Kirby Problem 4.57

Definitions and mathematical statement

For a closed oriented smooth $4$-manifold $X$, write $\chi(X)=\sum_i(-1)^i\dim H_i(X;\mathbb Q)$ for its Euler characteristic and $\sigma(X)$ for the signature of its intersection form on $H_2(X;\mathbb R)$. Let $X,Y$ have equal Euler characteristics and equal signatures.

A torus link in $X$ is a finite disjoint union of smoothly embedded copies $T_i$ of $T^2=S^1\times S^1$. Require every component to have trivial normal bundle, and choose disjoint tubular neighborhoods $\nu T_i\cong T^2\times D^2$. Simultaneous torus surgery removes their interiors and glues back copies of $T^2\times D^2$ using chosen boundary diffeomorphisms of $T^3$, with the orientation convention giving an oriented result.

Is there always such a torus link in $X$ and a choice of gluings whose result is diffeomorphic to $Y$? If this smooth conclusion is too strong, does the same assertion hold with "homeomorphic to $Y$" in its place?

All surgery tori must already be disjoint in the original $X$. A sequence in which later tori occur only in a previously altered manifold would not automatically give the simultaneous surgery asked for. The number of components and all surgery slopes may depend on the pair. No simple-connectivity, symplecticity, or prescribed homology class for the tori is assumed. The numerical equalities are the hypotheses, not a claimed complete diffeomorphism classification.

Short English statement

Can any two closed oriented smooth four-manifolds with equal Euler characteristic and signature be related by simultaneous surgery on a torus link in the first?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2933 (KP-4.57), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 4.57. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.