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Consider a finite simplicial complex $X$ and the simplicial cochain complex with real coefficients. The cochain groups are finite-dimensional vector spaces, they have a natural scalar product. The $n$-cocycles $Z^n$ are a linear subspace in the $n$-cochains $C^n$ and let $P:C^n\to Z^n$ be the orthogonal projection.

Is there an explicit formula for $P$ ?

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This is finite dimensional Hodge theory. You need to answer the following more general question: given a linear map $D: U\to V$, where $U, V$ are finite dimensional real Euclidean spaces find the projection onto $\ker D$. Here is how you do it.

Observe that $(\ker D)^\perp= {\rm range}\;D^*$. If $u\in U$ and $\bar{u}$ is the orthogonal projection onto $\ker D$, then

$$u= \bar{u}+ D^*v,\;\;v\in V, $$

so that

$$Du=DD^*v. $$

Find a solution $v$ of the above system. Then $\bar{u}=u-D^*v$. Indeed, note that

$$D(u-D^*v)= Du-DD^* v, $$

so that $u-D^*v\in \ker D$. Moreover $D^* v\perp \ker D$.

Liviu Nicolaescu
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  • Of course, the Hodge decomposition is one of the motivations here. Extracting an explicit formula for the projection out of the linear algebra above is not straightforward. This is the whole reason for the question. –  Feb 06 '14 at 14:10