How does non-isotopic embeddings affect the homology of manifolds and knots?

[Bacle]

Hi, everyone:

I think this should be simple, but I've been stuck for a while now:

Let M be an n-manifold and N an n-dim. subspace of M, of possibly lower dimension

than M , then M is knotted, or N is a knot in M if there are non-isotopic

embeddings of N in M.

Now:

Say there are non-isotopic embeddings of N in M. How does this affect the

n-th homology of M (n is the dimension of M) ? . Does it follow that if

H_n(M)==0 , that there are no knots, i.e., there is only one isotopy class

of embeddings of N in M? And, conversely, if f,g, are two non-isotopic embeddings

of N in M, does it follow that H_n(M) is not trivial?

Thanks .

 

[Bacle]

This is not a full answer by far, but I think we may be able to use the

map induced on homology by the inclusion of the subspace into the space.

I believe --please correct me if I am wrong -- that if the subspace N is

orientable , and the included image i(N) is a submanifold, then i(N) may be/represent

a homology class. And then the issue seems to be that of determining the

homology class of i*(N) , i'*(N) , for non-isotopic embeddings i, i'.