General Geometry/Manifolds

Definition (manifold):

Let $(C,G)$ be a site, and let $D$ be a subcategory of $C$. A manifold of type $D$ consists of the aforementioned site $(C,G)$ together with a class $(\varphi _{\alpha }:U_{\alpha }\to X_{\alpha })_{\alpha }$ of isomorphisms of $C$, where the $U_{\alpha }$'s are open in $G$ and the $X_{\alpha }$ are objects of $D$, such that

1. for all $U\in \operatorname {Obj} (C)$ there is a covering $\iota _{i}:U_{i}\to U$ of $U$ such that for each $i$, there exists an $\alpha$ with $U_{i}=U_{\alpha }$ and
2. whenever $U,V,W\in \operatorname {Obj} C$ st. $\iota _{1}:U\to W$ and $\iota _{2}:V\to W$ consitute a covering and $\varphi :U\to X$ and $\psi :V\to Y$ belong to the class $(\varphi _{\alpha }:U_{\alpha }\to X_{\alpha })_{\alpha }$, the maps $\varphi \upharpoonright (U\times _{W}V)$ and $\psi \upharpoonright (U\times _{W}V)$ guaranteed by the universal property of the pullback are isomorphisms onto their respective images and $(\psi \upharpoonright (U\times _{W}V))\circ (\varphi \upharpoonright (U\times _{W}V))^{-1}$ is a morphism of $D$.

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Proposition (fundamental induction lemma):

Let $P(M,A)$ be a logical statement, whose arguments are a topological manifold $M$ and a closed subset $A\subseteq M$. Suppose that the following are true:

1. Whenever $B\subseteq V_{\alpha }$ is compact and convex, where $\phi _{\alpha }:U_{\alpha }\to V_{\alpha }$ is a coördinate chart on $M$, then $P(M,\phi _{\alpha }^{-1}(B))$ is true
2. Whenever $P(M,A),P(M,B)$ and $P(M,A\cap B)$ are true, then $P(M,A\cup B)$ is true
3. Whenever $A_{1}\supseteq A_{2}\supseteq A_{3}\supseteq \cdots$ is a descending chain of compact subsets of $M$, $P(M,A_{j})$ is true for all $j\in \mathbb {N}$, then $P\left(M,\bigcap _{j\in \mathbb {N} }A_{j}\right)$ is true
4. Whenever $P(M,A\cap {\overline {W}})$ is true for all relatively compact, open $W\subseteq M$, then $P(M,A)$ is true

Then for all closed $A\subseteq M$, $P(M,A)$ is true.

Proof: First we prove by induction on $k$ that for all sets of the type $A_{j}=\phi _{\alpha }^{-1}(B_{j})$ for certain compact convex $B_{j}$ ($j\in \{1,\ldots ,k\}$), the statement $P(M,A_{1}\cup \cdots \cup A_{k})$ is true. We proceed by induction on $k$. For $k=1$, the statement is implied by the first assumption. Suppose now that $P(M,A_{1}\cup \cdots \cup A_{k-1})$ is true. Note that also $P(M,A_{k})$ is true by the first assumption. Also

$(A_{1}\cup \cdots \cup A_{k-1})\cap A_{k}=(A_{1}\cap A_{k})\cup \cdots \cup (A_{k-1}\cap A_{k})$,

and $A_{j}\cap A_{k}=\phi _{\alpha }^{-1}(B_{j}\cap B_{k})$, where $B_{j}\cap B_{k}$ is compact and convex as the intersection of two compact and convex sets. Thus, since the sets $A_{1}\cap A_{k},\ldots ,A_{1}\cap A_{k-1}$ are only $k-1$ many, by induction on $k$, we may also conclude that $P(M,(A_{1}\cup \cdots \cup A_{k-1})\cap A_{k})$ holds. By 2., we conclude $P(M,A_{1}\cup \cdots \cup A_{k})$.

Now, we prove that $P(M,A)$ is true whenever $A$ is a compact subset of a set of the type $\phi _{\alpha }^{-1}(B)$, where $B$ is compact and convex. Indeed, for each $m\in \mathbb {N}$, cover $\phi _{\alpha }(A)$ by all the cubes of sidelength $2/2^{m}$ centered at the points of $(1/2^{m})\mathbb {Z} ^{n}$ that intersect it. Then set

$A_{m}:=\bigcup _{j=1}^{k_{m}}B_{j,m}$, so that $A=\bigcap _{m\in \mathbb {N} }A_{m}$.

By the second assumption, $P(M,A_{m})$ holds for each $m\in \mathbb {N}$, and hence by the third assumption $P(M,A)$ holds.

Now we claim by induction on $k$ that whenever $A_{1},\ldots ,A_{k}$ are compact subsets of sets of the type $\phi _{\alpha }^{-1}(B)$ ($B$ compact and convex), then $P(M,A_{1}\cup \cdots \cup A_{k})$ holds. For $k=1$, this follows from what we just proved. For the induction step, suppose that $P(M,A_{1}\cup \cdots \cup A_{k-1})$ holds. Note that also $P(M,A_{k})$ holds by what we just proved. Then we have

$(A_{1}\cup \cdots \cup A_{k-1})\cap A_{k}=(A_{1}\cup A_{k})\cup \cdots \cup (A_{k-1}\cap A_{k})$,

and since $A_{j}\cap A_{k}$ is a compact subset of $A_{j}$, $P(M,A_{j}\cap A_{k})$ holds by what we just proved, and therefore, by induction, $P(M,(A_{1}\cup A_{k})\cup \cdots \cup (A_{k-1}\cap A_{k})$ holds. Hence, by 2., we get that $P(M,A_{1}\cup \cdots \cup A_{k})$ is true.

Now we are ready to prove that $P(M,A)$ is true whenever $A$ is compact. Indeed, let $A\subseteq M$ be compact. Then cover $A$ by sets $\phi _{\alpha }^{-1}(B_{\alpha })$, where $\phi _{\alpha }$ are some charts and $B_{\alpha }$ is compact and convex. By compactness of $A$, we may extract a finite subcover $\phi _{\alpha _{1}}^{-1}(B_{\alpha _{1}})=:A_{1},\ldots ,A_{n}:=\phi _{\alpha _{n}}^{-1}(B_{\alpha _{n}})$. By intersecting with $A$ and retaining compactness (as the intersection of two compact sets is compact), we may assume that $A_{1},\ldots ,A_{n}$ are contained within $A$ and in particular

$A=\bigcup _{j=1}^{n}A_{j}$.

Thus, by the previous step, $P(M,A)$ holds.

Finally, let $A$ be an arbitrary closed subset of $M$, and let $W\subseteq M$ be a relatively compact open subset. Then $P(M,A\cap {\overline {W}})$ is true since $A\cap {\overline {W}}$ is compact as a closed subset of a compact set. Hence, by the fourth assumption, $P(M,A)$ is true. $\Box$

Definition (vector bundle):

Let $R$ be a topological ring, and let $M$ be a manifold. An $n$-dimensional $R$-vector bundle over $M$ is a manifold $E$ together with a morphism of manifolds $\pi :E\to M$ such that:

1. for each $x\in M$, the set $E_{x}:=\pi ^{-1}(x)$ (which is called the fibre of $x$), is a finite-dimensional topological vector space over $R$
2. for each $x\in M$, there exists a neighbourhood $U$ of $x$, an $n\in \mathbb {N}$ and a map $\Psi :\pi ^{-1}(U)\to U\times R^{n}$ which is a fibre-wise TVS isomorphism such that the diagram

   

   commutes.