Homology of manifolds with boundary

Question

If $M$ is a compact topological manifold WITH boundary does it follow that its homology groups are finitely generated and zero almost all of them? I know it is true in case it has no boundary (i.e. is properly a manifold).

Answer 1

Yes, it is still true: any compact topological manifold (with or without boundary) is homotopy-equivalent to a finite CW-complex, which has finitely-generated homology groups, only finitely many of which are nonzero.

Answer 2

This is an addendum to Brad's answer (including references as requested).

Every topological manifold (with or without boundary) is locally contractible (this is immediate from the definition). In particular, it is ANR (absolute neighborhood retract) - this requires a bit of a proof, which can be found in Borsuk's book "Theory of retracts". K.Borsuk conjectured in 1940s that every compact ANR is homotopy-equivalent to a finite CW complex. This conjecture became a theorem in 1970s.

See references in my answer to this MSE question.

On the other hand, using these theorems for establishing finite generation of homology groups of compact manifolds is a bit of an overkill: One can prove it directly using sheaf cohomology (and local contractibility).