Smooth Structures

A topological manifold (7A.3 Manifolds and Surfaces) looks locally like $\mathbb{R}^n$, and that is enough to talk about continuity. The Ricci flow needs more: it differentiates a metric twice. To differentiate on a manifold, one must be able to say when a function is smooth, and the obvious definition, "smooth when written in a chart", only makes sense if the answer doesn't depend on the chart. So a smooth manifold is a topological manifold together with a choice of charts whose changes of coordinates are smooth. This chapter makes that precise, gives the standard examples, and explains a fact that matters for the whole guide: in most dimensions a topological manifold can carry several genuinely different smooth structures, but in dimension three it carries exactly one.

The anchor is an atlas in the everyday sense. No single flat map covers the Earth without tearing it, so mapmakers use many overlapping maps with rules for converting between them. That is a smooth atlas, and the rules are transition maps.

By the end of this chapter you will be able to:

• define charts, smooth atlases, transition maps and smooth structures;
• verify that $S^n$, $\mathbb{R}P^n$, tori, products and open subsets are smooth manifolds;
• define smooth maps and diffeomorphisms, and compute in charts;
• explain what it means for a manifold to have several smooth structures, with the examples of exotic spheres;
• state Moise's theorem and explain why it lets a smooth method prove a topological theorem.

1.1The UTM atlas

1.2Charts and atlases

Let $M$ be a topological $n$-manifold (7A.3 Manifolds and Surfaces).

A smooth atlas is contained in a unique maximal one, consisting of all charts compatible with every chart of the atlas (Exercise 1.7). A smooth structure on $M$ is a maximal smooth atlas, and a smooth manifold is a topological manifold with a smooth structure. In practice one specifies a smooth structure by giving any smooth atlas, often with very few charts, and then uses whichever compatible chart is convenient. Indices on coordinates are written as superscripts, $x^i$, in preparation for the index notation of 8A.7 Tensors and Index Notation.

The point of the definition is that smoothness becomes chart-independent. A function $f : M \to \mathbb{R}$ is smooth if $f\circ\varphi^{-1}$ is smooth for every chart in the smooth structure. If it is smooth in one chart around a point, it is smooth in every compatible chart around that point, because $f\circ\psi^{-1} = (f\circ\varphi^{-1})\circ(\varphi\circ\psi^{-1})$ is a composite of smooth maps (2B.8 Calculus in Several Variables).

Diffeomorphism is the notion of sameness for smooth manifolds, as homeomorphism is for topological ones. Every diffeomorphism is a homeomorphism; the converse fails ($x \mapsto x^3$ on $\mathbb{R}$ is a smooth homeomorphism whose inverse is not smooth at $0$).

1.3Examples

Open subsets and graphs. $\mathbb{R}^n$ with the single chart $\operatorname{id}$. Any open subset of a smooth manifold, with the restricted charts; for example the general linear group $GL(n, \mathbb{R})$, open in the space $\mathbb{R}^{n^2}$ of matrices since it is $\{\det \neq 0\}$. The graph of a smooth function $\mathbb{R}^n \to \mathbb{R}^k$, with one chart given by projection.

Spheres. $S^n \subset \mathbb{R}^{n+1}$ has the two stereographic charts (7A.2 Compactness and Compactification): $\sigma_N$, projecting from the north pole, defined on $S^n\setminus\{N\}$, and $\sigma_S$, projecting from the south pole, defined on $S^n\setminus\{S\}$. Their transition map on the overlap $\mathbb{R}^n\setminus\{0\}$ is the inversion

which is smooth away from $0$ and is its own inverse (Exercise 1.5, Figure 1.2). So the two charts form a smooth atlas. For $S^2 \subset \mathbb{R}^3$, identifying $\mathbb{R}^2$ with $\mathbb{C}$, the transition is $z \mapsto \frac{z}{|z|^2} = \frac{1}{\bar z}$, which is the Riemann sphere of 5A.5 Uniformization and the Two-Dimensional Ricci Flow up to complex conjugation.

Projective spaces. $\mathbb{R}P^n$ is the set of lines through $0$ in $\mathbb{R}^{n+1}$, a point written in homogeneous coordinates $[x^0 : \dots : x^n]$, defined up to a non-zero scalar multiple. On $U_i = \{x^i \neq 0\}$ the chart

is well defined, and the transition maps are rational functions with non-vanishing denominators, hence smooth (Exercise 1.6). This makes $\mathbb{R}P^n$ a smooth manifold of dimension $n$, and the quotient map $S^n \to \mathbb{R}P^n$ smooth.

Tori and products. If $M$ and $N$ are smooth manifolds, products of charts form a smooth atlas on $M\times N$. So $T^n = S^1\times\dots\times S^1$ and $S^2\times S^1$ are smooth manifolds. Quotients $\mathbb{R}^n/\mathbb{Z}^n$, and more generally quotients of smooth manifolds by smooth covering space actions, inherit smooth structures from the covering (7A.6 Covering Spaces, 8A.5 Lie Groups and Group Actions).

