**Title:** A properad action on homology that fails to lift to the chain level

**Speaker:** Theo Johnson-Freyd

**Speaker Info:** Northwestern

**Brief Description:**

**Special Note**: **Please note the unusual time**

**Abstract:**

A tenet of algebraic topology is that algebraic structures on the homology of a space should correspond to structures at the chain level, such that the axioms that hold on homology are weakened to coherent homotopies. For example, the homology of an oriented manifold is a Frobenius algebra --- what about the chains? In this talk, I will explain that for one-dimensional manifolds, the answer is No. I will save comenting on higher-dimensional manifolds for the 4pm talk.To make this precise, I will spend some time discussing the notion of "properad", which generalizes the notion of "operad" to allow many-to-many operations. I will recall the Koszul duality for properads, and how to compute cofibrant replacements. I will not assume that the words "Koszul duality" or "cofibrant replacement" are particularly familiar.

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