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Celebratio Mathematica

Walter D. Neumann

Splice diagrams and splice-quotient surface singularities

by Jonathan Wahl

It has been my great pleas­ure to col­lab­or­ate on many pa­pers with Wal­ter Neu­mann, a ter­rif­ic math­em­atician and friend. I’ll try to present an over­all ac­count of some of our work, primar­ily on splice dia­grams and splice-quo­tient sin­gu­lar­it­ies. Our goal was to be able to write down ex­pli­cit equa­tions of cer­tain kinds of nor­mal sur­face sin­gu­lar­it­ies, giv­en only the to­po­logy of their links. I men­tion our mo­tiv­a­tions, and why and how we were led to cer­tain kinds of ques­tions, leav­ing de­tailed proofs (and care­ful defin­i­tions) to the ori­gin­al pa­pers. One can also con­sult in­ter­est­ing more re­cent work of oth­ers, such as [e13], which uses a trop­ic­al geo­metry ap­proach.

1. Provocative early work of Walter

While I first be­came friendly with Wal­ter in 1972 at the In­sti­tute for Ad­vanced Study, our col­lab­or­a­tion did not be­gin un­til 1987, well after he had writ­ten two pa­pers about nor­mal sur­face sin­gu­lar­it­ies which I found very pro­voc­at­ive.

Re­call that if (X,0) is a com­plex nor­mal sur­face sin­gu­lar­ity, its link Σ is the ori­ented 3-man­i­fold which is its neigh­bor­hood bound­ary, i.e., X in­ter­sec­ted with a small sphere about 0. From the min­im­al good res­ol­u­tion π:(X~,E)(X,0), where π1(0)=E is a uni­on of smooth curves with strong nor­mal cross­ings, Σ is the bound­ary of a tu­bu­lar neigh­bor­hood of E. One as­so­ci­ates to E a plumb­ing graph Γ, en­cod­ing the gen­era, self-in­ter­sec­tions, and in­ter­sec­tions of the curves. From Γ, one can re­con­struct Σ by the plumb­ing con­struc­tion, as made clear in sem­in­al pa­pers [e1] and [e2] by Mum­ford and Hirzebruch. Thus the plumb­ing graph Γ de­term­ines the to­po­logy of Σ. Wal­ter’s ob­vi­ously im­port­ant the­or­em proves that not only does Γ de­term­ine the 3-man­i­fold Σ, but the con­verse is true as well.

The­or­em 1.1: [1] Sup­pose Σ is a sin­gu­lar­ity link. Then Σ de­term­ines the min­im­al good res­ol­u­tion graph Γ.

In fact, with the ex­cep­tion of lens spaces (from cyc­lic quo­tient sin­gu­lar­it­ies) and some two-tor­us bundles over the circle (from cusp sin­gu­lar­it­ies), already π1(Σ) de­term­ines Γ.

The the­or­em was proved by a cal­cu­lus for gen­er­al plumb­ing dia­grams (not ne­ces­sar­ily neg­at­ive-def­in­ite), and im­plied oth­er ba­sic res­ults (e.g., S1×S2 is not a sin­gu­lar­ity link). Thus, any in­vari­ant con­cocted from Γ is to­po­lo­gic­al, and not just a sin­gu­lar­ity in­vari­ant. On the oth­er hand, the geo­met­ric genus and Mil­nor num­ber de­pend in gen­er­al on ana­lyt­ic in­form­a­tion.

Figure 1.

A later res­ult of Wal­ter’s in­volved cer­tain sin­gu­lar­it­ies with a ra­tion­al ho­mo­logy sphere (or QHS) link; that is, E is a tree of smooth ra­tion­al curves. Then H1(Σ;Z)H1 is the fi­nite dis­crim­in­ant group D(Γ), cal­cu­lated for ex­ample as the coker­nel of the in­ter­sec­tion mat­rix (EiEj); thus, Σ has a fi­nite uni­ver­sal abeli­an cov­er­ing (or UAC) ΣΣ. That cov­er­ing is real­ized by a fi­nite map of nor­mal germs (X,0)(X,0), which is a quo­tient by an ac­tion of the group H1; this is the UAC of (X,0).

A weighted-ho­mo­gen­eous sin­gu­lar­ity with QHS link has res­ol­u­tion graph shown in Fig­ure 1.

Figure 2.

Here, the con­tin­ued frac­tion ex­pan­sion n/q=b11/b21/bs rep­res­ents a string of ra­tion­al curves em­an­at­ing from the cen­ter; see Fig­ure 2.

(We shall as­sume that t3.) The ana­lyt­ic type of (X,0) is uniquely de­term­ined by the graph plus the loc­a­tion of the t in­ter­sec­tion points on the cent­ral curve; in fact, one can write X=Speci=0Ai, where each Ai is com­put­able as a co­homo­logy group from data on the cent­ral P1 [e5], The­or­em 5.1. Still, equa­tions de­fin­ing the sin­gu­lar­ity might be quite com­plic­ated, where­as the UAC is simple.

