Building New Manifolds

Constructing New Manifolds Introduction Rather than studying manifolds in isolation, we often construct new manifolds from ones we already understand. This section explores the main techniques for building new manifolds: gluing spaces together via group actions, joining manifolds along their boundaries, taking products, and deciding whether to view manifolds as embedded in Euclidean space or as abstract spaces. These constructions are fundamental to understanding the landscape of manifold types. Gluing via Group Actions (Quotient Constructions) When we have a manifold and a group acting on it, we can identify points that are related by the group action. This identification creates a quotient space, which is often a manifold itself. What does this mean? Suppose a group $G$ acts on a manifold $M$. Two points $p, q \in M$ are in the same orbit if $q = g \cdot p$ for some $g \in G$. The quotient space $M/G$ treats each orbit as a single point. The natural projection map $\pi: M \to M/G$ assigns each point to its orbit. When is the quotient a manifold? The quotient space $M/G$ inherits a quotient topology from $M$. For the quotient to be a manifold, the group action must be free (no non-identity element fixes any point) and properly discontinuous (orbits are discrete). Under these conditions, the quotient is a smooth manifold, and the projection map is a submersion. Why should you care? Quotient constructions let us build new manifolds systematically and understand symmetries. They answer questions like: "If I identify certain points of a known manifold, what do I get?" Real Projective Space as a Key Example Real projective space $\mathbb{RP}^n$ is one of the most important examples and is defined via a quotient construction. Start with the sphere $S^n \subset \mathbb{R}^{n+1}$. Define a group action where the group is $\mathbb{Z}2 = \{1, -1\}$ under multiplication, acting on $S^n$ by: $$g \cdot p = g \cdot p \text{ (scalar multiplication)}$$ In other words, the non-identity element $-1$ sends each point to its antipodal point (the opposite side of the sphere). $$\mathbb{RP}^n = S^n / \mathbb{Z}2$$ Real projective space identifies antipodal points on the sphere. What does this look like? For $\mathbb{RP}^1$: starting with the circle $S^1$, we identify each point with its antipodal point. Topologically, this gives a circle again, but the quotient map $S^1 \to \mathbb{RP}^1$ is a 2-to-1 covering map. For $\mathbb{RP}^2$ (the projective plane), we start with $S^2$ and identify antipodal pairs. This is a famous non-orientable surface and cannot be smoothly embedded in $\mathbb{R}^3$. Key point: The action of $\mathbb{Z}2$ on $S^n$ is free (no point is fixed by $-1$) and properly discontinuous, so $\mathbb{RP}^n$ is a smooth $n$-dimensional manifold. Gluing Along Boundaries Another fundamental construction takes two manifolds with boundary and joins them by identifying their boundaries via a homeomorphism (or diffeomorphism for smooth manifolds). The setup: Let $M$ and $N$ be two smooth manifolds with boundary, where $\partial M$ and $\partial N$ are their boundaries. Suppose $f: \partial M \to \partial N$ is a diffeomorphism. We can create a new manifold by "gluing" $M$ and $N$ along their boundaries using $f$. How does this work? Form the disjoint union $M \sqcup N$, then identify points $p \in \partial M$ with $f(p) \in \partial N$. More formally, we take the quotient space: $$M \cupf N = (M \sqcup N) / \sim$$ where $p \sim f(p)$ for all $p \in \partial M$. Why is the result a manifold? The key is that the gluing is done smoothly along the boundary. The resulting quotient has a natural smooth structure inherited from $M$ and $N$. Interior points of $M$ and $N$ have neighborhoods within their original spaces, and points in the glued boundary have neighborhoods that overlap both pieces smoothly. Important consequence: The new manifold $M \cupf N$ has no boundary—we've sewn up the boundaries by identifying them. This construction produces a closed manifold (compact and without boundary). Intuition: Think of gluing two surfaces along their edges. If you glue a disc to another disc along their circular boundaries, you get a sphere. Cartesian Products The Cartesian product is the simplest way to build higher-dimensional manifolds from lower-dimensional ones. Definition: If $M$ is an $m$-dimensional manifold and $N$ is an $n$-dimensional manifold, their product $M \times N$ is the set of all ordered pairs $(p, q)$ with $p \in M$ and $q \in N$. Product topology: The product topology on $M \times N$ is generated by open sets of the form $U \times V$ where $U$ is open in $M$ and $V$ is open in $N$. Product manifold structure: The key observation is that $M \times N$ is itself an $(m+n)$-dimensional manifold. To see this, construct an atlas as follows: Take a chart $(U\alpha, \phi\alpha)$ on $M$ where $\phi\alpha: U\alpha \to \mathbb{R}^m$ Take a chart $(V\beta, \psi\beta)$ on $N$ where $\psi\beta: V\beta \to \mathbb{R}^n$ Define a chart on the product: $U\alpha \times V\beta \to \mathbb{R}^{m+n}$ by: $$(\phi\alpha \times \psi\beta)(p, q) = (\phi\alpha(p), \psi\beta(q))$$ The transition functions for the product are simply products of the transition functions from $M$ and $N$, so they are smooth. This collection of charts covers $M \times N$ and forms a smooth atlas. Dimension formula: $\dim(M \times N) = \dim(M) + \dim(N) = m + n$ Examples: $S^1 \times S^1$ is a 2-torus (the doughnut surface) $S^1 \times \mathbb{R}$ is an infinite cylinder $S^m \times S^n$ is a product of spheres Intrinsic vs. Extrinsic Perspectives A subtle but important distinction shapes how we think about manifolds: whether we view them as embedded in Euclidean space or as abstract spaces. The extrinsic view treats a manifold as a subset of some ambient Euclidean space $\mathbb{R}^N$. For example: The circle is viewed as $\{(x,y) \in \mathbb{R}^2 : x^2 + y^2 = 1\}$ The 2-sphere is $\{(x,y,z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 = 1\}$ From this viewpoint, the geometry of the manifold is "visible" from the embedding. Curvature, distance, and angles can be measured using the Euclidean geometry of the surrounding space. The intrinsic view treats the manifold as a space on its own, defined entirely by its charts and transition maps. We forget about any embedding and ask: "What is the geometry of this space as experienced by an inhabitant living on it?" For example: $\mathbb{RP}^2$ is most naturally described intrinsically (as $S^2/\mathbb{Z}2$) rather than as an embedded surface The description involves only the manifold itself: its topology and the smooth structure given by charts Why does this distinction matter? Many manifolds cannot be smoothly embedded in any Euclidean space, or require very high dimensions to do so. The intrinsic approach avoids this problem by working directly with the manifold structure. It also reveals which properties are intrinsic (properties that don't depend on the embedding) versus extrinsic (properties that depend on how we embed the manifold). Key insight: A manifold is the same object whether we view it extrinsically (embedded in $\mathbb{R}^N$) or intrinsically (via charts). The intrinsic view is more general and doesn't require an embedding to exist. Most modern differential geometry emphasizes the intrinsic approach because it's more natural and powerful.