1.4How many smooth structures?

On a given topological manifold there can be several maximal atlases. On $\mathbb{R}$, the single chart $\psi(x) = x^3$ defines a smooth structure different from the standard one: the transition $x \mapsto x^{1/3}$ is not smooth at $0$. But the two structures are diffeomorphic, by the map $x \mapsto x^{1/3}$ from one to the other. The question that matters is whether a topological manifold carries smooth structures that are not diffeomorphic to each other.

In many cases it does. John Milnor showed in 1956 that there are smooth manifolds homeomorphic to $S^7$ but not diffeomorphic to it, exotic spheres; Michel Kervaire and Milnor later showed that $S^7$ carries exactly $28$ smooth structures up to orientation-preserving diffeomorphism. In dimension $4$ the situation is wilder: $\mathbb{R}^4$ itself carries exotic smooth structures, as follows from work of Michael Freedman and Simon Donaldson in the early 1980s (Clifford Taubes later showed there are uncountably many), while every $\mathbb{R}^n$ with $n \neq 4$ has only one. And some topological manifolds of dimension $4$ carry no smooth structure at all.

In low dimensions there is no such freedom:

For $n = 1$ and $2$ this follows from the classifications (7A.3 Manifolds and Surfaces). For $n = 3$ it rests on Edwin Moise's 1952 theorem that every three-manifold has a triangulation, unique up to subdivision, combined with results of James Munkres and J. H. C. Whitehead relating triangulations to smooth structures. As 7A.9 The Poincaré Conjecture, Precisely explained, this is why the Ricci flow, which needs a smooth structure and a metric, can prove a topological theorem.

1.5History

Riemann's 1854 lecture spoke of $n$-fold extended manifolds described by coordinates; the modern definition of a manifold by charts and transition maps was given by Hermann Weyl in 1913 for Riemann surfaces and by Oswald Veblen and J. H. C. Whitehead in 1931–32 in general, and Hassler Whitney's papers of 1936 established smooth manifolds as the standard setting. Milnor's exotic 7-spheres appeared in 1956, and the count of $28$ in Kervaire and Milnor's 1963 paper on groups of homotopy spheres. Moise's triangulation theorem dates from 1952. The UTM system was developed by the US Army in the 1940s.

1.6Exercises

Solution

If $\varphi : S^n \to \varphi(S^n)$ were a homeomorphism onto an open subset of $\mathbb{R}^n$, then $\varphi(S^n)$ would be compact, hence closed and bounded, and also open and non-empty; a non-empty subset of $\mathbb{R}^n$ that is both open and closed is all of $\mathbb{R}^n$ (2B.4 Connectedness), which is not bounded.

Solution

If $y = \sigma_N(x)$ then $\sigma_S(x) = \frac{(x^1, \dots, x^n)}{1 + x^{n+1}} = y\cdot\frac{1 - x^{n+1}}{1 + x^{n+1}} = \frac{y}{|y|^2}$.

Solution

$\varphi_0[x^0 : x^1 : x^2] = (x^1/x^0, x^2/x^0) = (u, v)$, so $\varphi_0^{-1}(u, v) = [1 : u : v]$, and $U_0\cap U_1$ corresponds to $u \neq 0$. Then $\varphi_1[1 : u : v] = (1/u, v/u)$, smooth on $u \neq 0$.

Solution

(a) The coordinate representation of $\operatorname{id} : \mathbb{R} \to \mathbb{R}_3$ is $\psi\circ\operatorname{id}(x) = x^3$, smooth. That of its inverse $\operatorname{id} : \mathbb{R}_3 \to \mathbb{R}$ is $\operatorname{id}\circ\psi^{-1}(y) = y^{1/3}$, not smooth at $0$. (b) The representation of $G$ is $\psi\circ G(x) = (x^{1/3})^3 = x$, and that of $G^{-1}(u) = u^3$ is $G^{-1}\circ\psi^{-1}(y) = (y^{1/3})^3 = y$: both smooth. (c) $f$ is smooth on $\mathbb{R}_3$ iff $f\circ\psi^{-1}(y) = f(y^{1/3})$ is smooth, i.e. iff $f(x) = g(x^3)$ for a smooth $g$.

Solution

(a) In complex notation $w = 1/\bar y$, so $|dw| = |d\bar y|/|y|^2 = |dy|/|y|^2$. (b) $\frac{4|dw|^2}{(1 + |w|^2)^2} = \frac{4|dy|^2/|y|^4}{(1 + |y|^{-2})^2} = \frac{4|dy|^2}{(|y|^2 + 1)^2}$.