# Hodge star and harmonic simplicial differential forms

Is there a notion of harmonic forms and Hodge theory for Sullivan's piecewise smooth differential forms on a simplicial set?

Let me recall some background.

Hodge Theory on a Riemannian manifold A Riemannian metric $g$ an $n$-dimension closed manifold $M$ gives a Hodge star operator on the smooth differential forms $*: \Omega^k(M) \to \Omega^{n-k}(M)$, a nondegenerate inner product on $\Omega^k(M)$ given by $\langle \alpha, \beta \rangle = \int_M \alpha \wedge * \beta$, and a codifferential $\delta$ that is the adjoint of the usual exterior differential $d$. The Laplacian is $\Delta = \delta d + d \delta$, and the harmonic forms are those which are in the kernel of the Laplacian.

Hodge theory asserts that the space of harmonic forms is isomorphic to the real cohomology of $M$. I.e., every harmonic form is closed, and each cohomology class contains a unique harmonic representative.

Sullivan's piecewise smooth differential forms on a simplicial complex Let $K$ be a simplicial set. A differential form on $K$ is essentially a smooth differential form on each simplex of $K$ subject to compatibility conditions given by the face and degeneracy maps. In detail, $\Omega^*(\Delta^\bullet)$ is a simplicial object in commutative differential graded algebras, and the algebra of piecewise smooth forms on $K$ is
$A_{C^\infty}(K) = Hom_{\mathrm{SSet}}(K_\bullet, \Omega^*(\Delta^\bullet))$

The question Suppose now that $K$ is a finite simplicial set (i.e., a simplicial set with finitely many nondegenerate simplices) with an appropriate version of a Riemannian metric. Is there a notion of harmonic forms and Hodge theory for $A_{C^\infty}(K)$?

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Hi Jeff, the main step towards this is to understand Hodge theory for manifolds with boundary. If you can do this, then you can treat the standard simplex and then it should be a formal matter to extend the theory to finite simplicial sets. Gilkey's book "Heat Equation, invariance theory and the Atiyah-Singer theorem" contains a section on the de Rham complex on manifolds with boundary. There you probably find the correct boundary condition on the forms and the Hodge decomposition on manifolds with boundary. –  Johannes Ebert Nov 22 '10 at 9:56
You might be interested in the following paper : J. Cheeger. A vanishing theorem for piecewise constant curvature spaces. Curvature and topology of Riemannian manifolds (Katata, 1985), 33–40, Lecture Notes in Math., 1201, Springer, Berlin. (1986). –  Dmitri Nov 22 '10 at 10:17
hi Johannes. I think the extension to simplicial sets is not just a formal matter... An n-simplex is contractible, and rel boundary it looks like an n-sphere. For harmonic forms, Dirichlet boundary conditions give you (up to scalars) the constant function and only a single harmonic form in degree n, and Neumann boundary conditions leave you with only the constant function on each simplex. So it seems unlikely that one represent a middle degree cohomology class by a form that is harmonic on each top-dimensional simplex and satisfies either of these boundary conditions. –  Jeffrey Giansiracusa Nov 22 '10 at 15:44
Well, don't there exist forms that are harmonic in the interior of the simplex and coincide with a given form on the boundary (the classical boundary value problem)? The arguments you gave show that these forms are nearly uniquely determined by the boundary values, which to me seems promising rather than discouraging. –  Johannes Ebert Nov 22 '10 at 17:43