The­or­em 1.2: [2] With nota­tion as above, the UAC of the weighted-ho­mo­gen­eous sin­gu­lar­ity (X,0) is a Brieskorn–Pham com­plete in­ter­sec­tion (X,0)(Ct,0), defined by equa­tions j=1taijzjnj=0,i=1,,t2. In ad­di­tion, the dis­crim­in­ant group H1 acts on X di­ag­on­ally on the t co­ordin­ates zj.

The mat­rix (aij) in the The­or­em sat­is­fies the Hamm con­di­tion [e4], that all max­im­al minors are nonzero.

I found this a very sur­pris­ing res­ult! It is easy to show that (X,0) is Goren­stein; but why is it a com­plete in­ter­sec­tion? And, while its de­fin­ing equa­tions must be weighted-ho­mo­gen­eous, why are they of the very spe­cial Brieskorn–Pham type? Wal­ter’s proof was simply to write down these equa­tions, find an ap­pro­pri­ate rep­res­ent­a­tion of the dis­crim­in­ant group H1 on Ct, and then to prove that (for ju­di­ciously chosen aij) the quo­tient is the sin­gu­lar­ity we star­ted with.

At­tempts to find a nat­ur­al gen­er­al state­ment which in­cludes this the­or­em led years later to our defin­i­tion of splice-quo­tient sin­gu­lar­it­ies in [6] (see Sec­tion 5).

2. Casson invariant and splice diagrams

In 1987 I at­ten­ded a lec­ture giv­en by Sir Mi­chael Atiyah at the Weyl Centen­ni­al Con­fer­ence at Duke Uni­versity, en­titled “New in­vari­ants of three and four-man­i­folds” [e6]. An­drew Cas­son had dis­covered a new in­teger in­vari­ant λ(Σ) for a com­pact in­teg­ral ho­mo­logy 3-sphere (or ZHS); its mod 2 re­duc­tion is the Rokh­lin in­vari­ant (see, e.g., [e7], The­or­em 12.1). Nat­ur­ally I asked about its value for the link Σ(p,q,r) of the Brieskorn sin­gu­lar­ity xp+yq+zr=0, which is a ZHS when p,q,r are pair­wise re­l­at­ively prime. Atiyah answered that based on de­vel­op­ing work by Fin­tushelStern, λ was a com­plic­ated ex­pres­sion in­volving the num­ber of in­teger points in some three-di­men­sion­al re­gion. I guessed it was re­lated to the sin­gu­lar­ity it­self; a call to Wal­ter com­pletely cla­ri­fied the situ­ation.

First, λ(Σ) is minus one-half the “num­ber” of equi­val­ence classes of SU(2) rep­res­ent­a­tions of π1(Σ), where num­ber is an al­geb­ra­ic count done us­ing a Hee­gaard de­com­pos­i­tion of Σ. Wal­ter ex­plained that L. Green­berg had found an ac­tu­al count of such rep­res­ent­a­tions, giv­en by the num­ber of in­teger points in some re­gion, and Fin­tushel–Stern were prov­ing that λ(Σ(p,q,r)) in­volves simply the ac­tu­al count. But then we real­ized that a for­mula of Brieskorn [e3] im­plied that the cor­res­pond­ing ex­pres­sion for λ was equal to 1/8 the sig­na­ture of the in­ter­sec­tion pair­ing on the second ho­mo­logy of the Mil­nor fiber of the sin­gu­lar­ity. (As the pair­ing is even, nonde­gen­er­ate, and un­im­od­u­lar, a ba­sic res­ult im­plies that the sig­na­ture is di­vis­ible by 8.) Put­ting to­geth­er work of oth­ers, we had real­ized a beau­ti­ful and pro­voc­at­ive ex­pres­sion for the Cas­son in­vari­ant of these par­tic­u­lar links. Our pa­per [4] con­tained the fol­low­ing con­jec­ture, ap­peal­ing yet wildly op­tim­ist­ic.

Cas­son In­vari­ant Con­jec­ture: (CIC) Sup­pose (X,0) is the link of a hy­per­sur­face (or com­plete in­ter­sec­tion) sin­gu­lar­ity whose link Σ is an in­teg­ral ho­mo­logy sphere. Then the Cas­son in­vari­ant of Σ equals one-eighth the sig­na­ture of the Mil­nor fiber.
The­or­em 2.1: [4] The Cas­son In­vari­ant Con­jec­ture is true in the fol­low­ing cases:
  1. The Brieskorn–Pham com­plete in­ter­sec­tion j=1taijzjnj=0,i=1,,t2, with pair­wise re­l­at­ively prime ni.
  2. The sin­gu­lar germ zn+f(x,y)=0, where f(x,y)=0 is an ir­re­du­cible plane curve sin­gu­lar­ity and n is re­l­at­ively prime to all the num­bers among the Puiseux pairs.
  3. The com­plete in­ter­sec­tion sin­gu­lar­ity in C4 defined by xn=un+1+vny,yn=vn+1+unx.

