Special structures on some product of spheres
by
Maurizio Parton
University of Pisa, Italy

Poster

It is a classical result in Algebraic Topology that spheres Sⁿ are parallelizable if and only if n=1, 3, 7. When one looks at the product of two or more spheres S^n₁ x ... x S^n_r, r > 1, one gets instead a parallelizable manifold only under the condition that at least one of the n_i be odd. This was proved by Kervaire in the fifties [Kervaire56], but some homotopy theory considerations makes the proof not very suitable to write down explicit parallelizations. The only reference the author knows where explicit parallelization on some product of spheres are given is the article [Bruni92], where the frame is given whenever one of the factor sphere is S¹, S³, S⁵, S⁷, and the writing of frames in the general case is left as an open problem.

Another proof of Kervaire's theorem can be developed by following a series of hints contained in the book [Hirsh88]. We give the details of such a proof, as we developed it, and show how such a proof leads to explicit parallelization in the general case.

Next, we use the above frame to study the following structures on product of spheres: a family I of almost complex structures, a family H of almost hypercomplex structures, a G₂ and a Spin(7)-structure.

First, we look at the case of 2 spheres. If at least one sphere is S¹, all almost complex or hypercomplex structures in the families I and H turn out to be equivalent to the Hopf complex structure given by the action x --> e^2\pix on Cⁿ-0 and Hⁿ-0 respectively. As for G₂ and Spin(7)-structures on S⁶ x S¹ and S⁷ xS¹, we recover in this way the structures given in [Cabrera97] and [Cabrera95], respectively.

If at least one of the spheres is S³, the integrable almost complex structures in I are just the Calabi-Eckmann complex structures, whereas the almost hypercomplex structures in H are all non integrable.

Both the G₂ and Spin(7)-structures on S⁴ x S³ and S⁵ xS³, turns out (also with the help of Mathematica@) to belong to the general class W in the classification given by [Fernández-Gray82] and [Fernández86], respectively. Other such structures are similarly obtained in the product of three or more spheres. A Spin(9) case is also discussed.

Bruni92 M. Bruni. Sulla parallelizzazione esplicita dei prodotti di sfere. Rend. di Mat., serie VII, 12:405-423, 1992.
Cabrera95 F. M. Cabrera. On Riemannian manifolds with Spin(7)-structure.Publ. Math. Debrecen, 46:271-283, 1995.
Cabrera97 F. M. Cabrera. Orientable Hypersurfaces of Riemannian Manifolds with Spin(7)-Structure.Acta Math. Hung., 76(3):235-247, 1997.
Fern\'andez86 M. Fernández. A Classification of Riemannian Manifolds with Structure Group Spin(7). Ann. Mat. Pura e Appl., 148:101-122, 1986.
Fern\'andez-Gray82 M. Fernández and A. Gray. Riemannian manifolds with structure group G₂. Ann. Mat. Pura e Appl., 132:19-45, 1982.
Hirsch88 M. W. Hirsch. Differential Topology, volume 33 of GTM. Springer-Verlag, third edition, 1988.
Kervaire56 M. Kervaire. Courbure intégrale généralisée et homotopie. Math. Ann., 131:219-252, 1956.

Date received: May 30, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadq-89.