🔢Elementary Algebraic Topology Unit 5 – Covering Spaces

Key Concepts and Definitions

• Covering space $p: \tilde{X} \to X$ consists of a continuous surjective map $p$ from a topological space $\tilde{X}$ to another space $X$
  • For each point $x \in X$, there exists an open neighborhood $U$ such that $p^{-1}(U)$ is a disjoint union of open sets in $\tilde{X}$, each of which is mapped homeomorphically onto $U$ by $p$
• Covering map $p$ has the unique lifting property for paths and homotopies
  • Any path in the base space $X$ can be uniquely "lifted" to a path in the covering space $\tilde{X}$ given a starting point
• Deck transformations are homeomorphisms of the covering space $\tilde{X}$ that preserve the covering map $p$
  • Form a group under composition called the deck transformation group or group of covering transformations
• Universal covering space is a simply connected covering space that covers all other covering spaces (up to isomorphism)
• Evenly covered neighborhoods are open sets in the base space $X$ whose preimages under the covering map $p$ consist of disjoint copies of the neighborhood
• Regular covering space has a normal subgroup of the fundamental group of the base space as its deck transformation group

Motivation and Historical Context

• Covering spaces arise naturally in various branches of mathematics, including topology, geometry, and complex analysis
• Historically, covering spaces were studied in the context of Riemann surfaces and complex analysis in the 19th century
  • Riemann surfaces are one-dimensional complex manifolds that can be viewed as branched covering spaces of the complex plane
• In the early 20th century, the concept of covering spaces was generalized to arbitrary topological spaces by mathematicians such as Poincaré and Weyl
• Covering spaces provide a powerful tool for studying the fundamental group and other topological invariants of spaces
  • They allow for the "unwrapping" of complicated spaces into simpler, more manageable ones
• Applications of covering spaces extend beyond pure mathematics to areas such as physics (quantum mechanics, string theory) and crystallography

Properties of Covering Spaces

• Covering spaces inherit many topological properties from their base spaces
  • If $X$ is connected, path-connected, or locally path-connected, then so is any covering space $\tilde{X}$
  • If $X$ is a topological manifold, then any covering space $\tilde{X}$ is also a manifold of the same dimension
• Covering maps are local homeomorphisms, meaning they preserve local topological structure
• The number of sheets (preimages) of a covering map is constant on each connected component of the base space
  • This number is called the degree or fold number of the covering
• Covering spaces of a space $X$ correspond bijectively to conjugacy classes of subgroups of the fundamental group $\pi_1(X)$
  • The subgroup consists of loops in $X$ that lift to loops in the covering space
• Homotopy lifting property: homotopies of paths or maps in the base space can be uniquely lifted to the covering space given a starting lift
• Covering spaces are classified by their deck transformation groups, which act freely and properly discontinuously on the covering space

Construction and Examples

• Covering spaces can be constructed using various methods, such as quotients of group actions, fiber products, or by specifying local homeomorphisms
• The circle $S^1$ is a covering space of itself with covering map $z \mapsto z^n$ for any integer $n$
  • The degree of this covering is $|n|$, and the deck transformation group is cyclic of order $|n|$
• The real line $\mathbb{R}$ is the universal covering space of the circle $S^1$ with covering map $t \mapsto e^{2\pi i t}$
  • The deck transformation group is infinite cyclic, generated by the translation $t \mapsto t+1$
• The torus $T^2$ has a universal covering space given by the plane $\mathbb{R}^2$ with covering map $(x,y) \mapsto (e^{2\pi i x}, e^{2\pi i y})$
  • The deck transformation group is $\mathbb{Z}^2$, generated by translations in the $x$ and $y$ directions
• Finite-sheeted covering spaces of a graph correspond to finite-index subgroups of its fundamental group (the free group on its edges)
• Riemann surfaces are covering spaces of the complex plane or the Riemann sphere, branched over a finite set of points