The meth­od was cor­rect but math­em­at­ic­ally dis­ap­point­ing: we com­pute and com­pare the Cas­son in­vari­ant and sig­na­ture in each case. Wal­ter real­ized that the key was the use of splice dia­grams, in­tro­duced earli­er by Sieben­mann, and ex­tens­ively stud­ied by Eis­en­budNeu­mann [3]. Spli­cing refers to a dif­fer­ent to­po­lo­gic­al con­struc­tion than plumb­ing, al­low­ing one to make a new ZHS from two old ones. Spe­cific­ally, if (Σi,Ki) is a pair of a ho­mo­logy 3-sphere and a knot, one re­moves the in­teri­ors of closed tu­bu­lar neigh­bor­hoods K1×D2 of K1 and D2×K2 of K2, and pastes the Σi to­geth­er along the tor­us bound­ar­ies K1×S1 and S1×K2 — in oth­er words, switch­ing the roles of me­ridi­an and lon­git­ude (and pay­ing at­ten­tion to ori­ent­a­tions!). This pro­cess yields a new ho­mo­logy sphere, which in our situ­ation can be rep­res­en­ted by a splice dia­gram.

Figure 3.

A splice dia­gram Δ is a fi­nite tree with ver­tices only of valency 1 (leaves) or 3 (nodes) and with a col­lec­tion of in­teger weights at each node, as­so­ci­ated to the edges de­part­ing the node. The Brieskorn link Σ(p,q,r) is rep­res­en­ted by the dia­gram shown in Fig­ure 3.

The next ex­ample is the dia­gram from spli­cing Σ(2,3,7) with Σ(2,5,11) along the knots giv­en by the last co­ordin­ates, shown in Fig­ure 4.

Figure 4.

For an edge con­nect­ing two nodes in a splice dia­gram the edge de­term­in­ant is the product of the two weights on the edge minus the product of the weights ad­ja­cent to the edge. In the above ex­ample, the one edge con­nect­ing two nodes has edge de­term­in­ant 7760=17.

The­or­em 2.2: [3] The ho­mo­logy spheres that are sin­gu­lar­ity links are in one-to-one cor­res­pond­ence with splice dia­grams sat­is­fy­ing these con­di­tions:
  • The weights around a node are pos­it­ive and pair­wise coprime.
  • The weight on an edge end­ing in a leaf is great­er than 1.
  • All edge de­term­in­ants are pos­it­ive.

From the res­ol­u­tion dia­gram for a ho­mo­logy sphere sin­gu­lar­ity link, one as­so­ci­ates a splice dia­gram as fol­lows: dis­reg­ard all ver­tices of valency 2; at each node and out­go­ing edge, take the ab­so­lute value of the de­term­in­ant of the out­er dia­gram. For ex­ample, the splice dia­gram above arises from the res­ol­u­tion dia­gram shown in Fig­ure 5.

Figure 5.

There are meth­ods to com­pute the res­ol­u­tion dia­gram from the splice dia­gram; see a gen­er­al dis­cus­sion of plumb­ing versus spli­cing in the Ap­pendix to [6].

One reas­on to con­sider splice dia­grams for the CIC is that the Cas­son in­vari­ant is known to be ad­dit­ive un­der spli­cing. All ZHS sin­gu­lar­ity links ad­mit splice dia­grams built on the Σ(p,q,r) and more gen­er­al Brieskorn–Pham com­plete in­ter­sec­tions, so pro­du­cing these dia­grams for the ex­amples in The­or­em 2.1 al­lows a dir­ect cal­cu­la­tion of the Cas­son in­vari­ants. The sig­na­tures in these cases, however, are gen­er­ally ex­tremely dif­fi­cult to com­pute, and we had to use sev­er­al dif­fer­ent meth­ods.

One con­sequence of the CIC would be that for a com­plete in­ter­sec­tion sin­gu­lar­ity with ZHS link, the to­po­logy of the link de­term­ines the geo­met­ric genus. (We fool­ishly guessed ori­gin­ally that we could re­place “com­plete in­ter­sec­tion” by “Goren­stein”, but [e8] found a counter­example.) Such a sur­pris­ing res­ult should give one pause be­fore be­ing too con­fid­ent of its cor­rect­ness. On the oth­er hand, the Cas­son in­vari­ant of a ZHS Σ is one-eighth the sig­na­ture of some simply con­nec­ted spin-man­i­fold whose bound­ary is Σ; the CIC as­serts that the Mil­nor fiber is such a man­i­fold.

3. Singularities of splice type

After the com­ple­tion in 1989 of our pa­per on the Cas­son in­vari­ant, Wal­ter (and I) pur­sued oth­er top­ics in the 1990s, he work­ing primar­ily on hy­per­bol­ic 3-man­i­folds and geo­met­ric group the­ory. But in 1998 Wal­ter spent a semester at Duke Uni­versity, at which time we re­sumed our dis­cus­sion of two big ques­tions (bey­ond the CIC) left open for us from [4].

The first was the dearth of ex­pli­cit ex­amples of hy­per­sur­face and com­plete in­ter­sec­tion sin­gu­lar­it­ies with ZHS links (so, the in­ter­sec­tion mat­rix (EiEj) has de­term­in­ant ±1). Could one write down equa­tions of such sin­gu­lar­it­ies for at least some splice dia­grams from The­or­em 2.2? The com­plete in­ter­sec­tion ex­amples of The­or­em 2.1(3) gave a hint; we were lucky that Henry Laufer had provided equa­tions for us from the cor­res­pond­ing res­ol­u­tion graph in the simplest case n=2.

