , [4 , , n = 9 [4 , , , [2 , , , [2 , , , [2 , , , [2 , , , [3 , , , [3 , , , [3 , , , [2 , , n = 10 [2 , , , [3 , , , [2 , , , [2 , , , [5 , , , [2 , , , [2 , , , [3 , , , [2 , , , [2 , , n = 11 [2 , , , [3 , , , [2 , , , [2 , , , [3 , , , [2 , , , [4 , , n = 12 [5 , , , [4 , , , [4 , , , [2 , , , [2 , , , [3 , , , [3 , , , [2 , , , [2 , , , [2 , , , [2 , , , [2 , , , [3 , , , [3 , , , [2 , , , [5 , , , [3 , , , [2 , , , [3 , , Table 1. Seifert homology spheres with rank of reduced Heegaard Floer homologyless than or equal to 12.8. Maps between Seifert homology spheres It is our goal in this section to prove Theorem 1.9 and therefore we must study maps f : Y ′ → Y between Seifert homology spheres. First of all, it is clear that Theorem 1.9 trivially holds ifthe degree of f is zero. Thus, for the rest of this section, we only study non-zero degree maps.Furthermore we always choose orientations of Y and Y ′ such that deg f > 0. We will determinewhich pairs of Seifert homology spheres admit non-zero degree maps between them and what theirpossible degrees are. We will then use this in conjunction with the inequalities in Theorems 1.1and 1.2 to prove Theorem 1.9. It turns out that the problem of the existence of non-zero degreemaps between Seifert fibered spaces with infinite fundamental groups is settled [5, 22, 23], and wewill specialize this to the case of homology spheres.Before beginning our discussion we set up some conventions which will be used throughout thissection. We shall introduce and use unnormalized Seifert invariants (see for example [13, Sections2.14 and 2.15]). Let Y be a Seifert homology sphere. Let ( e , ( p ′ , p ) , . . . , ( p ′ t , p t )) be its normalizedSeifert invariants. We define the Euler number of Y to be e ( Y ) := e + t X i =1 p ′ i p i . The unnormalized Seifert invariants of Y is a set (cid:26) ω α , . . . , ω k α k (cid:27) with k ≥ t satisfying(1) α i = p i for all i = 1 , . . . , t ,(2) ω i ≡ − p ′ i mod p i for all i = 1 , . . . , t ,(3) gcd( ω i , α i ) = 1 for all i = 1 , . . . , t ,(4) e ( Y ) = − k X i =1 ω i α i . ANK INEQUALITIES FOR THE HEEGAARD FLOER HOMOLOGY OF SEIFERT HOMOLOGY SPHERES 23 Note that the unnormalized Seifert invariants are not unique, but they uniquely determine thenormalized Seifert invariants and hence the Seifert homology sphere Y . When referring to a Seiferthomology sphere Σ( p , . . . , p t ), we will also allow p i = 1, unless stated otherwise.We begin with the types of maps that can exist between Seifert fibered spaces. Theorem 8.1 (Rong, Theorem 3.2 of [22]) . Suppose that f : Y ′ → Y is a map between P -irreducible Seifert fibered spaces with infinite fundamental group. Then, f is homotopic to a map p ◦ g ◦ π , where π is a degree one map between Seifert fibered spaces, g is a fiber-preserving branchedcover branched along fibers, and p is a covering. Furthermore, if Y is not a Euclidean manifold,we can choose p = id . Rather than define P -irreducible manifolds, we point out that Seifert homology spheres arealways P -irreducible. Theorem 8.1 is actually more specific, stating that the map π is a specialtype of degree one map, called a vertical pinch which is defined below. Since Seifert homologyspheres never have Euclidean geometry, we may further assume f is homotopic to g ◦ π as above.The following is a standard fact about non-zero degree maps, but we give a proof for completeness. Proposition 8.2. If f : Y ′ → Y is a non-zero degree map between closed, connected, orientablethree-manifolds, the index