In mathematics, an Eells–Kuiper manifold is a compactification of R n {\displaystyle \mathbb {R} ^{n}} by a sphere of dimension n / 2 {\displaystyle n/2} , where n = 2 , 4 , 8 {\displaystyle n=2,4,8} , or 16 {\displaystyle 16} . It is named after James Eells and Nicolaas Kuiper. If n = 2 {\displaystyle n=2} , the Eells–Kuiper manifold is diffeomorphic to the real projective plane R P 2 {\displaystyle \mathbb {RP} ^{2}} . For n ≥ 4 {\displaystyle n\geq 4} it is simply-connected and has the integral cohomology structure of the complex projective plane C P 2 {\displaystyle \mathbb {CP} ^{2}} ( n = 4 {\displaystyle n=4} ), of the quaternionic projective plane H P 2 {\displaystyle \mathbb {HP} ^{2}} ( n = 8 {\displaystyle n=8} ) or of the Cayley projective plane ( n = 16 {\displaystyle n=16} ).
Properties These manifolds are important in both Morse theory and foliation theory: Theorem: Let M {\displaystyle M} be a connected closed manifold (not necessarily orientable) of dimension n {\displaystyle n} . Suppose M {\displaystyle M} admits a Morse function f : M → R {\displaystyle f\colon M\to \mathbb {R} } of class C 3 {\displaystyle C^{3}} with exactly three singular points. Then M {\displaystyle M} is a Eells–Kuiper manifold. Theorem: Let M n {\displaystyle M^{n}} be a compact connected manifold and F {\displaystyle F} a Morse foliation on M {\displaystyle M} . Suppose the number of centers c {\displaystyle c} of the foliation F {\displaystyle F} is more than the number of saddles s {\displaystyle s} . Then there are two possibilities:
c = s + 2 {\displaystyle c=s+2} , and M n {\displaystyle M^{n}} is homeomorphic to the sphere S n {\displaystyle S^{n}} ,
c = s + 1 {\displaystyle c=s+1} , and M n {\displaystyle M^{n}} is an Eells–Kuiper manifold, n = 2 , 4 , 8 {\displaystyle n=2,4,8} or 16 {\displaystyle 16} .
See also Reeb sphere theorem
References