Second, since the Cas­son in­vari­ant adds un­der spli­cing, per­haps the CIC is true be­cause one can “splice” sin­gu­lar­it­ies and their Mil­nor fibers, match­ing spli­cing on the link level, and so that on the Mil­nor fiber level the sig­na­tures add. Both of these is­sues were even­tu­ally dis­cussed in [6] and [7].

A splice dia­gram with a single node and t leaves arises from a Brieskorn–Pham com­plete in­ter­sec­tion in Ct as in The­or­em 2.1(1). A vari­able is as­signed to each of the leaves, and one defines a sin­gu­lar­ity by t2 gen­er­ic lin­ear com­bin­a­tions of monomi­als from each leaf. Weighted-ho­mo­gen­eity is giv­en by as­sign­ing to each leaf a weight which is the product of all the oth­er weights from the oth­er edges. In ad­di­tion, note that adding to the equa­tions terms of weight great­er than the weight n1n2nt gives oth­er sin­gu­lar­it­ies with the same res­ol­u­tion and splice dia­grams.

Figure 6.

Con­sider now the two-node splice dia­gram in Fig­ure 6, where we have already ad­ded a vari­able to each leaf.

Re­call that p,q,p,q are 2, the triples p,q,r and p,q,r are each pair­wise re­l­at­ively prime, and rr>pqpq.

Defin­i­tion 3.1: The dia­gram sat­is­fies the semig­roup con­di­tions if rN(p,q) (the semig­roup gen­er­ated by p and q) and rN(p,q).

Since r(p1)(q1) im­plies that rN(p,q), the edge de­term­in­ant con­di­tion does guar­an­tee that at least one of r,r is in the ap­pro­pri­ate semig­roup.

The­or­em 3.2: For the splice dia­gram above, as­sume the semig­roup con­di­tions are sat­is­fied, and write r=αp+βq,r=γp+δq. Then the isol­ated com­plete in­ter­sec­tion sin­gu­lar­ity giv­en by Xp+Yq+ZδWγ=0,Zp+Wq+XβYα=0 has ZHS link with the giv­en splice dia­gram.

In ana­logy with the one-node case, we have first as­signed a vari­able to each leaf. Next, to each node we as­so­ci­ate a weight which is the product of the weights of the sur­round­ing edges (so, pqr for the left node). Third, fix­ing a node we as­sign a weight to each leaf, by tak­ing the product of the weights ad­ja­cent to (but not on) the route from the node to that leaf; for the left node, this weight for the up­per left leaf is qr, while the weight for the up­per right leaf is pqq. Cru­cially, the semig­roup con­di­tion in­volving r al­lows one to write the first equa­tion, the sum of three monomi­als for the three edges of the node, each of which is ho­mo­gen­eous with re­spect to these weights, of total de­gree pqr.

Thus, with the weights from the left node, the first equa­tion is weighted ho­mo­gen­eous, while the second is not. Still, the as­so­ci­ated graded is an in­teg­ral do­main, and its nor­mal­iz­a­tion is the sin­gu­lar­ity Xp+Yq+Tr=0. One starts to re­solve the sin­gu­lar­ity with a weighted blow-up of C4. A key point is that set­ting the vari­able Z or W equal to 0 gives a monomi­al curve, cor­res­pond­ing to the “r” knot in Σ(p,q,r). But adding high­er weight terms to the equa­tions would not af­fect the to­po­logy.

One can sim­il­arly write down a sys­tem of equa­tions for an ar­bit­rary splice dia­gram Δ, as long as ap­pro­pri­ate semig­roup con­di­tions are as­sumed. Start by as­sign­ing a vari­able zw to every leaf w. Next, for each node v, take as weight dv the product of the weights dve on the t sur­round­ing edges. To every leaf w, define its v-weight to be lvw, the product of all weights ad­ja­cent to (but not on) the path from v to w; and define lvw to be the same product ex­clud­ing the weights around v. The semig­roup con­di­tion means: the weight on any edge e is in the semig­roup gen­er­ated by the lvw of all the out­er leaves on the e-side of v. Writ­ing dve=Σwαvwlvw, it fol­lows that the ad­miss­ible monomi­al Mve=wzwαvw has weight dv, the weight of the node. Now take t2 lin­ear com­bin­a­tions of those t monomi­als so that the coef­fi­cient mat­rix (avie) has all max­im­al minors nonzero, and add to each such lin­ear com­bin­a­tion a term Hvi of weight >dv: ΣeavieMve+Hvi=0,i=1,,t2. Fi­nally, choose such equa­tions for every node v of Δ. The fol­low­ing is a spe­cial case of The­or­em 5.4 be­low, proved ori­gin­ally in [6].

The­or­em 3.3: For any splice dia­gram Δ sat­is­fy­ing the semig­roup con­di­tions, the equa­tions above define an isol­ated com­plete in­ter­sec­tion sin­gu­lar­ity whose link is the ZHS cor­res­pond­ing to that splice dia­gram.

