As the question was asked the answer seems to be: no. Consifder R^n and a sequence of non-zero points converging to zero. Choose around each point in this sequence a small ball which does not meat the other points in the sequence and remove these closed balls. This is nour manifold M. Now consider the 0-cycle given by the point 0. All neighbourhoods of 0 contain infintely many holes and so H^{n-1} is non-zero.

If you ask the question, whether a given homology class has a representative with this property I agree with John that the answer should be yes. But I have not thought about a detailed argument.

Matthias Kreck

As the question was asked the answer seems to be: no. Consider R^n and a sequence of non-zero points converging to zero. Choose around each point in this sequence a small ball which does not meet the other points in the sequence and remove these closed balls. This is our manifold $M$. Now consider the 0-cycle given by the point 0. All neighbourhoods of 0 contain infinitely many holes and so $H^{n-1}(M)$ is non-zero.

If you ask the question, whether a given homology class has a representative with this property I agree with John that the answer should be yes. But I have not thought about a detailed argument.

Matthias Kreck