Does Z^n as the First Homology Group Imply R^n with Real Coefficients?

[petergreat]

If the first homology group of a manifold is Z^n, does it imply that the first homology group with real coefficients (obtained from singular chains with real coefficients) is R^n?

 

[mathwonk]

i think so. in general the group with integer coefficients should have more information than with other coefficients. look up universal coefficients:

in general, if the integral homology has no torsion, i.e. is free, then the other homologies are free of the same dimension.

please forgive me for answering in my dotage without verifying any of this. but i am probably not too far wrong.

 

[lavinia]

If the first homology group of a manifold is Z^n, does it imply that the first homology group with real coefficients (obtained from singular chains with real coefficients) is R^n?

Look at the Universal Coefficient Theorem for homology.