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Let $X^4$ be the 4-dimensional handlebody with $\partial X=S^3$ and $\pi_i(X)=\pi_i(B^4)$.

Is it true that we can always change $X^4$ with handlebody without 3-handle?

(I'm concerning about the Andrews-Curtis conjecture)

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What kind of "change" do you want to allow? For example, do you allow replacing $X$ with a 4-ball? That would do it. –  Ryan Budney Jun 13 '11 at 17:49
Are you allowing handle slides? Cancellations? Perhaps more or less? –  Ryan Budney Jun 13 '11 at 17:53
Fixing diffeomorphism type. –  Topologieee Jun 13 '11 at 17:54
Allowing handle slide and cancellation. –  Topologieee Jun 13 '11 at 17:54
A little confused -- is John Smith the same person as Topologieee? –  Ryan Budney Jun 13 '11 at 17:57

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