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Putting constraints on the nature of the interaction between nonsingular gradient-like fields and the boundary of a smooth compact manifold have profound implications for the topology of the manifold, and this book describes these implications. Along the way, this study extends into the spaces of flow trajectories, their complexity and intricate combinatorics. New invariants of gradient flows, as well as invariants of the underlying manifold are produced, based on the understanding of these structures (on the space of trajectories and on the flow-generated spines).The process leading to a discovery is equally if not more important than the discover itself and in this book, classical singularity theory guides our investigations, and algebraic topology provides the tools.
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