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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 Ndoi:10.1090/s0002-9939-2014-12231-3 fatcat:m4mtw36glzgonlrwnbs2qegg4q