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edited Apr 13, 2017 at 12:58

I would explain it like this:

Start with a framed link in $S^3$ and do Dehn-surgery along it to get a new $3$-manifold. In fact one can get every compact oriented $3$-manifold in this way as mentioned earlier.

To distinguish this $3$-manifolds one can use the fundamental group. Actually the fundamental group can distinguish almost almost all $3$-manifolds. It is easy to give a presentation of the fundamental group out of a surgery diagram:

First use the Wirtinger presentation of the link exterior and then add a new realtion (of the form $p\mu+q\lambda=1$) for every surgery.

Of course its difficult to distinguish two such presentations in general. But to convince someone that there are many different should not be hard. For example one can do the following:

make this groups abelian and use the classification of abelian groups.
compare the orders of the groups (this is how Poincaré proved the Poincaré sphere to be not $S^3$).
show that some of this groups are non-abelian (for example by finding surjective group homomorphisms in non-abelian groups) others are abelian.
compare orders of the elements.

I would explain it like this:

Start with a framed link in $S^3$ and do Dehn-surgery along it to get a new $3$-manifold. In fact one can get every compact oriented $3$-manifold in this way as mentioned earlier.

To distinguish this $3$-manifolds one can use the fundamental group. Actually the fundamental group can distinguish almost all $3$-manifolds. It is easy to give a presentation of the fundamental group out of a surgery diagram:

First use the Wirtinger presentation of the link exterior and then add a new realtion (of the form $p\mu+q\lambda=1$) for every surgery.

Of course its difficult to distinguish two such presentations in general. But to convince someone that there are many different should not be hard. For example one can do the following:

make this groups abelian and use the classification of abelian groups.
compare the orders of the groups (this is how Poincaré proved the Poincaré sphere to be not $S^3$).
show that some of this groups are non-abelian (for example by finding surjective group homomorphisms in non-abelian groups) others are abelian.
compare orders of the elements.

I would explain it like this:

Start with a framed link in $S^3$ and do Dehn-surgery along it to get a new $3$-manifold. In fact one can get every compact oriented $3$-manifold in this way as mentioned earlier.

First use the Wirtinger presentation of the link exterior and then add a new realtion (of the form $p\mu+q\lambda=1$) for every surgery.

Of course its difficult to distinguish two such presentations in general. But to convince someone that there are many different should not be hard. For example one can do the following:

make this groups abelian and use the classification of abelian groups.
compare the orders of the groups (this is how Poincaré proved the Poincaré sphere to be not $S^3$).
show that some of this groups are non-abelian (for example by finding surjective group homomorphisms in non-abelian groups) others are abelian.
compare orders of the elements.

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created Feb 29, 2016 at 17:23

Marc Kegel

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I would explain it like this:

Start with a framed link in $S^3$ and do Dehn-surgery along it to get a new $3$-manifold. In fact one can get every compact oriented $3$-manifold in this way as mentioned earlier.

First use the Wirtinger presentation of the link exterior and then add a new realtion (of the form $p\mu+q\lambda=1$) for every surgery.

Of course its difficult to distinguish two such presentations in general. But to convince someone that there are many different should not be hard. For example one can do the following:

make this groups abelian and use the classification of abelian groups.
compare the orders of the groups (this is how Poincaré proved the Poincaré sphere to be not $S^3$).
show that some of this groups are non-abelian (for example by finding surjective group homomorphisms in non-abelian groups) others are abelian.
compare orders of the elements.