One calls these sin­gu­lar­it­ies of splice type. The proof is by in­duc­tion on the num­ber of nodes; the key step starts with an “end node” (all but one of whose em­an­at­ing edges is a leaf), takes a weighted blow-up, and uses in­duct­ively the res­ult for a sub­dia­gram. Again, set­ting a vari­able equal to 0 gives a monomi­al curve cor­res­pond­ing to a knot in the link.

How gen­er­al are sin­gu­lar­it­ies of splice type among sin­gu­lar­it­ies with ZHS link? By an im­port­ant gen­er­al res­ult of Popes­cu-Pam­pu [e12], every splice dia­gram as in The­or­em 2.2 is the link of some Goren­stein sin­gu­lar­ity. One should keep in mind a class of ex­amples in the pa­per [e8] of Némethi, Lu­engo-Velasco, and Melle-Hernan­dez.

Ex­amples:
  1. There ex­ists a Goren­stein sin­gu­lar­ity, not a com­plete in­ter­sec­tion, whose link is the Brieskorn sphere Σ(2,13,31).
  2. There ex­ists a Goren­stein sin­gu­lar­ity, not a com­plete in­ter­sec­tion, whose link is a ho­mo­logy sphere but which does not sat­is­fy the semig­roup con­di­tions.

An ana­lyt­ic con­di­tion dis­tin­guishes sin­gu­lar­it­ies of splice type from oth­er sin­gu­lar­it­ies with the same link. Each leaf of the splice (or res­ol­u­tion) dia­gram gives a knot in Σ, unique up to iso­topy. A key point in prov­ing that splice dia­gram equa­tions give in­teg­ral ho­mo­logy sphere links is to show that the vari­able zi as­so­ci­ated to a leaf cuts out the cor­res­pond­ing knot in Σ. In oth­er words, the curve Ci giv­en by zi=0 is ir­re­du­cible, and its prop­er trans­form Di on the min­im­al good res­ol­u­tion is smooth and in­ter­sects trans­versely the ex­cep­tion­al curve cor­res­pond­ing to the leaf of the splice dia­gram. Then the ex­ist­ence of such “end-curve func­tions” im­plies the semig­roup con­di­tion on the splice dia­gram.

The­or­em 3.4: [7], Section 8.1. Let (X,0) be a nor­mal sur­face sin­gu­lar­ity whose link Σ is an in­teg­ral ho­mo­logy sphere. As­sume that for each of the t leaves wi of the splice dia­gram Δ of Σ, there is a func­tion zi in­du­cing the end knot as above. Then
  1. Δ sat­is­fies the semig­roup con­di­tion;
  2. X is a com­plete in­ter­sec­tion of em­bed­ding di­men­sion less than or equal to t;
  3. z1,,zt gen­er­ate the max­im­al ideal of the loc­al ring of X at 0, and X is a com­plete in­ter­sec­tion of splice type with re­spect to these gen­er­at­ors.

Thus, some nat­ur­al open ques­tions are as fol­lows:

Ques­tions:
  1. Does there ex­ist a com­plete in­ter­sec­tion sin­gu­lar­ity for the two-node splice dia­gram based on (2,3,1;2,3,37), where the semig­roup con­di­tion fails?
  2. Is every com­plete in­ter­sec­tion with ZHS link of splice type?
  3. As­sum­ing there are com­plete in­ter­sec­tions with ZHS link that are not of splice type, do they sat­is­fy the Cas­son In­vari­ant Con­jec­ture?

4. Casson Invariant and Milnor Fiber Conjectures

What about the CIC for these sin­gu­lar­it­ies of splice type? It was real­ized early on that it is easi­er to re­state it in terms of the geo­met­ric genus pg of the sin­gu­lar­ity (as op­posed to the sig­na­ture). The­or­em 2 of [7] proves the con­jec­ture for sin­gu­lar­it­ies for which the nodes of the splice dia­gram lie on one line. The gen­er­al case was proved by Némethi and Ok­uma, us­ing a more power­ful in­duc­tion.

The­or­em 4.1: [e10] The Cas­son In­vari­ant Con­jec­ture is true for sin­gu­lar­it­ies of splice type.

As in­dic­ated pre­vi­ously, the valid­ity of the CIC for sin­gu­lar­it­ies of splice type sug­gests that there is a con­struc­tion of “spli­cing Mil­nor fibers” for which the ad­dit­iv­ity of the sig­na­tures is geo­met­ric­ally clear.

Sup­pose the equa­tions fi(z1,,zn)=0, i=1,,n2 define a sin­gu­lar­ity X of splice type, cor­res­pond­ing to a splice dia­gram Δ. Then the curve zj=0 cuts out in Σ the knot Kj cor­res­pond­ing to the jth leaf of Δ. The Mil­nor fiber of the sin­gu­lar­ity at 0 of the com­plete in­ter­sec­tion curve (f1,,fn2,zj)1(0) can be con­sidered as a sur­face Gj in the Mil­nor fiber of X, with Gj=GjΣ=Kj. We pro­posed in [7], Sec­tion 6, a con­jec­tur­al it­er­at­ive de­scrip­tion of a Mil­nor fiber in terms of the Mil­nor fibers of sim­pler com­plete in­ter­sec­tion sur­face sin­gu­lar­it­ies and fibers Gj as above ly­ing in­side them.

