Chataur, DavidCirici, Joana2022-10-262022-10-2620220002-9947https://hdl.handle.net/2445/190227ABSTRACT. We describe the $E$-infinity algebra structure on the complex of singular cochains of a topological space, in the context of sheaf theory. As a first application, for any algebraic variety we define a weight filtration compatible with its $E$-infinity structure. This naturally extends the theory of mixed Hodge structures in rational homotopy to $p$-adic homotopy theory. The spectral sequence associated to the weight filtration gives a new family of algebraic invariants of the varieties for any coefficient ring, carrying Steenrod operations. As a second application, we promote Deligne's intersection complex computing intersection cohomology, to a sheaf carrying E-infinity structures. This allows for a natural interpretation of the Steenrod operations defined on the intersection cohomology of any topological pseudomanifold.36 p.application/pdfengcc-by-nc-nd (c) American Mathematical Society (AMS), 2022https://creativecommons.org/licenses/by-nc-nd/4.0/Topologia algebraicaHomologiaTeoria de HodgeAlgebraic topologyHomologyHodge theoryinfo:eu-repo/semantics/article7187092022-10-26info:eu-repo/semantics/openAccess