- #1
Bacle
- 662
- 1
Hi, Everyone:
I am curious as to the relationship between general knots and homology in the
following respect:
Say an orientable surface S is knotted in X^4, an orientable 4-manifold. Then there are
two non-isotopic embeddings e, e' of S in X^4 . Does it follow that H_2(X^4)=/0?
If e(S), e'(S) are both orientable in X^4, then they define respective homology
classes a and b. How can we tell if a~b? I imagine we would use the respective
induced maps e* and e'* , but I don't see where to go from there. I think there
may be an issue of bordism here ( AFAIK, for dimensions n<=4 , homology and
bordism coincide, i.e., if a~b , for a,b in H_k ; k<=4, then a-b bounds a (k+1)-manifold) -- and not
just some subspace. ) Conversely: if I knew that H_2(X^4)=0 . Does it follow that there aren't any S-knots, i.e., that there
is only one embedding of S in X^4 , up to isotopy? . Again, it seems to come down to determining if
we can have a 3-manifold whose boundaries are e(S) and e'(S). I don't see why this could not happen. Any Ideas?
Thanks.
I am curious as to the relationship between general knots and homology in the
following respect:
Say an orientable surface S is knotted in X^4, an orientable 4-manifold. Then there are
two non-isotopic embeddings e, e' of S in X^4 . Does it follow that H_2(X^4)=/0?
If e(S), e'(S) are both orientable in X^4, then they define respective homology
classes a and b. How can we tell if a~b? I imagine we would use the respective
induced maps e* and e'* , but I don't see where to go from there. I think there
may be an issue of bordism here ( AFAIK, for dimensions n<=4 , homology and
bordism coincide, i.e., if a~b , for a,b in H_k ; k<=4, then a-b bounds a (k+1)-manifold) -- and not
just some subspace. ) Conversely: if I knew that H_2(X^4)=0 . Does it follow that there aren't any S-knots, i.e., that there
is only one embedding of S in X^4 , up to isotopy? . Again, it seems to come down to determining if
we can have a 3-manifold whose boundaries are e(S) and e'(S). I don't see why this could not happen. Any Ideas?
Thanks.
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