Morse homology with differential graded coefficients / Jean-François Barraud, Mihai Damian, Vincent Humilière, Alexandru Oancea.

Format:

Book

Author/Creator:

Barraud, Jean-François, Author.

Damian, Mihai, Author.

Humilière, Vincent, Author.

Oancea, Alexandru, Author.

Series:

Progress in mathematics (Boston, Mass.) ; 0743-1643 v. 360.

Virtual series on symplectic geometry

Progress in mathematics, 0743-1643 ; volume 360

The virtual series on symplectic geometry

Language:

English

Subjects (All):

Morse theory.

Homology theory.

Physical Description:

xi, 229 pages : illustrations (black and white) ; 25 cm.

Place of Publication:

Cham, Switzerland : Birkhäuser, [2025].

©2025

Summary:

"The key geometric objects underlying Morse homology are the moduli spaces of connecting gradient trajectories between critical points of a Morse function. The basic question in this context is the following: How much of the topology of the underlying manifold is visible using moduli spaces of connecting trajectories? The answer provided by “classical” Morse homology as developed over the last 35 years is that the moduli spaces of isolated connecting gradient trajectories recover the chain homotopy type of the singular chain complex. The purpose of this monograph is to extend this further: the fundamental classes of the compactified moduli spaces of connecting gradient trajectories allow the construction of a twisting cocycle akin to Brown’s universal twisting cocycle. As a consequence, the authors define (and compute) Morse homology with coefficients in any differential graded (DG) local system. As particular cases of their construction, they retrieve the singular homology of the total space of Hurewicz fibrations and the usual (Morse) homology with local coefficients. A full theory of Morse homology with DG coefficients is developed, featuring continuation maps, invariance, functoriality, and duality. Beyond applications to topology, this is intended to serve as a blueprint for analogous constructions in Floer theory. " - publisher

Contents:

Introduction and Main Results

Morse versus DG Morse Homology Toolset

Comparison of the Barraud–Cornea Cocycle and the Brown Cocycle

Algebraic Properties of Twisted Complexes

Morse Homology with DG-Coefficients: Construction

Morse Homology with DG-Coefficients: Invariance

Fibrations

Functoriality: General Properties

Functoriality: First Definition

Functoriality: Second Definition

Cohomology and Poincaré Duality

Shriek Maps and Poincaré Duality for Non-Orientable Manifolds

Beyond the Case of Manifolds of Finite Dimension.

Notes:

Includes bibliographical references (pages 227-229) and indexes.

ISBN:

9783031880193

3031880196

OCLC:

1527659906