Classification of Compact One-Manifolds

= Theorem =

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

Corollary: It follows trivially from the theorem that any compact one-manifold has an even number of points in its boundary.

= Proof =


 * Lemma 1: Let f be a function on [a,b] that is smooth and have 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 have positive derivative everywhere.

Proof: 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 $$k=f(b)-f(a)-\int_{a}^{b} f'(s)(1-r(s))ds$$. QED

Let f be a Morse function on [a,b], and S the union of the critical points of f and a,b. Since S is finite, [a,b]-S consists of a finite number of one-manifolds, L1, L2, ..., Ln.


 * Lemma 2: f maps each Li diffeomorphically onto an open interval in $$\mathbb{R}$$

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