Kirby Problem 4.16

Definitions and mathematical statement

For a closed connected oriented smooth four-manifold $X$, the real intersection form on $H_2(X;\mathbb R)$ has positive and negative indices $b_2^+(X)$ and $b_2^-(X)$. Its signature is $\sigma(X)=b_2^+(X)-b_2^-(X)$ and its Euler characteristic is $\chi(X)=\sum_i(-1)^i b_i(X)$. Write \[ c_1^2(X):=2\chi(X)+3\sigma(X) \] as a numerical abbreviation, without requiring an almost-complex structure.

Call $X$ smoothly irreducible if every smooth connected-sum decomposition has a summand that is a homotopy four-sphere. (a) Does an irreducible $X$ exist with $b_2^+(X)>1$ and $2\chi(X)+3\sigma(X)<0$?

(b) For some integer $n>9$, does the topological manifold \[ \mathbb{CP}^2\#n\,\overline{\mathbb{CP}}^{2} \] have an irreducible exotic smooth structure? The bar reverses the standard complex orientation; exotic means homeomorphic but not diffeomorphic to the standard connected sum, with the indicated orientation. For this topology the displayed numerical invariant is $9-n$.

Part (b) concerns positive index one, whereas part (a) explicitly requires positive index greater than one; neither is a literal special case of the other. A blow-up introduces a projective-plane summand and therefore does not provide the irreducible examples being requested. The orientation reversal in (b) is necessary for the intended negative-square geography. Neither question assumes that the desired manifold is symplectic.

Short English statement

Can irreducible smooth four-manifolds have the specified negative Chern-square number, including exotic rational surfaces beyond nine blow-ups?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2892 (KP-4.16), 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 4.16. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.