🔁Elementary Differential Topology Unit 14 – Degree of a Map: Applications & Significance

Definition and Basics

• The degree of a map is a fundamental concept in topology that measures the number of times a continuous map wraps around its target space
• Formally, given a continuous map $f: M \to N$ between oriented, compact, connected manifolds of the same dimension, the degree of $f$ is an integer that represents the signed count of preimages of a regular value
• The degree is independent of the choice of regular value, making it a global property of the map
• The degree captures important topological information about how the map behaves and how it relates the spaces $M$ and $N$
• The concept of degree can be generalized to maps between manifolds with boundary and to maps between manifolds of different dimensions
  • In the case of manifolds with boundary, the degree is defined using relative homology
  • For maps between manifolds of different dimensions, the degree is defined using the induced map on homology

Topological Foundations

• The degree of a map relies on several key topological concepts, including manifolds, orientability, compactness, and connectedness
• Manifolds are topological spaces that locally resemble Euclidean space and provide a suitable setting for studying the degree of a map
  • Examples of manifolds include spheres, tori, and projective spaces
• Orientability is a property of manifolds that allows for a consistent choice of orientation, which is crucial for defining the signed count of preimages
• Compactness ensures that the preimage of a regular value consists of a finite number of points, making the degree well-defined
• Connectedness is required to ensure that the degree is a global invariant of the map and does not depend on the choice of regular value
• The degree of a map is closely related to the induced map on homology, which captures the effect of the map on the topological structure of the spaces

Computing the Degree

• There are several methods for computing the degree of a map, depending on the specific context and available information
• One common approach is to choose a regular value $y \in N$ and count the preimages of $y$ under $f$, taking into account the local orientation of $M$ and $N$ at each preimage point
  • The local orientation is determined by the sign of the Jacobian determinant of $f$ at each preimage point
• Another method involves using the induced map on homology, $f_*: H_n(M) \to H_n(N)$, where $n$ is the dimension of the manifolds
  • The degree of $f$ is the integer $d$ such that $f_*([M]) = d[N]$, where $[M]$ and $[N]$ are the fundamental classes of $M$ and $N$, respectively
• In some cases, the degree can be computed using algebraic methods, such as the winding number for maps between circles or the Brouwer degree for maps between spheres
• Computational tools from algebraic topology, such as simplicial homology and the Mayer-Vietoris sequence, can be employed to calculate the degree in more complex situations

Properties and Theorems

• The degree of a map satisfies several important properties and is the subject of various theorems in topology
• The degree is a homotopy invariant, meaning that if two maps $f, g: M \to N$ are homotopic, then they have the same degree
  • This property allows for the classification of maps up to homotopy using the degree
• The degree is multiplicative under composition: if $f: M \to N$ and $g: N \to P$ are continuous maps between oriented, compact, connected manifolds, then $\deg(g \circ f) = \deg(g) \cdot \deg(f)$
• The degree of the identity map is always 1, and the degree of a constant map is always 0
• The Hopf degree theorem states that for any integer $d$, there exists a map $f: S^n \to S^n$ of degree $d$, where $S^n$ is the $n$-dimensional sphere
• The Brouwer fixed point theorem can be proved using the degree of a map, showing that any continuous map from a ball to itself has a fixed point

Applications in Topology

• The degree of a map has numerous applications in various branches of topology and related fields
• In algebraic topology, the degree is used to study the relationships between manifolds and their mappings, providing a powerful tool for classification and understanding the structure of topological spaces
  • For example, the degree can be used to prove the Borsuk-Ulam theorem, which states that any continuous map from an $n$-sphere to $\mathbb{R}^n$ must map some pair of antipodal points to the same point
• In differential topology, the degree appears in the statement and proof of the Poincaré-Hopf theorem, which relates the Euler characteristic of a manifold to the indices of a vector field's zeros
• The degree is also employed in the study of covering spaces and the fundamental group, helping to characterize the behavior of lifts and the relationship between the base space and the covering space
• In physics and other applied fields, the degree of a map arises in the context of topological invariants, such as the winding number and the Chern number, which have important physical interpretations

Examples and Calculations

• Consider the map $f: S^1 \to S^1$ given by $f(z) = z^n$, where $S^1$ is the unit circle in the complex plane and $n$ is an integer. The degree of this map is $n$, as each point in the codomain is wrapped around $n$ times by the map
• Let $f: S^2 \to S^2$ be the antipodal map, defined by $f(x) = -x$ for all $x \in S^2$. The degree of the antipodal map is $-1$, as it reverses the orientation of the sphere
• For the map $f: T^2 \to S^2$ given by $f(x, y) = (\cos(2\pi x)\sin(\pi y), \sin(2\pi x)\sin(\pi y), \cos(\pi y))$, where $T^2$ is the torus, the degree is 0. This is because the image of $f$ is not surjective, as it misses the poles of the sphere
• Consider the map $f: \mathbb{RP}^2 \to S^2$ induced by the quotient map $q: S^2 \to \mathbb{RP}^2$, where $\mathbb{RP}^2$ is the real projective plane. The degree of this map is 2, as each point in the codomain has two preimages (antipodal points) in the domain

Significance in Differential Topology

• The degree of a map plays a crucial role in differential topology, where it is used to study smooth manifolds and their mappings
• In the context of smooth manifolds, the degree of a map can be defined using differential forms and integration, providing a link between topology and analysis
  • The degree of a smooth map $f: M \to N$ between oriented, compact, connected smooth manifolds can be computed as the integral of the pullback of a volume form on $N$ over $M$
• The degree appears in the formulation and proof of several important theorems in differential topology, such as the Poincaré-Hopf theorem and the Gauss-Bonnet theorem
• The concept of degree is generalized to the intersection number in differential topology, which measures the signed count of intersections between submanifolds of complementary dimensions
• The degree of a map is related to the concept of the Jacobian determinant in local coordinates, which captures the local behavior of the map and its effect on orientations
• In Morse theory, the degree of a map between manifolds can be expressed in terms of the critical points of a Morse function, establishing a connection between the topology of the manifolds and the behavior of smooth functions on them

Related Concepts and Extensions

• The degree of a map is closely related to other topological invariants, such as the Euler characteristic and the fundamental group
  • The Euler characteristic of a manifold can be expressed as the degree of a certain map (e.g., the Gauss map for surfaces in $\mathbb{R}^3$)
  • The degree of a map between manifolds is related to the induced homomorphism between their fundamental groups
• The concept of degree can be extended to more general settings, such as maps between manifolds with boundary and maps between singular spaces
  • For manifolds with boundary, the degree is defined using relative homology and takes into account the behavior of the map on the boundary
  • In the case of singular spaces, the degree can be defined using homology or cohomology theories, such as singular homology or Čech cohomology
• The degree of a map has applications in fixed point theory, where it is used to prove existence and multiplicity results for fixed points of continuous maps
  • The Lefschetz fixed point theorem relates the fixed points of a map to its induced map on homology and the Lefschetz number, which is defined in terms of the degree
• In dynamical systems, the degree of a map appears in the study of the rotation number for circle homeomorphisms and the Poincaré index for planar vector fields
• The degree of a map has generalizations in algebraic topology, such as the Hopf invariant for maps between spheres and the degree of a fibration in the context of fiber bundles