We say that two $S^{1}$ -central extensions $S^{1}\to{\widetilde{\Gamma}}\to\Gamma\rightrightarrows M$ and $S^{1}\to{\widetilde{\Gamma}}^{\prime}\to\Gamma^{\prime}\rightrightarrows M^{\prime}$ are Morita equivalent if there exists an $S^{1}$ -equivariant ${\widetilde{\Gamma}}_{\scriptscriptstyle\bullet}$ - ${\widetilde{\Gamma}}^{\prime}_{\scriptscriptstyle\bullet}$ -bitorsor $Z$ , by which we mean that $Z$ is a ${\widetilde{\Gamma}}_{\scriptscriptstyle\bullet}$ - ${\widetilde{\Gamma}}^{\prime}_{\scriptscriptstyle\bullet}$ -bitorsor endowed with an $S^{1}$ -action such that

whenever $(\lambda,r,r^{\prime},z)\in S^{1}\times{\widetilde{\Gamma}}\times{\widetilde{\Gamma}}^{\prime}\times Z$ and the products make sense.

A homotopy operator between two chain complexes $f,g:X_{\bullet}\rightarrow Y_{\bullet}$ is a linear map $h:X_{\bullet}\rightarrow Y_{\bullet+1}$ such that

In that case, $f$ and $g$ are called chain homotopic and we denote it by $f\simeq g$ .
Two chain maps $f:X_{\bullet}\rightarrow Y_{\bullet}$ and $g:Y_{\bullet}\rightarrow X_{\bullet}$ are called homotopy inverse if $g\circ f\simeq id_{X}$ and $f\circ g\simeq id_{Y}$ are both homotopic to the identity. If $f:X_{\bullet}\rightarrow Y_{\bullet}$ admits a homotopy inverse, it is called a homotopy equivalence. In particular every homotopy equivalence is a quasi-isomorphism.

The Hodge star operator $\star:\Omega^{j}(M)\to\Omega^{n-j}(M)$ is defined by:

Let $K$ be a triangulation of an $n$ -dimensional manifold $M$ . The mesh $\eta=\eta(K)$ of a triangulation is:

where $r$ means the geodesic distance in $M$ and the supremum is taken over all pairs of vertices $p$ , $q$ of a $1$ -simplex in $K$ .

The fullness $\Theta=\Theta(K)$ of a triangulation $K$ is

where the $\inf$ is taken over all $n$ -simplexes $\sigma$ of $K$ and $vol(\sigma)$ is the Riemannian volume of $\sigma$ , as a Riemannian submanifold of $M$ .