Essential manifold

From formulasearchengine
Jump to navigation Jump to search

Essential manifold a special type of closed manifolds. The notion was first introduced explicitly by Mikhail Gromov.[1]


A closed manifold M is called essential if its fundamental class [M] defines a nonzero element in the homology of its fundamental group π, or more precisely in the homology of the corresponding Eilenberg–MacLane space K(π, 1), via the natural homomorphism


where n is the dimension of M. Here the fundamental class is taken in homology with integer coefficients if the manifold is orientable, and in coefficients modulo 2, otherwise.


is injective in homology, where
is the Eilenberg-MacLane space of the finite cyclic group of order 2.



  1. Gromov, M.: Filling Riemannian manifolds, J. Diff. Geom. 18 (1983), 1–147.

See also

Template:Systolic geometry navbox