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Polar Spaces

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Points and Lines

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Abstract

The goal of this chapter is a complete teachable proof from first principles of the classification of all polar spaces of polar rank at least four. Polar spaces are defined by the well-known “one or all” axiom on points and lines. Each such space is uniquely associated with a non-degenerate polar space whose singular subspaces are projective spaces. If a maximal singular subspace of the latter has finite rank, this number is called the polar rank of the space. The classical objects on the scene are the polar spaces defined by sesquilinear forms and pseudoquadratic forms. The classification of polar spaces of rank at least four proceeds in two steps: (1) showing that a polar space that can be embedded in a projective space must be classical and (2) showing that a non-degenerate polar space of rank at least four is in fact embeddable. The technical results from the end of the preceding chapter are used in both steps, while a homotopy-proof of Tits’ unpublished result that polar spaces of rank at least three are characterized by the cone over a point facilitates the proof (due to Cuypers, Johnson, and Pasini) of (2).

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Authors and Affiliations

  1. Kansas State University, Denison Avenue 419, Manhattan, KS, 66502, USA

    Ernest E. Shult

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  1. Ernest E. Shult
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Correspondence to Ernest E. Shult .

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© 2011 Springer-Verlag Berlin Heidelberg

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Shult, E.E. (2011). Polar Spaces. In: Points and Lines. Universitext. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15627-4_7

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