BLOW-UP FORMULAS FOR ( 2)-SPHERES

Transcription

1 BLOW-UP FORMULAS FOR 2)-SPHERES ROGIER BRUSSEE In this note we give a universal formula for the evaluation of the Donalson polynomials on 2)-spheres, i.e. smooth spheres of selfintersection 2. Note that the blow-up formulas can be consiere as formulas for the evaluation of Donalson polynomials on 1)-spheres. Such formulas come almost for free using Fintushel an Sterns metho to etermine the orinary blow-up formulas [FS1]. Fintushel an Stern informe me that they foun similar formulas with similar methos. In fact comparing their formulas with my own ones, I realise that remark 4 coul be use to get clean formulas in the main theorem. Acknowlegment. Special thanks to both Fintushel an Stern for informing me about their results an encouraging me to publish my own ones. Further thanks to Stefan Bauer for getting me in the Bielefel vector bunle work group BVP, an Viktor Pistrigatch an again Stefan Bauer for useful iscussions. Our conventions follow [FS1]. We consier the stable) Donalson polynomial D c as a linear form on AX) = S H X, Q) etermine by a class c H 2 X, Z/4Z). The generator of H 0 X) is enote x, an Q[x] is consiere as a subalgebra of AX). We also use the convention that D c a) means either evaluation of D c on a AX) or the linear form efine by D c a)b) = D c ab) epening on the context. Finally we exten D c linearly to the formal series AX)[[t]] or the formal Laurent series AX)[[t]][t 1 ] whenever convenient. Now the main result of [FS1] is that there are even respectively o power series Bt), St) Q[x][[t]] such that for every 1)-sphere e in a compact close simply connecte 4-manifol with b + > 1 1) 2) D c e te ) = D c Bt)), Bt) = σ 3 t)e x t2 /6 if c e is even D c e te ) = D c e St)), St) = σt)e x t2 /6 if c e is o on the subalgebra Ae ) generate by the orthogonal complement of e this formulation is equivalent to the one where c e = 0). Here σ 3 an σ are certain quasi elliptic functions with a power series expansion in Q[x][[t]]. Actually Fintushel an Stern mention that the simply connecteness hypothesis is not necessary, an the simply connecte) in our main theorem below is meant to mean whatever they nee. Theorem 1. Let X be a simply connecte) close compact oriente four manifol with o b + > 1 containing a 2) sphere τ, then on Aτ ) 3) 4) D c e tτ ) = D c B 2 t)) + D c+τ S 2 t)) if c τ is even, D c e tτ ) = D c BS B S)t) + τbst)) if c τ is o. Date: g-ga/ originally poste, 23 Dec 1994, this version 2 Mar 2014, upate to run with moern AMS latex an current aress). 1

