Birman-Hilden theory for reducible 3-manifolds

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• Trent Lucas (Brown University)
• Friday 14 June 2024, 14:30-15:00
• External.

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TRHW01 - Workshop on topology, representation theory and higher structures

For a manifold M, we discuss its mapping class group Mod(M) = Homeo+(M)/isotopy.  Given a finite branched cover of manifolds M → N, one can lift mapping classes from N to M to obtain a (virtual) homomorphism of mapping class groups.  A celebrated theorem of Birman-Hilden and MacLachlan-Harvey says that if M is a hyperbolic surface, then this lifting map is injective for all regular covers.  Following a question of Margalit-Winarski, we show that this lifting map is not injective for many branched covers of reducible 3-manifolds, and we study the kernel for the 3-manifold analog of the hyperelliptic involution.  In this case, the lifting map is closely related to symmetric outer automorphism groups of free products.

This talk is part of the Isaac Newton Institute Seminar Series series.

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