Essential manifold

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

:$H\_n(M)\backslash to\; H\_n(K(\backslash pi,1)),$

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.

• All closed surfaces (i.e. 2-dimensional manifolds) are essential with the exception of the 2-sphere S².
• Real projective space RPⁿ is essential since the inclusion
• :$\backslash mathbb\{RP\}^n\; \backslash to\; \backslash mathbb\{RP\}^\backslash infty$

:is injective in homology, where

::$\backslash mathbb\{RP\}^\backslash infty\; =\; K(\backslash mathbb\{Z\}\_2,\; 1)$

:is the Eilenberg–MacLane space of the finite cyclic group of order 2.

• All compact aspherical manifolds are essential (since being aspherical means the manifold itself is already a K({{pi}}, 1))
• In particular all compact hyperbolic manifolds are essential.
• All lens spaces are essential.

• The connected sum of essential manifolds is essential.
• Any manifold which admits a map of nonzero degree to an essential manifold is itself essential.

{{Reflist}}

• Gromov's systolic inequality for essential manifolds
• Systolic geometry

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