This paper contains several results concerning circle action on almost-complex and smooth manifolds. More precisely, we show that, for an almost-complex manifold M^{2mn} (resp. a smooth manifold M^{2mn} ), if there exists a partition M^{2mn} of weight M^{2mn} such that the Chern number M^{2mn} (resp. Pontrjagin number M^{2mn} ) is nonzero, then\emph {any} circle action on M^{2mn} (resp. M^{2mn} ) has at least M^{2mn} fixed points. When an even-dimensional smooth manifold M^{2mn} admits a semi-free action with isolated fixed points, we show that M^{2mn} bounds, which generalizes a well-known fact in the free case. We also provide a topological obstruction, in terms of the first Chern class, to the existence of semi-free circle action with\emph {nonempty} isolated fixed points on almost-complex manifolds. The main ingredients of our proofs are Bott's residue formula and rigidity theorem.

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In this paper we introduce a path complex that can be regarded as a generalization of the notion of a simplicial complex. The main motivation for considering path complexes comes from directed graphs (digraphs). We obtain a new notion of the path homology and cohomology of a digraph.

We construct a cohomology theory of transitive digraphs and use it to give a new proof of a theorem of Gerstenhaber and Schack about isomorphism between simplicial cohomology and Hochschild cohomology of a certain algebra associated with the simplicial complex.

We use a differential form cohomology theory on transitive digraphs to give a new proof of a theorem of Gerstenhaber and Schack about isomorphism between simplicial cohomology and Hochschild cohomology of a certain algebra associated with the simplicial complex.