Let Δ be a splice dia­gram sat­is­fy­ing the semig­roup con­di­tions, and write Σ=Σ1K1K2Σ2 to rep­res­ent two ho­mo­logy spheres de­term­ined by cut­ting Δ at an edge to form two dia­grams with dis­tin­guished leaves. Then Δ1 and Δ2 also sat­is­fy the semig­roup con­di­tion, so Σ1 and Σ2 are both com­plete in­ter­sec­tion sin­gu­lar­ity links giv­en by equa­tions of splice type. They thus have Mil­nor fibers, say F1 and F2, with Fi=Σi.

For the knot K1Σ1, con­sider as above G1F1, as well as a tu­bu­lar neigh­bor­hood G1×D2F1. Sim­il­arly, con­sider D2×G2F2.

De­note F1o:=F1(G1×(D2)o),F2o:=F2((D2)o×G2), so F1o is the uni­on of G1×S1 and the ex­ter­i­or (com­ple­ment of an open tu­bu­lar neigh­bor­hood) of the knot K1Σ1, and sim­il­arly for F2o.

Mil­nor Fiber Con­jec­ture: The Mil­nor fiber F is homeo­morph­ic to the res­ult F of past­ing: F:=F1oG1×S1(G1×G2)S1×G2F2o, where we identi­fy G1×S1 with G1×G2 and S1×G2 with G1×G2.

The con­struc­tion of F ex­tends spli­cing on the bound­ary links. It is not hard to see that sig­na­tures add, spe­cific­ally signF=sign F1+sign F2. Thus the Mil­nor Fiber Con­jec­ture im­plies the Cas­son In­vari­ant Con­jec­ture.

In [7], The­or­em 8.2, the Mil­nor Fiber Con­jec­ture is veri­fied for the pre­vi­ously dis­cussed sin­gu­lar­it­ies {zn+f(x,y)=0} with ZHS link. In [e11], the con­jec­ture was proved for it­er­ated sus­pen­sions. Re­cently, M. A. Cueto, P. Popes­cu-Pam­pu, and D. Stepan­ov [e13], [e14] have an­nounced a proof of the gen­er­al con­jec­ture, which com­bines tools from trop­ic­al and log­ar­ithmic geo­metry (in the sense of Fon­taine and Il­lusie).

5. Splice-quotient singularities

Figure 7.

In 2000, as we worked through the ba­sics of sin­gu­lar­it­ies of splice type, we began to look for gen­er­al­iz­a­tions of the un­ex­pec­ted res­ult of The­or­em 1.2, that the UAC of a weighted-ho­mo­gen­eous sin­gu­lar­ity with QHS link is a com­plete in­ter­sec­tion.

From the point of view of clas­si­fic­a­tion of sin­gu­lar­it­ies, the easi­est ex­amples not covered by The­or­em 1.2 are the quo­tient-cusps. These are sin­gu­lar­it­ies whose res­ol­u­tion graphs have the form shown in Fig­ure 7 for k2,ei3 and some ej>2.

They are ra­tion­al sin­gu­lar­it­ies, log-ca­non­ic­al, and taut, i.e, the to­po­logy of the link de­term­ines the ana­lyt­ic type. The above quo­tient-cusp is double-covered by the cusp sin­gu­lar­ity whose res­ol­u­tion graph is shown in Fig­ure 8.

Figure 8.

It fol­lows that the uni­ver­sal abeli­an cov­er is also a cusp.

It is easy to de­term­ine, giv­en the res­ol­u­tion graph, when a cusp is a com­plete in­ter­sec­tion; and, this was the case for the UAC for the few ex­amples we checked. However, it re­quired a de­tailed and lengthy ana­lys­is of the un­im­od­u­lar mat­rix clas­si­fy­ing the quo­tient-cusp to prove the fol­low­ing.

The­or­em 5.1: [5] The uni­ver­sal abeli­an cov­er of a quo­tient-cusp is a com­plete in­ter­sec­tion of em­bed­ding di­men­sion 4.

Though we wrote down equa­tions and group ac­tion for the UAC, at the time (2001) we’d not yet for­mu­lated a gen­er­al meth­od for do­ing so, and our res­ult seemed ad hoc.

Giv­en the res­ol­u­tion dia­gram Γ of a sin­gu­lar­ity with QHS link Σ, the meth­od de­scribed after The­or­em 2.2 al­lows one to pass from Γ to a splice dia­gram Δ. Only in the ZHS case can one in­sure that the weights around a node are re­l­at­ively prime, and only in that case can one re­cov­er Γ from Δ.

Non­ethe­less, the same ap­proach as above in the ZHS case means that as­sum­ing the semig­roup con­di­tions on the as­so­ci­ated splice dia­gram, one can write down a set of splice dia­gram equa­tions [6], Defin­i­tion 2.5, which give isol­ated com­plete in­ter­sec­tion sur­face sin­gu­lar­it­ies in Ct, where t is the num­ber of ends of Γ or Δ.

