# Tuesday, January 12

## Background

From Phil's email:

Personally, I found the following online references particularly useful: 

- Dietmar Salamon: Spin Geometry and Seiberg-Witten Invariants [@Dietmar99]

- Richard Mandelbaum: Four-dimensional Topology: An Introduction [@Mandelbaum1980]

  - This book has a nice introduction to surgery aspects of four-manifolds, but as a warning: It was published right before Freedman's famous theorem. 
  For instance, the existence of an exotic R^4 was not known. 
  This actually makes it quite useful, as a summary of what was known before, and provides the historical context in which Freedman's theorem was proven.

- Danny Calegari: Notes on 4-Manifolds [@Calegari]

- Yuli Rudyak: Piecewise Linear Structures on Topological Manifolds [@Rudyak]

- Akhil Mathew: The Dirac Operator [@Matthew]

- Tom Weston: An Introduction to Cobordism Theory [@Weston]

  A wide variety of lecture notes on the Atiyah-Singer index theorem, which are available online.

## Introduction

:::{.definition title="Topological Manifold"}
Recall that a **topological manifold** (or $C^0$ manifold) $X$ is a Hausdorff topological space *locally homeomorphic* to $\RR^n$ with a countable topological base, so we have charts $\phi_u: U\to \RR^n$ which are homeomorphisms from open sets covering $X$.

:::

:::{.example title="The circle"}
$S^1$ is covered by two charts homeomorphic to intervals:

\begin{tikzpicture}
\node (node_one) at (0,0) { \fontsize{45pt}{1em} \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Spring/FourManifolds/sections/figures/}{2021-01-16_21-54.pdf_tex} };
\end{tikzpicture}

:::

:::{.remark}
Maps that are merely continuous are poorly behaved, so we may want to impose extra structure.
This can be done by imposing restrictions on the transition functions, defined as 
\[
t_{uv} \da \varphi_V \to \varphi_U ^{-1} : \varphi_U(U \intersect V) \to \varphi_V(U \intersect V)
.\]
:::

:::{.definition title="Restricted Structures on Manifolds"}
\envlist

- We say $X$ is a **PL manifold** if and only if $t_{UV}$ are piecewise-linear.
  Note that an invertible PL map has a PL inverse.

- We say $X$ is a **$C^k$ manifold** if they are $k$ times continuously differentiable, and **smooth** if infinitely differentiable.

- We say $X$ is **real-analytic** if they are locally given by convergent power series.

- We say $X$ is **complex-analytic** if under the identification $\RR^n \cong \CC^{n/2}$ if they are holomorphic, i.e. the differential of $t_{UV}$ is complex linear.

- We say $X$ is a **projective variety** if it is the vanishing locus of homogeneous polynomials on $\CP^N$.

:::

:::{.remark}
Is this a strictly increasing hierarchy? 
It's not clear e.g. that every $C^k$ manifold is PL.
:::

:::{.question}
Consider $\RR^n$ as a topological manifold: are any two smooth structures on $\RR^n$ diffeomorphic?
:::


:::{.remark}
Fix a copy of $\RR$ and form a single chart $\RR \mapsvia{\id} \RR$.
There is only a single transition function, the identity, which is smooth.
But consider 
\[
X &\to \RR \\
t &\mapsto t^3
.\]
This is also a smooth structure on $X$, since the transition function is the identity.
This yields a different smooth structure, since these two charts don't like in the same maximal atlas.
Otherwise there would be a transition function of the form $t_{VU}: t\mapsto t^{1/3}$, which is not smooth at zero.
However, the map 
\[
X &\to X \\
t &\mapsto t^3
.\]
defines a diffeomorphism between the two smooth structures.

:::


:::{.claim}
$\RR$ admits a unique smooth structure.
:::


:::{.proof title="sketch"}
Let $\tilde \RR$ be some exotic $\RR$, i.e. a smooth manifold homeomorphic to $\RR$.
Cover this by coordinate charts to the standard $\RR$:

\begin{tikzpicture}
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Spring/FourManifolds/sections/figures}{2021-01-16_22-31.pdf_tex} };
\end{tikzpicture}


:::{.fact}
There exists a cover which is *locally finite* and supports a *partition of unity*: a collection of smooth functions $f_i: U_i \to \RR$ with $f_i \geq 0$ and $\supp f \subseteq U_i$ such that $\sum f_i = 1$ (*i.e., bump functions*).
It is also a purely topological fact that $\tilde \RR$ is orientable.
:::

So we have bump functions:

\begin{tikzpicture}
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2021/Spring/FourManifolds/sections/figures}{2021-01-16_22-37.pdf_tex} };
\end{tikzpicture}

Take a smooth vector field $V_i$ on $U_i$ everywhere aligning with the orientation.
Then $\sum f_i V_i$ is a smooth nowhere vector field on $X$ that is nowhere zero in the direction of the orientation.
Taking the associated flow 
\[
\RR &\to \tilde \RR \\
t &\mapsto \varphi(t)
.\]
such that \( \varphi'(t) = V(\varphi(t)) \).
Then \( \varphi \) is a smooth map that defines a diffeomorphism.
This follows from the fact that the vector field is everywhere positive.
:::


:::{.slogan}
To understand smooth structures on $X$, we should try to solve differential equations on $X$.
:::


:::{.remark}
Note that here we used the existence of a global frame, i.e.  a trivialization of the tangent bundle, so this doesn't quite work for e.g. $S^2$.
:::


:::{.question}
What is the difference between all of the above structures?
Are there obstructions to admitting any particular one?
:::


:::{.answer}
\envlist

1. (Munkres) Every $C^1$ structure gives a unique $C^k$ and $C^ \infty$ structure.[^note_note_c0]

2. (Grauert) Every $C^ \infty$ structure gives a unique real-analytic structure.

3. Every PL manifold admits a smooth structure in $\dim X \leq 7$, and it's unique in $\dim X\leq 6$, and above these dimensions there exists PL manifolds with no smooth structure.

4. (Kirby-Siebenmann) Let $X$ be a topological manifold of $\dim X\geq 5$, then there exists a cohomology class $\ks(X) \in H^4(X; \ZZ/2\ZZ)$ which is 0 if and only if $X$ admits a PL structure.
  Moreover, if $\ks(X) = 0$, then (up to concordance) the set of PL structures is given by $H^3(X; \ZZ/2\ZZ)$.

5. (Moise) Every topological manifold in $\dim X\leq 3$ admits a unique smooth structure.

6. (Smale et al.): In $\dim X\geq 5$, the number of smooth structures on a topological manifold $X$ is finite.
  In particular, $\RR^n$ for $n \neq 4$ has a unique smooth structure.
  So dimension 4 is interesting!

7. (Taubes) $\RR^4$ admits uncountably many non-diffeomorphic smooth structures.

8. A compact oriented smooth surface \( \Sigma \), the space of complex-analytic structures is a complex orbifold [^orbifold]
of dimension $3g-2$ where $g$ is the genus of \( \Sigma \), up to biholomorphism (i.e. *moduli*).

[^note_note_c0]: 
Note that this doesn't start at $C^0$, so topological manifolds are genuinely different! 
There exist topological manifolds with no smooth structure.


[^orbifold]: 
Locally admits a chart to $\CC^n/ \Gamma$ for \( \Gamma \) a finite group.

:::

:::{.remark}
Kervaire-Milnor: $S^7$ admits 28 smooth structures, which form a group.
:::