Smooth gluing of group actions and applications

Parkhe, Kiran

doi:10.1090/S0002-9939-2014-12231-3

Smooth gluing of group actions and applications
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by Kiran Parkhe PDF

Proc. Amer. Math. Soc. 143 (2015), 203-212 Request permission

Abstract:

Let $M_1$ and $M_2$ be two $n$-dimensional smooth manifolds with boundary. Suppose we glue $M_1$ and $M_2$ along some boundary components (which are, therefore, diffeomorphic). Call the result $N.$ If we have a group $G$ acting continuously on $M_1,$ and also acting continuously on $M_2,$ such that the actions are compatible on glued boundary components, then we get a continuous action of $G$ on $N$ that stitches the two actions together. However, even if the actions on $M_1$ and $M_2$ are smooth, the action on $N$ probably will not be smooth.

We give a systematic way of smoothing out the glued $G$-action. This allows us to construct interesting new examples of smooth group actions on surfaces and to extend a result of Franks and Handel (2006) on distortion elements in diffeomorphism groups of closed surfaces to the case of surfaces with boundary.

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