15.2 Connections between Morse theory and Floer homology
3 min read•august 7, 2024
Morse theory and share deep connections. Both use chain complexes built from critical points or objects, with differentials or curves between them. These structures reveal topological and geometric information about manifolds and symplectic spaces.
Compactness and gluing are key in both theories. They ensure well-defined counts and allow reconstruction of . Algebraic tools like and help compute homology and show its invariance under different choices.
Morse and Floer Complexes
Morse-Smale Pairs and Morse Complex
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- consists of a and a Riemannian metric satisfying transversality conditions
- Stable and unstable manifolds of critical points intersect transversely
- Ensures well-defined between critical points
- is a chain complex generated by critical points of a Morse function
- Differential counts between critical points (descending manifolds)
- Homology of the Morse complex is isomorphic to the of the manifold
- Gradient trajectories are integral curves of the negative gradient vector field of the Morse function
- Connect critical points of adjacent indices (Morse index)
- Represent the flow lines between critical points in the Morse complex
Floer Complex and Pseudoholomorphic Curves
- Floer complex is an infinite-dimensional analogue of the Morse complex
- Generated by certain objects in symplectic geometry (Lagrangian intersections, periodic orbits)
- Differential counts connecting these objects
- Pseudoholomorphic curves are solutions to the Cauchy-Riemann equations in symplectic manifolds
- Generalize the notion of holomorphic curves in complex geometry
- Serve as the analogues of gradient trajectories in the Floer complex
- Satisfy a perturbed version of the Cauchy-Riemann equations involving an almost complex structure
Compactness and Gluing
Compactification of Moduli Spaces
- is necessary to ensure well-defined counts in the Morse and Floer complexes
- Moduli spaces of gradient trajectories or pseudoholomorphic curves may not be compact
- Compactification adds limit points corresponding to broken trajectories or curves
- Compactified moduli spaces include degenerate configurations
- Broken gradient trajectories consist of concatenations of trajectories between intermediate critical points
- Broken pseudoholomorphic curves involve multiple-level curves connected at nodal points (bubbling)
Gluing Theorems
- provide a way to reconstruct the compactified moduli spaces from their boundary strata
- Describe how broken trajectories or curves can be glued together to form smooth configurations
- Crucial for establishing the chain complex structure and the continuation maps
- Gluing of broken gradient trajectories involves connecting trajectories at common critical points
- Requires careful analysis of the exponential decay properties near critical points
- Gluing of pseudoholomorphic curves involves resolving nodal singularities
- Uses techniques from elliptic regularity theory and the implicit function theorem
Algebraic Structures
Spectral Sequences
- Spectral sequences are algebraic tools for computing homology groups
- Arise naturally in the context of filtered complexes or double complexes
- Provide a systematic way to extract information from the Morse or Floer complex
- Spectral sequence of the Morse complex relates the to the singular homology
- Converges to the singular homology of the manifold
- Successive pages reveal finer information about the topology of the manifold
- Spectral sequences in Floer theory relate different versions of Floer homology
- Example: Leray-Serre spectral sequence relates the Floer homology of a fibration to its base and fiber
Continuation Maps
- Continuation maps are homomorphisms between Morse or Floer complexes for different data
- Relate complexes defined using different Morse functions, Riemannian metrics, or almost complex structures
- Induced by counting gradient trajectories or pseudoholomorphic curves for a homotopy of data
- Continuation maps establish the invariance of Morse or Floer homology
- Show that the homology is independent of the choice of auxiliary data
- Provide isomorphisms between different versions of the homology theories
- Continuation maps are crucial for defining product structures and other algebraic operations
- Example: Pair-of-pants product in Floer homology is defined using continuation maps
Key Terms to Review (28)
Action Functional: The action functional is a mathematical tool used in variational calculus that measures the 'action' of a path taken by a system in a given space. It connects classical mechanics to modern mathematical physics, particularly in contexts like Morse theory and Floer homology, where it plays a key role in understanding critical points and their significance in the study of dynamical systems and geometry.
Andreas Floer: Andreas Floer was a mathematician known for his groundbreaking work in the field of symplectic geometry and Floer homology, which connects analysis and topology. His contributions provided tools for studying the topology of manifolds by using the critical points of functionals on infinite-dimensional spaces, significantly advancing the understanding of Morse theory and its applications in mathematical physics.
