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Gluing conjecture for Donaldson invariants along surfaces

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Let (X1,Σ1)(X_1, \Sigma_1) and (X2,Σ2)(X_2,\Sigma_2) be permissible 4-manifolds with b1(Xi)=0b_1(X_i)=0, and let X~i=Xi#ΣiBg\tilde X_i=X_i\#_{\Sigma_i}B_g. Write

DX~1w(eα)=eQ(α)/2∑ja~j,weK~j⋅α,DX~2w(eα)=eQ(α)/2∑kb~k,weL~k⋅α.{\Bbb D}^w_{\tilde X_1}(e^{\alpha})=e^{Q(\alpha)/2}\sum_j \tilde a_{j,w}e^{\tilde K_j\cdot\alpha},\qquad {\Bbb D}^w_{\tilde X_2}(e^{\alpha})=e^{Q(\alpha)/2}\sum_k \tilde b_{k,w}e^{\tilde L_k\cdot\alpha}.

Choose an identification ϕ\phi, set X=X1#ΣX2X=X_1\#_{\Sigma}X_2, and choose compatible classes w∈H2(X;Z)w\in H^2(X;\mathbb Z) and wi∈H2(X~i;Z)w_i\in H^2(\tilde X_i;\mathbb Z) with wi∣Bgo=k P.D.⁡[T1]∣Bgow_i|_{B_g^o}=k\,\operatorname{P.D.}[T_1]|_{B_g^o}. For D∈H2(X)D\in H_2(X), write D=D1o+D2oD=D_1^o+D_2^o and choose linear cappings Di∈H2(X~i)D_i\in H_2(\tilde X_i). Gluing conjecture for Donaldson invariants. The manifolds X~i\tilde X_i and XX are of simple type, and

DXw(etD)=ϵeQ(tD)/2(∑K~j⋅Σ=L~k⋅Σ=2g−2−2−3g+5a~j,wb~k,we(K~j⋅D1+L~k⋅D2)t+∑K~j⋅Σ=L~k⋅Σ=−(2g−2)(−1)g2−3g+5a~j,wb~k,we(K~j⋅D1+L~k⋅D2)t),{\Bbb D}^w_X(e^{tD})=\epsilon e^{Q(tD)/2}\left(\sum_{\substack{\tilde K_j\cdot\Sigma=\tilde L_k\cdot\Sigma=2g-2}}-2^{-3g+5}\tilde a_{j,w}\tilde b_{k,w}e^{(\tilde K_j\cdot D_1+\tilde L_k\cdot D_2)t}+\sum_{\substack{\tilde K_j\cdot\Sigma=\tilde L_k\cdot\Sigma=-(2g-2)}}(-1)^g2^{-3g+5}\tilde a_{j,w}\tilde b_{k,w}e^{(\tilde K_j\cdot D_1+\tilde L_k\cdot D_2)t}\right),

where ϵ=(−1)(g−1)(w2−w12−w22)/2\epsilon=(-1)^{(g-1)(w^2-w_1^2-w_2^2)/2}. The conjecture gives a universal formula expressing the Donaldson invariant of the connected sum along a surface in terms of those of the capped manifolds; the paper's preceding results establish the corresponding genus-22 case, while the general statement remains open.

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  1. Gluing conjecture for Donaldson invariants along surfaces

    Let Xˉi\bar X_i have b1=0b_1=0 and be of simple type. Suppose that there are embedded surfaces Σi⊂Xˉi\Sigma_i\subset\bar X_i of genus gg and self-intersection zero. Form

    X=Xˉ1#ΣXˉ2.X=\bar X_1\#_{\Sigma}\bar X_2.

    Let Y=S1×ΣY={\mathbb S}^1\times\Sigma be the gluing boundary, and let κ\kappa range over the basic classes of XX. The gluing conjecture. The manifold XX is of simple type, and every basic class κ\kappa intersects YY in nS1n{\mathbb S}^1, where nn is an even integer satisfying

    −(2g−2)≤n≤2g−2.-(2g-2)\leq n\leq 2g-2.

    Moreover, the sum of the coefficients cκc_\kappa of the basic classes agreeing with (K,L)∈H2(Xˉ1;Z)/Z[Σ1]⊕H2(Xˉ2;Z)/Z[Σ2](K,L)\in H^2(\bar X_1;\mathbb Z)/\mathbb Z[\Sigma_1]\oplus H^2(\bar X_2;\mathbb Z)/\mathbb Z[\Sigma_2] is zero unless (K,L)(K,L) comes from basic classes KiK_i and LjL_j of Xˉ1\bar X_1 and Xˉ2\bar X_2 satisfying Ki⋅Σ1=Lj⋅Σ2=±(2g−2)K_i\cdot\Sigma_1=L_j\cdot\Sigma_2=\pm(2g-2); in that case the sum is ±27g−9\pm2^{7g-9} times the product of the coefficients of KiK_i and LjL_j. The conjecture extends the genus-two gluing theorem and is motivated by the relationship between Donaldson and Seiberg–Witten invariants, but its resolution is not supplied here.

    source: Vicente Muñoz, “Donaldson invariants for some glued manifolds”, arXiv:dg-ga/9511013 (1995).

References

Primary source

Vicente Munoz, “Gluing formulae for Donaldson invariants for connected sums along surfaces”, arXiv:dg-ga/9702002 (1997).

4-manifoldsdonaldson-invariantsfour-manifoldsgauge-theory
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