2 2 ROGIER BRUSSEE Remark 2. Note that respectively Rubermann s relation [FS1, Th 2.2] 5) D c τ 2 ) = 2D c+τ on Aτ ), c τ even follows from the formulas in the main theorem. In particular, formula 3 of the main theorem implies 6) D c e tτ ) = D c B 2 t) + S 2 t)τ/2) on Aτ ), c τ even. In this form it also implies Wieczorek s relation [FS1, Cor. 2.5] 7) D c τ 4 ) = D c 4 4xτ 2 ). In fact to get these two relations we only nee the existence an form of the universal formulas for S an B up to orer 4. These two relations in turn etermine the full series for S an B. Thus we will see that the formulas for B an S in [FS1] can be prove from the magic formula 8) for 3)-spheres an the expansion of B an S up to orer 4 as gauge theoretic input the sign in 8) can easily be etermine from B an S up to orer 4 instea of formula 5) as in [FS1]). Also note that the evenness of D c e tτ ) if c τ is even is clear a priori from the invariance of the Donalson polynomial uner the reflection in τ. The formulas simplify when X is of simple type, i.e. if Dx 2 ) = 4D. The elliptic functions σ an σ 3 then egenerate to respectively sinh an cosh. The formulas are particularly nice if we express them in terms of the Donalson series D c h) = D c 1 + x/2)e h ), h H 2 X) this expression is a priori formal, but in fact D c is an entire real analytic function at least uner the assumption b 1 X) = 0, [KM],[FS2]). Corollary 3. If in aition to the assumptions of the theorem, X is of simple type an h τ, then D c h + tτ) = e t2 cosh 2 t)d c h) + sinh 2 t)d c+τ h)), if c τ is even, D c h + tτ) = e t2 D c h) sinh2t) t t=0d c h + tτ)) if c τ is o. Proof of Theorem 1. Theorem 1 is an easy consequence of the following magic formula for 3)-spheres [FS1, Theorem 2.4]. If c τ 0 mo 2) then 8) D c+τ3 = D c τ 3 ) on τ 3 As in [FS1] we blow-up to get relations for the 2)-sphere. In X# P 2 the exceptional line is a 1)-sphere e, τ + e is represente by a 3)-sphere, τ 2e) τ + e), an τ e. First consier the case c τ even. Then formula 8) gives the ientity D c+τ+e e tτ 2e) ) 2) = D c+τ S 2t)e tτ ) 8) = D c τ + e)e tτ 2e) ) 1) = D c B 2t)τe tτ + B 2t)e tτ ). on τ e = τ H 2 X). However, Bt) is invertible in Q[x][[t]] Aτ ). Thus without loss of generality we can multiply by 1/B uner D c. With this remark an using the parity of B an S we get 9) t D ce tτ ) = D c B B 2t)etτ ) + D c+τ S B 2t)etτ ).

3 BLOW-UP FORMULAS FOR 2)-SPHERES 3 Now in this formula we can replace c by c + τ, an use that D c+2τ = D c. Hence if we efine D c,± = D c ± D c+τ, formula 9) is equivalent to the two equations 10) t D c,±e tτ ) = D c,± B ± S B 2t)etτ ) This ifferential equation etermines D c,± e tτ ) uniquely in terms of D c,± τ, i.e. there exists a universal power series 2 B ± t) with coefficients in Q[x] such that D c,± e tτ ) = D c,± 2 B ± t)) on Aτ ). The series is only etermine moulo the intersection of the kernels of D c,± z) for all z Aτ )), but clearly the following series is a universal representative 11) 2B ± t) = e t 0 B ±S B 2s) s = B2t)e ±1/2) 2t S 0 B s) s Q[x][[t]]. Formula 11) can then be rewritten to a universal formula 12) D c e tτ ) = D c 2 B 0 t)) + D c+τ 2 B τ t)) on Aτ ). where 2 B 0 = 1/2) + B + B) an 2 B τ = 1/2) + B B). To get the clean formulas 3) an 4) in the theorem we simply apply the universal formula 12) to τ = e 1 + e 2 in X#2 P 2, an restrict to H 2 X). For example, a small computation with the blow-up formulas gives D c S 2 t)) = D c+e1+e 2 e te1+e2) ) AX) = D c 2 B τ ) vali on all of AX). This proves the even case. In case c τ is o we procee similarly. We use the magic formula 8) for D c+e an τ 3 = τ + e. Now the reflection R τ in τ is represente by an orientation an homology orientation preserving iffeomorphism, so by the invariance of the polynomials. Rτ D c = D c±τ = 1) τ 2 D c+τ = D c+τ This grante, Some reworking then gives D c+τ+2e e tτ 2e) ) = D c+τ B 2t)e tτ ) = D c B2t)e tτ ) = D c+e τ + e)e tτ 2e) = D c S2t)τ + S 2t))e tτ ). t D B + S ccoshtτ)) = D c 2t) coshtτ)) S t D csinhtτ)) = D c B + S 2t) sinhtτ)). S Since B0) = S 0) = 1, the first formula can be hanle as above, an we get a universal formula D c coshtτ)) = D c 2 S 0 t) on Aτ ) with an explicit expression 13) 2S 0 = e 1/2) 2t 0 B+S S s) s. In the latter formula we have a ifferential equation with regular singularities. It has a unique solution with initial conitions D c sinhtτ)) t=0 = 0, t t=0d c sinhtτ)) = D c τ) which yiels a universal formula D c sinhtτ)) = D c τ 2 S 1 t)) an an explicit expression 2S 1 t) = t e 1/2) 2t 0 B+S S s) 2 s s

