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1 vote

4-manifold obtained from a ribbon disk exterior by attaching a 2-handle is simply-connected if its boundary is a homology sphere

This is false in general, I’ll prove the existence of a counterexample. Since $D$ is a ribbon disk, $X$ has a handle structure with one $0$-handle, $n$ $1$-handles and $n-1$ $2$-handles (as described ...
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6 votes

Small examples of exceptional hyperbolic Dehn Filling of hyperbolic manifolds

Here is a variant of Ian's answer. We cut along the round component in his diagram, perform a full twist (with the correct handedness), and reglue. This gives the following three-component link. If ...
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7 votes

Small examples of exceptional hyperbolic Dehn Filling of hyperbolic manifolds

Another sort of example is a geodesic that is not embedded. In the Whitehead link complement, there is a thrice-punctured sphere (pants) which is totally geodesic. A figure 8 curve in such a surface ...
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9 votes

Small examples of exceptional hyperbolic Dehn Filling of hyperbolic manifolds

Here are two examples of exceptional hyperbolic fillings. In the first the core curve becomes parabolic after filling (so cannot be isotopic to a geodesic). In the second, the core curve is homotopic (...
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1 vote

Pullback of $w_1$ for 3-manifolds

For manifolds $M,N$ of dimension $<3$ I know the complete answer for the similar question. If $M,N$ are closed $1$-manifolds, obviously, $f$ exists if and only if $\alpha=0$. If $M,N$ are closed $...
4 votes

Pullback of $w_1$ for 3-manifolds

This basically boils down to Understanding what the class $w_1$ represents, and Doing obstruction theory. This is generally not an easy question to answer, since obstruction theory requires you to ...
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1 vote
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0-framed smoothly slice knot that can be obtained by blowing down successively a link of unknots

Yes, you can always do this. In fact, I think you can do it for any framed link, regardless of the framings, the number of components, and the slice assumption. (For simplicity, I will write the ...
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