Relationship to Fundamental Groups

• The fundamental group $\pi_1(X)$ of a space $X$ acts on the fiber $p^{-1}(x_0)$ of any covering space $p: \tilde{X} \to X$ by path lifting
  • This action is transitive if and only if the covering space is connected
• The stabilizer of a point in the fiber under this action is a subgroup of $\pi_1(X)$ isomorphic to the fundamental group of the covering space $\pi_1(\tilde{X})$
• Galois correspondence: there is a one-to-one correspondence between connected covering spaces of $X$ (up to isomorphism) and conjugacy classes of subgroups of $\pi_1(X)$
  • Universal covering space corresponds to the trivial subgroup
  • Regular covering spaces correspond to normal subgroups
• The deck transformation group of a covering space is isomorphic to the quotient $\pi_1(X) / p_*(\pi_1(\tilde{X}))$, where $p_*$ is the induced homomorphism on fundamental groups
• The fundamental group of a covering space can be computed using the Galois correspondence and the fundamental group of the base space

Lifting Theory

• Lifting problem: given a continuous map $f: Y \to X$ and a covering map $p: \tilde{X} \to X$, when does there exist a lift $\tilde{f}: Y \to \tilde{X}$ such that $p \circ \tilde{f} = f$?
  • Existence of lifts depends on the topology of $Y$ and the induced homomorphism $f_*: \pi_1(Y) \to \pi_1(X)$
• Unique lifting property: if a lift exists, it is unique provided that $Y$ is connected and a starting point in the fiber is specified
• Homotopy lifting property: if $f_t: Y \to X$ is a homotopy of maps and $\tilde{f}_0$ is a lift of $f_0$, then there exists a unique lifted homotopy $\tilde{f}_t: Y \to \tilde{X}$ starting at $\tilde{f}_0$
• Path lifting property: any path in the base space can be uniquely lifted to a path in the covering space given a starting point in the fiber
• Lifting criteria: a map $f: Y \to X$ lifts to a covering space $p: \tilde{X} \to X$ if and only if $f_*(\pi_1(Y))$ is contained in a conjugate of $p_*(\pi_1(\tilde{X}))$

Applications in Topology

• Covering spaces are used to compute fundamental groups and higher homotopy groups of spaces
  • Van Kampen's theorem relates the fundamental group of a space to those of its covering spaces and their deck transformation groups
• Covering spaces provide a way to study the action of the fundamental group on the higher homotopy groups (the homotopy groups of the universal covering space)
• The theory of covering spaces is central to the classification of surfaces and the study of 3-manifolds
  • Every compact surface admits a universal covering space that is either the sphere, the plane, or the hyperbolic plane
• Covering spaces play a role in the formulation and proof of the Poincaré conjecture and the geometrization theorem for 3-manifolds
• In knot theory, covering spaces of the complement of a knot or link are used to define invariants and study properties of the knot or link
  • The cyclic covering spaces correspond to the cyclic subgroups of the knot group and are related to the Alexander polynomial

Advanced Topics and Extensions

• Orbifold covering spaces: allow for branched covering maps and quotients by group actions with fixed points
  • Useful in the study of orbifolds and the geometry and topology of moduli spaces
• Equivariant covering spaces: covering spaces equipped with a group action that is compatible with the covering map and the action on the base space
  • Relevant in the study of symmetric spaces and transformation groups
• Generalized covering maps: continuous maps that satisfy a weaker lifting property, such as the unique path lifting property
  • Examples include branched covering maps and covering maps of non-locally path-connected spaces
• Étale fundamental groups: a generalization of the fundamental group using the category of finite étale coverings of a scheme or a stack
  • Important in algebraic geometry and number theory, related to the absolute Galois group of a field
• Homotopy theory of covering spaces: the study of covering spaces and their maps up to homotopy equivalence
  • Leads to the notion of $\infty$-groupoids and higher topos theory, which provide a general framework for homotopy theory and algebraic topology