DoubleCover

In mathematics, particularly in algebraic topology and geometry, a double cover (or two-fold cover) is a special case of a covering map. A covering map is a continuous function $p$ from a topological space $C$ to a topological space $X$ such that each point in $X$ has a neighborhood evenly covered by $p$. In the case of a double cover, each point in $X$ has exactly two preimages in $C$.

More formally, a double cover of a topological space $X$ is a topological space $C$ and a continuous surjective map

$p : C \to X$

such that for every point $x$ in $X$, there is an open neighborhood $U$ of $x$ in $X$ for which there exists a homeomorphism $\phi : p^{-1}(U) \to U \times {1, 2}$ such that for every $y$ in $p^{-1}(U)$, $p(y)$ equals the first component of $\phi(y)$.

The set ${1, 2}$ is just a two-point space, and $U \times {1, 2}$ looks locally like two copies of $U$. Therefore, in a double cover, locally around every point $x$ in $X$, the space $C$ looks like two copies of $X$ glued together along $X$.

Here are some examples:

1. The complex numbers $\mathbb{C}$ are a double cover of the real line $\mathbb{R}$, where the map is given by $p(z) = \text{Re}(z)$. Each real number $x$ is the real part of two complex numbers: $x + i0$ and $x - i0$.

2. The unit circle $S^1$ in the complex plane is a double cover of itself, where the map is given by $p(z) = z^2$. Each complex number on the circle, except for -1, is the square of exactly two numbers on the circle.

3. The sphere $S^2$ in three-dimensional space is a double cover of the projective plane $\mathbb{RP}^2$, where the map sends antipodal points to the same point.

Double covers play a central role in various areas of mathematics, including Galois theory, Riemann surface theory, and the study of Lie groups and algebraic groups. They are especially important in the study of symmetries and geometric transformations, where they often correspond to "square roots" of transformations.

Particle Spin Groups And the Mediation of Gravity

Double covers and spin groups are united around the concept of particle spin in quantum mechanics.

Spin Groups

To understand the connection, let's start with spin groups which are extensions of the special orthogonal groups SO(n) describing rotations in n-dimensional Euclidean space.

Spin groups, denoted Spin(n), are double covers of the special orthogonal group which provides the additional structure needed to describe quantum mechanical spin correctly because the representation theory of SO(n) does not model quantum mechanical spin which has things like spin-1/2 particles which have to be rotated by twice (720 degrees) to return to their original state.

Particle Spin

Now let's talk about particle spin. In quantum mechanics, "spin" is a form of angular momentum carried by elementary particles. Spin is a deeply quantum mechanical property and has no perfect classical analogue. It's quantized, meaning it can only take on certain discrete values.

The "spin" of a particle determines its behavior under rotation and, more broadly, its behavior under the symmetries of space and time described by the theory of relativity. These symmetries are described mathematically by the [Poincaré group](Poincaré group), and the different possible spins correspond to the different possible representations of this group. In other words, the different possible spins are like the different possible "shapes" that something can have while still respecting the symmetries of space and time.

The Mediator of Gravity Manifests as The Spin-2 Graviton

By 'manifest', don't freak out and think I'm going all woo-meister on you; I simply mean, 'produces observables'. The graviton, which is a hyppthesized particle that has not yet been claimed to be observed but is predicted by the theory of general relativity, is purported to be a spin-2 particle. This is because it its role as the mediator of the force of gravity is a direct consequence of the mathematical form of the Einstein field equations which describe how matter and energy influence the curvature of spacetime.

The spin-2 nature of the graviton should manifest in its interactions with other particles via its effects on the polarization of gravitational waves. Let us recall that the Einstein field equations do not restrict the global topology or geometry of of the Einstein field equations which are only local conditions that solutions must satisfy to exist and turn our attention to proposed Globally inhomogeneous 'spliced' universe model where a specific meromorphic function is proposed whose real roots correspond to different universes with different connectivity where the splicing is precisely defined via sympathetic manifold...