D. Barden, C. Thomas: An introduction to differential manifolds, 2003. Imperial College Press, London. ISBN:?1-86094-354-3; 1-86094-355-1. xii+218 pp.
Chapters 1-2, 4-6 and 9, minus sections 2.4, 6.6, 6.7, but plus sections 3.2, 7.1 and Poincare duality (you can use e.g. M. Spivak, A comprehensive introduction to differential geometry. Vol. I, or I. Madsen, J. Tornehave, From calculus to cohomology).