of f ∗ ( π ( Y ′ )) as a subgroup of π ( Y ) divides the degree of f , and thus isfinite. In particular, if Y ′ is an integer homology sphere and the degree of f is one, then Y is aninteger homology sphere as well.Proof. Let e π : e Y → Y be the covering corresponding to the subgroup f ∗ ( π ( Y ′ )). Then we can lift f to a map e f : Y ′ → e Y such that f = e π ◦ e f . If the index of f ∗ ( π ( Y ′ )) is not finite then e Y is notcompact hence H ( e Y ) is trivial. This implies that f : H ( Y ′ ) → H ( Y ) factors through the trivialgroup. Therefore, deg f = 0. If the index of f ∗ ( π ( Y ′ )) is finite then it is equal to deg e π and wehave deg f = deg e π deg e f . In particular, the index divides deg f . In the case that deg f = 1, wesee that f ∗ : π ( Y ′ ) → π ( Y ) is surjective. Abelianizing this map shows that H ( Y ′ ) surjects onto H ( Y ) if deg f = 1. Hence, Y is an integral homology sphere if Y ′ is. (cid:3) We also note that the only Seifert homology spheres with finite fundamental group are S andΣ(2 , , HF red ( S ) = HF red (Σ(2 , , π ( Y )is finite. Thus, it suffices to consider the case that π ( Y ) is infinite. By Proposition 8.2, we mayassume that π ( Y ′ ) is also infinite.Therefore, given a non-zero degree map, f , between Seifert homology spheres with infinite fun-damental group, we may factor f ≃ g ◦ π as in Theorem 8.1 where the codomain of π /domain of g isalso a Seifert homology sphere with infinite fundamental group. In order to prove Theorem 1.9, wetherefore analyze the pairs of Seifert homology spheres with infinite fundamental group which admitnon-zero degree maps between them via the next two propositions. We will use the abbreviationISHS for a Seifert homology sphere with infinite fundamental group. Proposition 8.3. Let π : Y ′ → Y be a degree one map between ISHS’s. Then Y can be obtainedfrom Y ′ by a sequence of moves (up to permutation of the multiplicities of the singular fibers) ofthe form Σ( p , . . . , p l ) → Σ( p , . . . , p k , p k +1 · · · p l ) . Proposition 8.4. If f : Y ′ → Y is a fiber-preserving branched cover between ISHS’s, then Y ′ admits a map to Y with the same degree as f which is obtained by a composition (up to permutationof the multiplicities of the singular fibers) of branched covers of the form: • φ : Σ( p , . . . , np l ) → Σ( p , . . . , p l ) or • ψ : Σ( p , . . . , p l , n ) → Σ( p , . . . , p l ) , where φ (respectively ψ ) is the n -fold cyclic branched cover, branched over the singular fiber of order p l (respectively a regular fiber). Before giving the proofs, we will see how these two propositions lead to a proof of Theorem 1.9. Proof of Theorem 1.9. As discussed, we only need to consider the case of a non-zero degree map f : Y ′ → Y between ISHS’s. We therefore factor f as g ◦ π as in Theorem 8.1. Since degreeis multiplicative under composition, we can prove Inequality (1.4) for π and g separately. Theinequality for π follows from Proposition 8.3 and Theorem 1.2. On the other hand, for g , weobserve that by combining Theorems 1.1 and 1.2, we obtain n rank HF red (Σ( p , . . . , p l )) ≤ rank HF red (Σ( p , . . . , np l )) ≤ rank HF red (Σ( p , . . . , p l , n )) . The result now follows from Proposition 8.4, since an n -fold branched covering is a degree n map. (cid:3) The rest of this section is devoted to the proofs of Proposition 8.3 and Proposition 8.4, whichwill thus complete the proof of Theorem 1.9.In order to prove Proposition 8.3, we recall a special kind of degree one map. We begin by fixinga