Poincare Lemma and Homotopy Invariance

The Poincare lemma states that every closed form on a contractible manifold is exact. It is the local foundation of de Rham cohomology and follows from the more general homotopy invariance of cohomology.

The Poincare Lemma

Theorem5.5Poincare lemma

On a star-shaped open subset $U \subset \mathbb{R}^n$ (or more generally, any contractible manifold), every closed $k$ -form with $k \geq 1$ is exact:

H^k_{\mathrm{dR}}(U) = 0 \quad \text{for all } k \geq 1.

Proof via the Homotopy Operator

Proof

We construct an explicit homotopy operator $K: \Omega^k(U) \to \Omega^{k-1}(U)$ satisfying $dK + Kd = \mathrm{id}$ (for $k \geq 1$ ).

Assume $U$ is star-shaped with center at the origin. Define

(K\omega)_x(v_1, \ldots, v_{k-1}) = \int_0^1 t^{k-1} \omega_{tx}(x, v_1, \ldots, v_{k-1}) \, dt

for $\omega \in \Omega^k(U)$ .

Verification: A direct computation shows that for any $k$ -form $\omega$ ,

(dK\omega + Kd\omega)(x) = \omega(x) - (\omega|_0)(x),

where $\omega|_0$ denotes evaluation at the origin. For $k \geq 1$ , the term $\omega|_0$ vanishes (a $k$ -form with $k \geq 1$ at a single point is determined by its action on tangent vectors, and the integral over $t$ evaluates to the identity). Thus $dK + Kd = \mathrm{id}$ .

If $\omega$ is closed ( $d\omega = 0$ ), then $\omega = dK\omega + Kd\omega = d(K\omega)$ , so $\omega$ is exact with primitive $K\omega$ . $\blacksquare$

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Homotopy Invariance

Theorem5.6Homotopy invariance of de Rham cohomology

If $f, g: M \to N$ are smoothly homotopic (i.e., there exists a smooth map $H: M \times [0,1] \to N$ with $H(\cdot, 0) = f$ and $H(\cdot, 1) = g$ ), then $f^* = g^*: H^k_{\mathrm{dR}}(N) \to H^k_{\mathrm{dR}}(M)$ .

Proof

Construct a chain homotopy $h: \Omega^k(N) \to \Omega^{k-1}(M)$ by

h(\omega) = \int_0^1 \iota_{\partial/\partial t}(H^*\omega) \, dt,

where $\iota_{\partial/\partial t}$ is interior multiplication by the $t$ -direction. Then $dh + hd = g^* - f^*$ . For closed $\omega$ : $g^*\omega - f^*\omega = dh\omega$ , so $[g^*\omega] = [f^*\omega]$ in cohomology. $\blacksquare$

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Consequences

ExampleCohomology of $\mathbb{R}^n$ and star-shaped domains

Since any star-shaped domain $U$ is homotopy equivalent to a point (via the homotopy $H(x,t) = (1-t)x$ ), homotopy invariance gives $H^k(U) = H^k(\{0\}) = 0$ for $k \geq 1$ , recovering the Poincare lemma.

RemarkExplicit primitives

The Poincare lemma is constructive: the homotopy operator $K$ gives an explicit formula for a primitive. For example, on $\mathbb{R}^3$ , if $\omega = P\,dy \wedge dz + Q\,dz \wedge dx + R\,dx \wedge dy$ is a closed 2-form (i.e., $\operatorname{div}(P,Q,R) = 0$ ), then $K\omega$ is a 1-form whose curl equals $(P, Q, R)$ .

RemarkFailure on non-contractible spaces

The Poincare lemma fails on spaces with nontrivial topology. The classic example is $\omega = \frac{-y\,dx + x\,dy}{x^2 + y^2}$ on $\mathbb{R}^2 \setminus \{0\}$ : it is closed but not exact, since $\int_{S^1} \omega = 2\pi \neq 0$ . This form generates $H^1(\mathbb{R}^2 \setminus \{0\}) \cong \mathbb{R}$ .