When Γ is star-shaped, Neu­mann’s The­or­em 1.2 proves that these equa­tions give the UAC of the weighted ho­mo­gen­eous sin­gu­lar­ity of graph Γ; also, his res­ult de­scribes the group ac­tion. Neg­lect­ing this group ac­tion in the gen­er­al QHS case, the fol­low­ing sug­gest­ive as­ser­tion ap­pears not to have been proved in full gen­er­al­ity:

Con­jec­ture 5.2: If the graph Γ sat­is­fies the semig­roup con­di­tions, then the link of the com­plete in­ter­sec­tion sin­gu­lar­ity defined by splice dia­gram equa­tions is the UAC of Σ.

A dif­fi­culty is that splice dia­gram equa­tions de­pend only on the splice dia­gram Δ, while the cov­er­ing group of the UAC is the dis­crim­in­ant group D(Γ), not com­put­able from Δ. Re­call its con­struc­tion: let E=j=1nZEj be the lat­tice spanned by all the ex­cep­tion­al curves, and E the dual lat­tice, with dual basis {ei} defined by ei(Ej)=δij. Then D(Γ)=E/EH1(Σ); view­ing Σ as the bound­ary of a tu­bu­lar neigh­bor­hood of E, ei rep­res­ents an ori­ented circle over a point of Ei. The pair­ing of E/E in­to Q/Z is the to­po­lo­gic­al link­ing pair­ing on H1(Σ). The key is to con­struct a nat­ur­al rep­res­ent­a­tion of D(Γ).

Pro­pos­i­tion 5.3: ([6], Proposition 5.2) Let e1,,et be the ele­ments of \/ E cor­res­pond­ing to the t leaves of Γ.
  1. The ho­mo­morph­ism EQt defined by e(ee1,,eet) in­duces an in­jec­tion D(Γ)=E/E  (Q/Z)t.
  2. Ex­po­nen­ti­at­ing each Q/ZC via rexp(2πir) provides a faith­ful di­ag­on­al rep­res­ent­a­tion D(Γ)(C)t.

Now, as­sume Γ is a graph with t ends. We have a di­ag­on­al rep­res­ent­a­tion of the dis­crim­in­ant group in Ct. If Δ sat­is­fies the semig­roup con­di­tions, one has splice type equa­tions in Ct. What we need is to be able to choose those equa­tions which be­have equivari­antly with re­spect to the group ac­tion.

Defin­i­tion 5.4: (congruence conditions) Let Γ be a res­ol­u­tion dia­gram, yield­ing a splice dia­gram Δ sat­is­fy­ing the semig­roup con­di­tions. Then Γ sat­is­fies the con­gru­ence con­di­tions if for each node v, one can choose for every ad­ja­cent edge e an ad­miss­ible monomi­al Mve so that D(Γ) trans­forms each of these monomi­als ac­cord­ing to the same char­ac­ter.

Clearly, the semig­roup plus con­gru­ence con­di­tions to­geth­er mean that the dis­crim­in­ant group acts on ap­pro­pri­ate splice dia­gram com­plete in­ter­sec­tion sin­gu­lar­it­ies (high­er or­der terms must also be D(Γ)-equivari­ant). Here is the main res­ult:

The­or­em 5.5: (splice-quotient singularities [6], Theorem 7.2) Sup­pose Γ sat­is­fies the semig­roup and the con­gru­ence con­di­tions. Then:
  1. Splice dia­gram equa­tions for Γ define an isol­ated com­plete in­ter­sec­tion sin­gu­lar­ity (X,0).
  2. The dis­crim­in­ant group D(Γ) acts freely on a punc­tured neigh­bor­hood of 0 in X.
  3. Y=X/D(Γ) has an isol­ated nor­mal sur­face sin­gu­lar­ity, and a good res­ol­u­tion whose as­so­ci­ated dual graph is Γ.
  4. XY is the uni­ver­sal abeli­an cov­er­ing.
  5. XY maps the curve zw=0 to an ir­re­du­cible curve, whose prop­er trans­form on the good res­ol­u­tion of Y is smooth and in­ter­sects the ex­cep­tion­al curve trans­vers­ally, along Ew. In fact the func­tion zwdet(Γ)xw, which is D(Γ)-in­vari­ant and hence defined on Y, van­ishes to or­der det(Γ) on this curve.

A sin­gu­lar­ity (Y,0) which arises in this way is called a splice-quo­tient sin­gu­lar­ity.

Of course, there are many things to be checked! The proof is by in­duc­tion on the num­ber of nodes, and one needs to con­sider cer­tain non­min­im­al res­ol­u­tions.

We con­jec­tured that ra­tion­al sur­face sin­gu­lar­it­ies were splice quo­tients; but the first proof of this fact was due to T. Ok­uma, who com­bined the semig­roup and con­gru­ence con­di­tions in a nov­el way.

The­or­em 5.6: [e9] Ra­tion­al sur­face sin­gu­lar­it­ies, and min­im­ally el­lipt­ic sin­gu­lar­it­ies with QHS links, are splice quo­tient sin­gu­lar­it­ies.