Seifert fibered space M with a separating torus T which is vertical (i.e. T is foliated by fibers).Decompose M along T into two components, M and M . Furthermore, suppose that there existsan essential simple closed curve on T which bounds a 2-sided surface in M . A degree one map f from M to M ∪ T D × S is a vertical pinch if f | M is the identity and f maps M onto D × S .It is straightforward to check that there exists a vertical pinch Σ( p , . . . , p l , q, r ) → Σ( p , . . . , p l , qr )(in this case M = D ( p , . . . , p l ) and M = D ( q, r )). With this definition, we are now able todescribe the degree one maps that appear in Theorem 8.1. Theorem 8.5 (Rong, Corollary 3.3 of [22]) . Let f : Y ′ → Y be a degree one map between closed, P -irreducible Seifert fibered spaces with infinite fundamental group. Then, f is a composition ofvertical pinches.Proof of Proposition 8.3. Let f : Y ′ → Y be a degree one map between ISHS’s. We write theunnormalized Seifert invariants of Y ′ as { q ′ q , . . . , q ′ k q k } . By Theorem 8.5, f is a composition ofvertical pinches. We now apply [23, Theorem 3.2] which states that there are unnormalized Seifertinvariants { p ′ p , . . . , p ′ l p l } for Y and a partition { , . . . , k } = I ∪ . . . ∪ I l such that for each j ,(8.1) X n ∈ I j q ′ n q n = p ′ j p j , lcm n ∈ I j { q n } = p j . Since q i (respectively p i ) are relatively prime, it is straightforward to check that Condition (8.1)implies that p j = Q n ∈ I j q n . Therefore, the relation between Y ′ and Y is as in the statement of theproposition. (cid:3) Now suppose that f : Y ′ → Y is a fiber-preserving branched cover between ISHS’s. Fix un-normalized Seifert invariants for Y ′ and Y as { q ′ q , . . . , q ′ k q k } and { p ′ p , . . . , p ′ l p l } respectively, arrangingthat p j ≥ p ′ j p j = 0 for each j . We use S ′ and S to denote the base orbifolds of Y ′ and Y respectively. The orders of the orbifold points correspond to the multiplicities of the singular fibersof the Seifert fibered space sitting over it. Since f is fiber-preserving, we obtain an induced mapbetween the base orbifolds F : S ′ → S . We call the degree of F the orbifold degree of f . Fix aregular fiber, h ′ , of Y ′ which f maps to a regular fiber, h , in Y , not contained in the branch setof f . The degree of f | h ′ : h ′ → h is called the fiber degree of f . It follows that the degree of f is the product of the fiber degree and the orbifold degree. While there are many fiber-preservingbranched covers between Seifert fibered spaces with arbitrary orbifold degree, we will see this does ANK INEQUALITIES FOR THE HEEGAARD FLOER HOMOLOGY OF SEIFERT HOMOLOGY SPHERES 25 not happen for ISHS’s. The following argument was shown to us by Ian Agol. We point out thatthe ideas are also similar to those used in the proof of [5, Lemma 2.1]. Proposition 8.6. Let f : Y ′ → Y be a fiber-preserving branched cover between ISHS’s. Then, thefiber degree of f equals the degree of f .Proof. We argue by contradiction. Therefore, we assume that the map F : S ′ → S on the underlyingtopological spaces has degree at least two, and thus is a non-trivial branched cover. Let d denotethis degree. The first claim is that the orbifold points of S must be contained within the branchset. Otherwise, there would exist an orbifold point x ∈ S with order at least two which would liftto more than one orbifold point in S ′ with the same order. Therefore, Y ′ would have two singularfibers with the same multiplicity (of at least two), contradicting