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Pullback invariants of Thurston maps

Associated to a Thurston map $f: S^2 \to S^2$ with postcritical set $P$ are several different invariants obtained via pullback: a relation on the set of free homotopy classes of curves in $S^2- P$, a linear operator on the free $\R$-module generated by these homotopy classes of curves, a virtual endomorphism on the pure mapping class group, an analytic self-map of an associated Teichmueller space, and an analytic self-correspondence on an associated moduli space. Viewing all of these objects as invariants of $f$, we investigate harmonious relationships between their properties.