What is the name of this invariant, the number of non-homotopic diffemorphism form a manifold to itself. What is this number for the closed ball B^n, and for euclidean space R^n and for the n-sphere?
2026-04-28 20:21:40.1777407700
Number of non-homotopic diffemorphism form a manifold to itself
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What you are describing (homotopy classes of diffeomorphisms, where in the homotopy "each target point follows a path in the manifold and no two paths cross each other simultaneously") is the mapping class group of your manifold, i.e. the isotopy classes of diffeomorphisms of $M$. The number you seek is the cardinality of the mapping class group; I do not think this number has a special name.