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I'm reading R.E. Gompf and A.I. Stipsicz, 4-Manifolds and Kirby Calculus. I don't understand Remarks 4.4.1 on page 116-117 Google books here.

At first I can't understand why we take immersed disk $D \subset Y$ with boundary $K \subset \partial _{+}Y$.

Next, I can't understand on the counterexamples

$X \approx D^4 , X ' \approx S^2\times D^2-intD^4$ .

If you can help me, then please teach me the reason for this

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