ON MANIFOLDS HOMEOMORPHIC TO SPHERES

Characterize the topology of a smooth manifold admitting a Morse-Bott function with two critical submanifolds, analogous to Reeb's theorem.

Generalizes Reeb’s theorem and results by Hai Bao and Rees in the context of Morse-Bott functions.

Not clearly stated in the paper

. We prove a result analogous to Reeb’s theorem in the context of Morse-Bott functions: if a closed, smooth manifold M admits a Morse-Bott function having two critical submanifolds Sk and Sl (k̸=l), then M hasdimensionk+l+1 andishomeomorphictothestandardsphere Sk+l+1 butnotnecessarilydiffeomorphicto it. Wealsoprovesimilarresultsforprojectivespacesovertherealnumbers, complexnumbersandquaternions.