Y ′ being an integer homologysphere.We write x , . . . , x l to denote the orbifold points of S ′ , where we write the order of x i as p i . Wealso let B i = f − ( x i ) and write B i = { a i , . . . , a i | B i | } . Denote by m ij the order of the orbifold point a ij (which may be 1). In other words, Y ′ = Σ( q , . . . , q k ) ∼ = Σ( m , . . . , m | B | , . . . , m l , . . . , m l | B l | ) . We would like to see how the m ij , p i , and d are related. Let d ij denote the local degree of F (between the underlying topological spaces) at a ij ; clearly P j d ij = d for each i . We also have m ij = p i gcd( d ij ,p i ) for all i and j .Fix some 1 ≤ i ≤ l and let r i denote the smallest prime which divides p i . We claim that(8.2) | B i | ≤ d + r i − r i . This is seen as follows. Since Y ′ is an integer homology sphere, gcd( m ij , m ij ′ ) = 1 for all j, j ′ . Inparticular, there is at most one j such that d ij = 1. We denote this index by j ∗ , should it exist.For all j = j ∗ , some non-trivial factor of d ij must divide p i . Therefore, we must have that d ij ≥ r i .Thus, we see d = X j d ij = X j = j ∗ d ij + d ij ∗ ≥ ( | B i | − r i + 1 , which gives Inequality (8.2). We will apply this relation to contradict the Riemann-Hurwitz formulaapplied to F , which states(8.3) l X i =1 | B i | ≥ ( l − d + 2 , because S and S ′ are spheres.Without loss of generality, we may assume that 2 ≤ p < . . . < p l . First, suppose that l ≥ 4. ByInequality (8.2), we have | B | ≤ d + 12 , | B | ≤ d + 23 , | B | ≤ d + 45 , and | B i | ≤ d + 67 for i ≥ . Therefore, we have l X i =1 | B i | ≤ l X i =1 d + r i − r i ≤ d + 12 + d + 23 + d + 45 + ( l − 3) ( d + 6)7= 31 d 30 + 5930 + ( l − 3) ( d + 6)7= 31 d 30 + 5930 + ( l − d + 6( l − − d )7= ( l − d + 2 + d − 130 + 6( l − − d )7= ( l − d + 2 − ( d − (cid:20) l − − (cid:21) < ( l − d + 2 , since l − ≥ d ≥ 2. This contradicts Inequality (8.3).Therefore, we assume that l ≤ 3. Since Y is an ISHS, we have l = 3. We begin with the casethat p > 2. In this case, Inequality (8.2) implies that for all i , | B i | ≤ d + 23 , and this inequality fails to be strict for at most one i (namely, equality is only possible if i = 1 and p = 3, since we put the p i in increasing order). Therefore, l X i =1 | B i | < d + 23 = d + 2 , contradicting Inequality (8.3). Thus, we let p = 2. Note that since π ( Y ) is infinite by assumption,we cannot have ( p , p , p ) = (2 , , p ≥ p ≥ 7. We againapply Inequality (8.2) to see X | B i | ≤ d + 12 + d + 23 + d + 67= 41 d 42 + 8542 < d + 2 , contradicting Inequality (8.3). (cid:3) Proof of Proposition 8.4. We let g : Y ′ → Y be a fiber-preserving branched cover between ISHS’s,which necessarily has degree equal to the fiber degree by Proposition 8.6. Let d denote this degree.We suppose that Y ′ (respectively Y ) has unnormalized Seifert invariants { q ′ q , . . . , q ′ k q k } (respectively { p ′ p , . . . , p ′ l p l } ), where we again allow p j = 1. We now apply [5, Theorem 2.2], which describes howthe Seifert invariants change under fiber-preserving branched covers, to see that { dq ′ q , . . . , dq ′ k q k } alsogives a set of unnormalized Seifert invariants for Y . Let d i = gcd( d, q i ). Therefore, we may write Y as Σ( q d , . . . , q l d l ). Next, [14, Theorem 1.2] states that if f : Y ′ → Y is a fiber-preserving branchedcover between closed, aspherical, oriented Seifert fibered spaces, then we have that e ( Y ′ ) = dd fib e ( Y ), ANK INEQUALITIES FOR THE HEEGAARD FLOER HOMOLOGY OF SEIFERT HOMOLOGY SPHERES 27 