This book gives a detailed presentation of twisted Morse homology and cohomology on closed finite-dimensional smooth manifolds. It contains a complete proof of the Twisted Morse Homology Theorem, which says that on a closed finite-dimensional smooth manifold the homology of the Morse-Smale-Witten chain complex with coefficients in a bundle of abelian groups G is isomorphic to the singular homology of the manifold with coefficients in G. It also includes proofs of twisted Morse-theoretic versions of well-known theorems such as Eilenberg's Theorem, the Poincaré Lemma, and the de Rham Theorem. The effectiveness of twisted Morse complexes is demonstrated by computing the Lichnerowicz cohomology of surfaces, giving obstructions to spaces being associative H-spaces, and computing Novikov numbers. Suitable for a graduate level course, the book may also be used as a reference for graduate students and working mathematicians or physicists.

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Augustin Banyaga is a Professor of Mathematics and a Distinguished Senior Scholar at Penn State University in the Eberly College of Science and a Fellow of the African Academy of Sciences. He has authored at least 70 peer reviewed papers and 3 books, including Lectures on Morse Homology published by Springer.

David Hurtubise is a Professor of Mathematics at Penn State Altoona. He has authored at least 14 peer reviewed papers, 140 Mathematical Reviews, 45 Zentralblatt Reviews, and the book Lectures on Morse Homology published by Springer.

Peter Spaeth is a Senior Research Scientist at NASA's Langley Research Center. He has authored over 20 peer reviewed papers in mathematics, materials science, and nondestructive evaluation. In 2023 he was awarded the NASA Early Career Achievement Medal.