  This paper contains a purely topological theorem and a geometric application.
The topological theorem states that if M is a simple closed orientable
3-manifold such that \pi_1(M) contains a genus g surface group and H_1(M;Z/2Z)
has rank at least 4g-1 then M contains a closed incompressible surface of genus
at most g. This result should be viewed as an analogue of Dehn's Lemma for
\pi_1-injective singular surfaces. The geometric application states that if M
is a closed orientable hyperbolic 3-manifold with volume less than 3.08 then
the rank of H_1(M;Z/2Z) is at most 6. The proof of the geometric theorem
combines the topological theorem with several deep geometric results, including
the Marden tameness conjecture,recently established by Agol and by
Calegari-Gabai; a co-volume estimate for 3-tame, 3-free Kleinian groups due to
Anderson, Canary, Culler and Shalen; and a volume estimate for hyperbolic Haken
manifolds recently proved by Agol, Storm and W. Thurston.