Compactification: Compactification is a mathematical process that transforms a non-compact space into a compact space, often by adding 'points at infinity' or other types of boundaries. This technique is essential in various areas of mathematics, as it allows for the application of tools and theorems that are valid only for compact spaces, enhancing the study of topological properties and structures.
Continuation Maps: Continuation maps are functions that connect different Morse functions or critical level sets across a parameter space, facilitating the study of the behavior of these functions as one varies the parameter. They allow for the comparison and analysis of Morse theory across different contexts, particularly when examining how critical points change and how they relate to the topology of manifolds. In the context of Floer homology, continuation maps play a crucial role in establishing connections between different Morse complex levels, helping to define invariants that arise from these constructions.
Counting trajectories: Counting trajectories refers to the process of enumerating the paths taken by critical points in a manifold under the influence of a gradient flow, specifically in relation to Morse theory and Floer homology. This concept plays a crucial role in understanding the relationships between different critical points and how they interact through flows, thereby contributing to the overall topology of the manifold.
Critical Point: A critical point is a point on a manifold where the gradient of a function is zero or undefined, indicating a potential local maximum, local minimum, or saddle point. Understanding critical points is crucial as they help determine the behavior of functions and the topology of manifolds through various mathematical frameworks.
Floer homology: Floer homology is a powerful mathematical tool used to study the topology of manifolds by analyzing the solution spaces of certain partial differential equations. It connects the critical points of a smooth function on a manifold, like those found in Morse theory, to algebraic invariants that reveal deeper geometric structures. This concept plays a crucial role in areas such as symplectic geometry and provides insights into the relationships between different topological spaces.
Gauge theory: Gauge theory is a framework in theoretical physics that describes how forces interact with matter through fields defined on a manifold, using the concept of symmetry to govern these interactions. The essential idea is that certain transformations can be made without altering the physical state of the system, leading to conservation laws and fundamental interactions. This notion is crucial in connecting different areas of mathematics and physics, particularly in understanding various topological aspects and their relationship with Morse theory and Floer homology.
Gluing Theorems: Gluing theorems are fundamental results in mathematics that allow one to construct new objects from existing ones by identifying certain parts. These theorems are particularly important in Morse theory and Floer homology, as they provide a way to relate different spaces or manifolds by piecing them together along specified regions. They facilitate the understanding of how local properties can combine to form global properties, which is essential in both fields.
Gradient Flow: Gradient flow refers to the flow generated by following the negative gradient of a function, effectively describing how a system evolves over time towards its critical points. This concept is crucial in understanding the dynamics of functions, particularly in relation to their critical points, where local minima and maxima exist, and connects deeply with various topological and geometrical properties of manifolds.
Gradient Trajectories: Gradient trajectories are paths in a manifold that are determined by the gradient flow of a function, illustrating how the values of that function change over time. They connect critical points and play a crucial role in both Morse theory and Floer homology, where they help analyze the topology of the underlying space by examining the dynamics of the gradient flow.
H-principle: The h-principle is a concept in mathematics that refers to a condition under which certain kinds of geometric or topological problems can be solved by means of homotopy or deformation. It suggests that if a certain map can be approximated by smooth maps, then there exists a smooth map that solves the problem. This principle is particularly relevant when connecting Morse theory and Floer homology, as it provides insights into how critical points of functions can be understood in the context of smooth structures and invariants.
Homological invariants: Homological invariants are algebraic structures that provide essential information about the topological properties of a space through the use of homology theory. They are crucial for distinguishing between different topological spaces and play a significant role in connecting Morse theory and Floer homology, as they help in understanding how the critical points of Morse functions relate to the underlying topology of manifolds.
Lefschetz Thimbles: Lefschetz thimbles are specific types of chains in the context of Morse theory that arise from studying critical points of a function defined on a manifold. These thimbles are used to connect different levels of Morse functions and are crucial for understanding how these critical points contribute to the topology of the manifold, especially when examining connections with Floer homology and symplectic geometry.
Marcel Grossmann: Marcel Grossmann was a Swiss mathematician and physicist known for his contributions to the fields of differential geometry and general relativity. He is particularly recognized for his collaboration with Albert Einstein, providing crucial mathematical support that facilitated the formulation of Einstein's theory of general relativity, which has deep connections to various aspects of topology and geometry.
Moduli Spaces: Moduli spaces are geometric spaces that parametrize families of mathematical objects, allowing for the classification and study of these objects up to certain equivalences. They serve as a bridge between algebraic geometry and topology, facilitating the understanding of how different structures relate to each other, particularly in the context of Morse theory and Floer homology, where they help analyze critical points and their stability.