4 4 ROGIER BRUSSEE Finally to get the clean formulas in the main theorem we blow-up twice an compute D c+e1 e te1+e2) ) AX) an D c e te1+e2) e 1 e 2 )) AX) in two ifferent ways as in the even case. Remark 4. It is by no means clear that say 2 B 0 t) = B 2 t) in Q[x][[t]], i.e. without iviing out ker D c. In fact, I hope to get universal ientities for D c x n ) like the simple type conition D c x 2 ) = 4D c this way. Alas, expaning both sies by computer, I foun the series agree at least up to orer 28 in t for all series consiere here. It shoul be expecte that B 2 t) ± S 2 t) satisfies the ifferential equation 10) before evaluation with D c In the same spirit, formulas for 2)-spheres give us relations for B an S by writing two orthogonal 1)-spheres as a 2)-sphere. For example when c H 2 X) H 2 X#2 P 2 ) we get the ientity 14) D c Bu + v)bu v)) = D c B 2 u)b 2 v) S 2 u)s 2 v)) on AX). Likewise, writing a 3)-sphere as three 1)-spheres an plugging this into formula 8) gives ientities like 15)D c Su)Sv)Su + v)) = D c B u)bv)bu + v) + Bu)B v)bu + v) Bu)Bv)B u + v)) Fintushel an Stern informe me, that they reuce these ientities to ientities between elliptic functions. Appenix 1. Explicit series For the convenience of the reaer we give the first few terms of the expansion of the relevant series:

5 BLOW-UP FORMULAS FOR 2)-SPHERES 5 Bt) = 1 + 2) t4 + 8 x)t6 4! 6! x 2) t 8 8! + 96 x x 3) t 10 10! x x 4) t 12 12! x x x 5) t 14 14! x x x 6) t 16 16! + St) = t + x) t3 3! x 2) t 5 5! + 6 x x 3) t 7 7! x 2 + x 4) t 9 9! x 20 x 3 x 5) t 11 11! x x 4 + x 6) t 13 13! x x 3 42 x 5 x 7) t 15 15! + 2B 0 t) = B 2 t) = 1 + 4) t x)t6 4! 6! x 2) t 8 8! x x 3) t 10 10! x x 4) t 12 12! x x x 5) t 14 14! + 2B τ t) = S 2 t) = 2 t2 + 8 x)t4 2! 4! x 2) t 6 6! x 128 x 3) t 8 8! x x 4) t 10 10! x x x 5) t 12 12! + 2S 0 t) = BS SB )t) = 1 + x) t2 2! x 2) t 4 4! + 56 x x 3) t 6 6! x 2 + x 4) t 8 8! x 3728 x 3 x 5) t 10 10! x x 4 + x 6) t 12 12! + 2S 1 t) = BSt) = t + x) t3 3! x 2) t 5 5! x x 3) t 7 7! x 2 + x 4) t 9 9! x x 3 x 5) t 11 11! x x 4 + x 6) t 13 13! +

6 6 ROGIER BRUSSEE References [FS1] Fintushel R. an Stern R.J. The blow up formula for Donalson invariants. e-print alggeom/ May [FS2] Fintushel R. an Stern R.J. The Donalson invariants of 4-manifols of simple type preprint [KM] Kronheimer, P.B. an Mrowka, T.S. Embee surfaces an the structure of the Donalson polynomial invariants preprint Merton College University of Applie Sciences Utrecht, PO Box 8611, 3503 RP Utrecht aress: