Hodge star and harmonic simplicial differential forms

2010-11-21T16:39:39Z

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 $A_{C^\infty}(K)$?

Hodge star and harmonic simplicial differential forms - MathOverflow

Hodge star and harmonic simplicial differential forms