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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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1  
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
4  
A little confused -- is John Smith the same person as Topologieee? – Ryan Budney Jun 13 '11 at 17:57

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