Abstract.

I will discuss the role of the non-branching assumption in different notions of Ricci curvature lower bounds in metric measure spaces. I will briefly recall results on CD(K,N) and MCP(K,N) spaces that were proven originally under the non-branching assumption. After this we will see how the simple example of a branching space satisfying CD(0,2), namely the plane with the supremum norm, can be modified to give counter-examples to some of these results in the branching case. We will obtain a counter-example to the local-to-global property on CD(K,N) spaces and counter-examples to topological splitting and maximal diameter theorems in MCP(K,N) spaces. This last set of examples were observed together with Christian Ketterer.