where d fib denotes the fiber degree of f . In our current setting, d fib = d , which gives e ( Y ′ ) = e ( Y ) d .For a Seifert homology sphere, we have e (Σ( r , . . . , r m )) = − r ··· r m . We therefore see that d · Y q i d i = q · · · q l . In particular, we have d = d · · · d l . In other words, Y can be obtained from Y ′ by a sequence of themoves described in the statement of the proposition and the degree of the map is as predicted. (cid:3) Discussion and further questions We now speculate about some possible generalizations of Theorem 1.4 and Theorem 1.9.9.1. Botany. Computational evidence suggests that Theorem 1.4 should hold for d HF . Therefore,we state this as a conjecture. Conjecture 9.1. Suppose that ( p , . . . , p l ) ≤ ( q , . . . , q l ) . Then, rank d HF (Σ( p , . . . , p l )) ≤ rank d HF (Σ( q , . . . , q l )) . Assuming Conjecture 9.1, it is in fact easy to see that the above inequality can be made strictif at least one q i is much larger than p i by pushing the argument in the proof of Proposition 4.3.Therefore, Conjecture 9.1 would imply that there are only finitely many Seifert homology sphereswith a fixed rank of d HF . A similar argument as in the proof of Theorem 1.6 would then guaranteean algorithmic solution to the botany problem for the hat-flavor of the Heegaard Floer homologyof Seifert homology spheres. Conjecture 9.2. Given n ≥ , there are at most finitely many Seifert homology spheres Y suchthat rank d HF ( Y ) = n . Moreover, there is an algorithm to find all such Y . Non-zero degree maps. As pointed out in the introduction, we did not directly make useof the branched covers or vertical pinches in the proofs of Theorem 1.1 and Theorem 1.2. It wouldbe interesting to see the roles of these maps in the rank inequalities. Problem 9.3. Prove Theorem 1.1 and Theorem 1.2 by directly using the corresponding maps. Proving Theorem 1.1 and Theorem 1.2 this way could also yield a generalization of Theorem 1.9.Now, we would like to ask about possible generalizations of Theorem 1.9. First, we point outthat the statement is too strong to generalize to all three-manifolds. One example can be seen bytaking Y = Σ g × S , the product of a genus g surface with a circle, for g ≥ Y covers itself non-trivially by a degree k map f k , for any k , an analogueof Theorem 1.9 would imply that rank HF red (Σ g × S ) = 0. However, this has been computed tobe non-trivial [18, 6]. We cannot have an analogous inequality even if we restrict to individualSpin c structures. By choosing s ∈ Spin c (Σ g × S ) such that rank HF red (Σ g × X , s ) = 0, we seethat | k | rank HF red (Σ g × S , s ) ≤ rank HF red (Σ g × S , f ∗ k s ) cannot hold for | k | ≫ 0, since for all k , rank HF red (Σ g × S , f ∗ k s ) ≤ rank HF red (Σ g × S ), the latter of which is finite. It also seemsunlikely that this is something special to having non-trivial first homology. We expect that thereare self-maps between integer homology spheres with non-trivial HF red which have deg ≥ HF red , again,the inequality in Theorem 1.9 still seems very unlikely. Therefore, we instead propose a weakerinequality. Conjecture 9.4. If f : Y ′ → Y is a non-zero degree map between integer homology spheres, then rank HF red ( Y ′ ) ≥ rank HF red ( Y ) , rank d HF ( Y ′ ) ≥ rank d HF ( Y ) . Correction terms. It is also natural to expect that there should be inequalities for thecorrection terms of Seifert homology spheres analogous to those in Theorems 1.1 and 1.4. 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