Classification of Compact One-Manifolds

Theorem
Every compact one-dimensional manifold is diffeomorphic to either a circle or a closed interval.

Corollary
Any compact one-manifold has an even number of points in its boundary.

Lemma 1
Let $$f$$ be a function on $$[a,b]$$ that is smooth and has a positive derivative everywhere except one interior point, $$c$$. Then there exists a globally smooth function $$g$$ that agrees with $$f$$ near $$a$$ and $$b$$ and has a positive derivative everywhere.

Proof of Lemma 1
Let $$r$$ be a smooth nonnegative function that vanishes outside a compact subset of $$(a,b)$$, which equals $$1$$ near $$c$$, and which satisfies $$\int_{a}^{b} r = 1$$.

Define $$g(x) = f(a) + \int_{a}^{x} (k r(s)+f'(s)(1-r(s)))ds$$ where the constant $$k=f(b)-f(a)-\int_{a}^{b} f'(s)(1-r(s))ds$$.

Now, let $$f$$ be a Morse function on a one-manifold $$X$$, and let $$S$$ be the union of the critical points of $$f$$ and $$\partial X$$. Since $$S$$ is finite, $$X-S$$ consists of a finite number of one-manifolds, $$L_1,L_2,\cdots,L_n$$.

Lemma 2
$$f$$ maps each $$L_i$$ diffeomorphically onto an open interval in $$\R$$

Proof of Lemma 2
Let $$L$$ be any of the $$L_i$$. Because $$f$$ is a local diffeomorphism and $$L$$ is connected, $$f(L)$$ is open and connected in $$\R$$. We also have $$f(L) \in f(X)$$, the latter of which is compact, so there are numbers $$c$$ and $$d$$ such that $$f(L) = (c,d)$$.

It suffices to show $$f$$ is one to one on $$L$$, because then $$f^{-1}:(c,d) \to L$$ is defined and locally smooth. Let $$p$$ be any point of $$L$$ and set $$q=f(p)$$.

It suffices to show that every other point $$z \in L$$ can be joined to $$p$$ by a curve $$\gamma : [q,y] \to L$$, such that $$f \circ \gamma$$ is the identity and $$\gamma (y) = z$$.

Since $$f(z) = y \neq q = f(p)$$, this result shows $$f$$ is one to one. So let $$Q$$ be the set of points $$x$$ that can be so joined. Since $$f$$ is a local diffeomorphism, $$Q$$ is both open and closed and hence $$Q=L$$.

Lemma 3
Let $$L$$ be a subset of $$X$$ diffeomorphic to an open interval in $$\R$$, where $$\dim X = 1$$. Then the closure $$Cl(L)$$ contains at most two points not in $$L$$.

Proof of Lemma 3
Let $$g$$ be a diffeomorphism $$g:(a,b) \to L$$ and let $$p \in Cl(L)-L$$. Let $$J$$ be a closed subset of $$X$$ diffeomorphic to $$[0,1]$$ such that $$1$$ corresponds to $$p$$ and $$0$$ corresponds to some $$g(t)$$ in $$L$$.

Consider the set $$\left\{{ s \in (a,t) | g(s) \in J }\right\}$$. This set is both open and closed in $$(a,b)$$, hence $$J$$ contains either $$g((a,t))$$ or $$g((t,b))$$.

Proof of Corollary
Follows trivially