ANNALES DE L'INSTITUT FOURIER

We define the natural extension $C_{•,•}$ of the complex $C$ generated by a simplicial set $\Gamma$. This enables us to define the notion of a ribbon base on a cycle of $C$. The direct sum of the homologies of the columns of $C_{•,•}$ contains, in addition to the homology of $C$, some groups where the obstructions to the existence of ribbons do live. In the case $\Gamma$ is a stable under subdivision simplicial subset of the set of singular simplexes of a topological space, the existence of ribbons implies the invariance of the homology of $C$ under subdivision. The fundamental lemma of the paper is proved in the general case of a complex $C$ generated by a (topological) simplicial space $\Gamma$ without degeneracy operators: if $\Gamma$ satisfies a local extension property, and also some “isotopy of stars” property, then these obstruction groups vanish, and the homology of $C$ satisfies the invariance under isotopy. As a consequence one obtains in an unified way both a new proof of the theorem of Lalonde about the homology of embedded simplexes meeting transversally a foliation of a differentiable manifold, and the extension of Lalonde’s theorem to the homology of immersed simplexes.