Classification of Compact One-Manifolds

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Theorem

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

Corollary

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

Proof

Lemma 1

Let $f$ be a function on $\closedint a b$ 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.

$\Box$

Let $f$ be a Morse function on a one-manifold $X$.

Let $S$ be the union of the critical points of $f$ and $\partial X$.

As $S$ is finite, $X \setminus S$ 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 $\R$.

$\Box$

Lemma 3

Let $L$ be a subset of $X$ diffeomorphic to an open interval in $\R$, where $\dim X = 1$.

Then the closure $\map \cl L$ contains at most two points not in $L$.

$\Box$

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