On Seifert fibered spaces bounding definite manifolds

Abstract

We establish an inequality which gives strong restrictions on when the standard definite plumbing intersection lattice of a Seifert fibered space over S 2 can embed into a standard diagonal lattice, and give two applications. First, we answer a question of Neumann-Zagier on the relationship between Donaldson's theorem and Fintushel-Stern's R-invariant. We also give a short proof of the characterisation of Seifert fibered spaces which smoothly bound rational homology S 1 × D 3 's.

Cyclic Branched Coverings of Brieskorn Spheres Bounding Acyclic 4-Manifolds

“…We recall that the weight at the central node will always be δ = −1 whenever (pa, b, c) bounds an acyclic 4-manifold (see Neumann and Zagier[26] and Issa and McCoy[17, Theorem 8]).The linear plumbing action on M( ) contributes both fixed and invariant 2-spheres which introduce constraints derived from the global orientation of the Yang-Mills moduli spaces.…”

Equivariant 4‐genera of strongly invertible and periodic knots

“…The intersection form on Σ(𝐵 4 , F) is isomorphic to  𝐹 by [17,Theorem 3]. Hence Σ(𝐵 4 , F) is positive definite by hypothesis and therefore 𝑋 is positive definite (see, for example, [20,Proposition 7]).…”

“…The intersection form on $$\Sigma (B^4,\hat{F})$$ is isomorphic to $$\mathcal {G}_F$$ by [17, Theorem 3]. Hence $$\Sigma (B^4,\hat{F})$$ is positive definite by hypothesis and therefore $$X$$ is positive definite (see, for example, [20, Proposition 7]). Since $$X$$ is smooth and positive definite, Donaldson's theorem [9] implies that the intersection form $$(H_2(X),Q_X)$$ is isomorphic to $$(\mathbb {Z}^k,\mbox{Id})$$, where $$\begin{equation*} k = \mbox{rank}\, H_2(X) = \mbox{rank}\, H_2(\Sigma (B^4,S)) + \mbox{rank}\, H_2(\Sigma (B^4,\hat{F})) = -\sigma (K) + b_1(F).$$…”

Classical and new plumbed homology spheres bounding contractible manifolds

“…This condition is even enough for the obstruction from bounding rational homology balls[IM20]. 5 Note that ∂X(1) = Σ(2, 5, 7) and ∂X (1) = Σ(3, 4, 5), compare with [AK79],[CH81], and[Şav20].…”