Morse Complex: The Morse complex is a combinatorial object that arises from a Morse function defined on a manifold, capturing the topological features of the manifold by analyzing critical points and their indices. It consists of cells corresponding to the critical points, organized in such a way that it reflects the topology of the underlying space through its critical level sets. This construction allows for deeper insights into both the topology of the manifold and its relationship with homology theory.
Morse Function: A Morse function is a smooth real-valued function defined on a manifold that has only non-degenerate critical points, where the Hessian matrix at each critical point is non-singular. These functions are crucial because they provide insights into the topology of manifolds, allowing the study of their structure and properties through the behavior of their critical points.
Morse Homology: Morse homology is a branch of algebraic topology that studies the topology of manifolds using Morse functions, which are smooth real-valued functions that have critical points. This theory connects critical points of these functions to the structure of the manifold, revealing important features about its topology and allowing for the computation of topological invariants.
Morse-Bott Theory: Morse-Bott Theory is an extension of Morse Theory that deals with functions whose critical points form manifolds instead of isolated points. This approach allows for a more sophisticated understanding of the topology of manifolds and leads to the construction of Morse homology while also connecting with various areas, including symplectic geometry and Floer homology.
Morse-Smale Pair: A Morse-Smale pair consists of a smooth function and a gradient-like flow on a manifold that exhibit specific properties for analyzing the topology of the manifold. This concept combines elements of Morse theory, which studies critical points of smooth functions, and Smale's theory of dynamical systems, focusing on the behavior of trajectories in relation to these critical points, thus bridging the gap between static topology and dynamic analysis.
Pseudo-holomorphic curve: A pseudo-holomorphic curve is a smooth map from a Riemann surface into a symplectic manifold that satisfies a certain differential equation, making it a critical point of a specific energy functional. These curves play a crucial role in the study of Floer homology by providing a way to connect the topology of manifolds with their geometric properties. Essentially, they bridge the gap between classical Morse theory and modern symplectic geometry, revealing deep insights about the structure of the manifolds involved.
Pseudoholomorphic curves: Pseudoholomorphic curves are smooth maps from a Riemann surface into a symplectic manifold that satisfy a certain nonlinear partial differential equation called the Cauchy-Riemann equation, which is adapted to the symplectic structure. These curves play a crucial role in Floer homology as they help to count holomorphic disks, which leads to invariants that connect different areas of geometry and topology, particularly relating Morse theory to quantum mechanics.
Riemannian manifold: A Riemannian manifold is a differentiable manifold equipped with a Riemannian metric, which allows for the measurement of lengths and angles of curves on the manifold. This structure provides a way to generalize the concepts of distance and curvature from Euclidean spaces to more complex geometries. Understanding Riemannian manifolds is crucial for analyzing geometric properties and is directly linked to various applications in differential geometry, physics, and advanced calculus.
Singular Homology: Singular homology is an algebraic topology tool that assigns a sequence of abelian groups or modules to a topological space, capturing information about its shape and structure through continuous maps called singular simplices. This concept allows for the comparison of different spaces and has deep connections to various mathematical theories, especially in the context of Morse theory and its applications in understanding manifold structures and critical points.
Spectral sequences: Spectral sequences are powerful computational tools in algebraic topology and homological algebra that provide a method for calculating homology and cohomology groups. They are used to systematically organize complex calculations into a sequence of simpler ones, enabling mathematicians to derive deep results about the structure of spaces and their invariants, particularly in contexts like Morse theory and Floer homology.
Symplectic manifold: A symplectic manifold is a smooth, even-dimensional manifold equipped with a closed, non-degenerate differential 2-form known as the symplectic form. This structure allows for the study of geometric properties and dynamics of Hamiltonian systems, making it crucial in areas like classical mechanics and mathematical physics. The interactions between symplectic geometry and Morse theory reveal deep connections, particularly in how critical points of functionals relate to the topology of the manifold.
Topological Quantum Field Theory: Topological Quantum Field Theory (TQFT) is a theoretical framework that combines concepts from quantum field theory with topology, focusing on the topological aspects of spaces rather than their geometric properties. It captures the behavior of quantum fields in a way that is invariant under continuous transformations, making it a powerful tool for studying topological invariants and their applications in mathematical physics and geometry.