The homology analogue of this question is answered affirmatively by Proposition 2.1 in math/0605535. By The proof provides $U$ with $H_p(U;{\mathbb Z})$ torsion-free and so implies the Universal Coefficient Theorem, it answers affirmative answer to the original question affirmatively with ${\mathbb Z}$ replaced by a fieldthe Universal Coefficient Theorem.

The proof of Proposition 2.1 is a version of the argument John Klein described, phrased in terms of transverse triangulations of Lemma 2.3 (which is proved in math/1012.3979, after Matthias Kreck had pointed out that this had not been in the literature).

The homology analogue of this question is answered affirmatively by Proposition 2.1 in math/0605535; by . By the Universal Coefficient Theorem, it answers the original question affirmatively as wellwith ${\mathbb Z}$ replaced by a field.

The proof of Proposition 2.1 is a version of the argument John Klein described, phrased in terms of transverse triangulations of Lemma 2.3 (which is proved in math/1012.3979, after Matthias Kreck had pointed out that this had not been in the literature).

The homology analogue of this question is answered affirmatively by Proposition 2.1 in math/0605535; by the Universal Coefficient Theorem, it answers the original question affirmatively as well.

The proof is a version of the argument John Klein described, phrased in terms of transverse triangulations of Lemma 2.3 (which is proved in math/1012.3979, after Matthias Kreck had pointed out that this had not been in the literature).