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For (3), here is a nice counterexample if we ignore contractibility (taken from an article from The American Mathematical Monthly): http://www.jstor.org/stable/pdfplus/2695643.pdf Hopefully someone can embed the image (Figure 13). It is two topologically distinct manifolds with the torus as a boundary... one being a solid [knot] torus, and the other being a 'cylinder' with a knotted inner-hole. |
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For (3), here is a nice counterexample (taken from an article from The American Mathematical Monthly): http://www.jstor.org/stable/pdfplus/2695643.pdf Hopefully someone can embed the image (Figure 13). It is two topologically distinct manifolds with the torus as a boundary... one being a solid [knot] torus, and the other being a 'cylinder' with a knotted inner-hole. |
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