I'm seeking for a proof of Theorem 9.2.12. in the Gompf-Stipsicz "4-Manifolds and Kirby Calculus" (for the statement, see the following image). But the textbook omits any proofs and only gives a weaker version (Theorem 9.2.13.). My question is:

How can we find a proof of the theorem?

Any references will be fine. Thank you for your help.

enter image description here

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    $\begingroup$ See, e.g., The Wide World of 4-Manifolds by A. Scorpan (google books) for a recent exposition of the Wall reference and its corollaries (p. 155, linked). $\endgroup$ Jul 10, 2016 at 9:20

1 Answer 1


The weaker version is a result due to Wall:

C.T.C. Wall, On simply connected 4-manifolds, J. London Math. Soc., 39 (1964), 141–149

A stronger statement has been proven by Kreck (who also gives some references), but I don't know about the original reference Gompf and Stipsicz were thinking about.

Matthias Kreck, h-cobordisms between 1-connected 4-manifolds Geom. Topol., 5 (2001), 1-6

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    $\begingroup$ @Golla Yes, I have already learned that, if two simply-connected closed 4-manifolds have the same intersection form, then they are h-cobordant. But I really need now is "functorial h-cobordism" (satisfying $\varphi_W=\varphi$ , see the attached image). Where can we find this statement in the Wall's paper? $\endgroup$ Jul 11, 2016 at 1:47
  • $\begingroup$ You're right. I added another reference (to a stronger statement). $\endgroup$ Jul 11, 2016 at 8:27
  • $\begingroup$ @Golla Thank you for your kindness. $\endgroup$ Jul 12, 2016 at 1:22

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