Even­tu­ally, we were able to prove a gen­er­al res­ult, which in­cluded Ok­uma’s, identi­fy­ing the cru­cial ana­lyt­ic prop­erty of a splice-quo­tient (Y,0): for every end of the graph, there is an “end-curve func­tion”on Y, as in (5) of The­or­em 5.5. Our res­ult (be­low) gen­er­al­ized The­or­em 3.4, but the proof was far more dif­fi­cult be­cause of the group ac­tion.

End Curve The­or­em: [8] Let (Y,0) be a nor­mal sur­face sin­gu­lar­ity with QHS link Σ. Sup­pose that for each leaf w of the res­ol­u­tion dia­gram Γ there ex­ists a cor­res­pond­ing end curve func­tion xw:YC which cuts out (pos­sibly with mul­ti­pli­city) an end knot KwΣ (or end curve) for that leaf. Then (Y,0) is a splice-quo­tient sin­gu­lar­ity, and a choice of a suit­able root zw of xw for each w gives co­ordin­ates for the splice-quo­tient de­scrip­tion.

We close with a caveat: As in­dic­ated in The­or­em 10.1 of [6], the gen­er­al splice type equa­tion can be found us­ing any choice of ad­miss­ible monomi­als, as long as one al­lows the ad­di­tion of high­er or­der equivari­ant terms. This im­plies that in the ZHS situ­ation, what look like “equisin­gu­lar” de­form­a­tions of sin­gu­lar­it­ies of splice type are also of splice type. However, even for a weighted ho­mo­gen­eous QHS sin­gu­lar­ity, a pos­it­ive weight de­form­a­tion of the de­fin­ing equa­tions might no longer be a splice-quo­tient. For ex­ample, z2=x4+y9+txy7 ([8], Ex­ample 10.4) is a pos­it­ive weight de­form­a­tion, but not a de­form­a­tion of splice-quo­tients; func­tions on the res­ol­u­tion lift un­der de­form­a­tion, but for t0 they are no longer end-curve func­tions.

Jonath­an Wahl is Emer­it­us Pro­fess­or of Math­em­at­ics at the Uni­versity of North Car­o­lina at Chapel Hill. His col­lab­or­a­tion with Wal­ter Neu­mann for over 30 years res­ul­ted in sev­en pub­lished pa­pers.

Works

[1] W. D. Neu­mann: “A cal­cu­lus for plumb­ing ap­plied to the to­po­logy of com­plex sur­face sin­gu­lar­it­ies and de­gen­er­at­ing com­plex curves,” Trans. Am. Math. Soc. 268 : 2 (1981), pp. 299–​344. MR 632532 Zbl 0546.​57002 article

[2] W. D. Neu­mann: “Abeli­an cov­ers of quasi­ho­mo­gen­eous sur­face sin­gu­lar­it­ies,” pp. 233–​243 in Sin­gu­lar­it­ies (Ar­cata, CA, 20 Ju­ly–7 Au­gust 1981), part 2. Edi­ted by P. Or­lik. Pro­ceed­ings of Sym­po­sia in Pure Math­em­at­ics 40. Amer­ic­an Math­em­at­ic­al So­ci­ety (Provid­ence, RI), 1983. MR 713252 Zbl 0519.​32010 incollection

[3] D. Eis­en­bud and W. Neu­mann: Three-di­men­sion­al link the­ory and in­vari­ants of plane curve sin­gu­lar­it­ies. An­nals of Math­em­at­ics Stud­ies 110. Prin­ceton Uni­versity Press, 1985. MR 817982 Zbl 0628.​57002 book

[4] W. Neu­mann and J. Wahl: “Cas­son in­vari­ant of links of sin­gu­lar­it­ies,” Com­ment. Math. Helv. 65 : 1 (1990), pp. 58–​78. MR 1036128 Zbl 0704.​57007 article

[5] W. D. Neu­mann and J. Wahl: “Uni­ver­sal abeli­an cov­ers of quo­tient-cusps,” Math. Ann. 326 : 1 (2003), pp. 75–​93. MR 1981612 Zbl 1032.​14010 ArXiv math/​0101251 article

[6] W. D. Neu­mann and J. Wahl: “Com­plete in­ter­sec­tion sin­gu­lar­it­ies of splice type as uni­ver­sal abeli­an cov­ers,” Geom. To­pol. 9 : 2 (2005), pp. 699–​755. MR 2140991 Zbl 1087.​32017 ArXiv math/​0407287 article

[7] W. D. Neu­mann and J. Wahl: “Com­plex sur­face sin­gu­lar­it­ies with in­teg­ral ho­mo­logy sphere links,” Geom. To­pol. 9 : 2 (2005), pp. 757–​811. MR 2140992 Zbl 1087.​32018 ArXiv math/​0301165 article

[8] W. D. Neu­mann and J. Wahl: “The end curve the­or­em for nor­mal com­plex sur­face sin­gu­lar­it­ies,” J. Eur. Math. Soc. 12 : 2 (2010), pp. 471–​503. MR 2608949 Zbl 1204.​32019 ArXiv 0804.​4644 article