Twisted Morse Complexes

Morse Homology and Cohomology with Local Coefficients

Augustin Banyaga, David Hurtubise, Peter Spaeth

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English

Springer International Publishing AG
02 November 2024

Cybernetics & systems theory; Calculus & mathematical analysis; Algebraic topology

Series: Lecture Notes in Mathematics

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.

By: Augustin Banyaga, David Hurtubise, Peter Spaeth
Imprint: Springer International Publishing AG
Country of Publication: Switzerland
Edition: 2024 ed.
Volume: 2361
Dimensions: Height: 235mm, Width: 155mm,
ISBN: 9783031716157
ISBN 10: 3031716159
Series: Lecture Notes in Mathematics
Pages: 158
Publication Date: 02 November 2024
Audience: Professional and scholarly , College/higher education , Undergraduate , Further / Higher Education
Format: Paperback
Publisher's Status: Active

- 1. Introduction.- 2. The Morse Complex with Local Coefficients.- 3. The Homology Determined by the Isomorphism Class of G.- 4. Singular and CW-Homology with Local Coefficients.- 5. Twisted Morse Cohomology and Lichnerowicz Cohomology.- 6. Applications and Computations.

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.