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Topology of the intersections of quadrics II

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Abstract

We study the topology of the affine real variety given by the intersection with the unit sphere of the zero set in \({\mathbb R}^n\) of a pair of quadratic forms. We give a complete topological description of this variety in the generic case: when non-empty, it is always diffeomorphic to either the unit tangent bundle of a sphere, the product of two or three spheres, or the connected sum of an odd number of manifolds, each of them a product of two spheres. With this we complete the partial description given in (López de Medrano, 1989). Starting with the cases described in that article and other elementary ones, the proofs are based on the study of three geometric operations that give new of these varieties from simpler ones. For each operation, the topology of the new variety can be described, under appropriate conditions, from that of the old one. The global structure of the proof consists in an elaborate partial ordering of all the cases, in such a way that at each step those conditions are satisfied.

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Notes

  1. As well as a list of many fields where these varieties appear as interesting examples.

  2. This property was first defined in [2]. The concept was implicited in [8] and independently discovered in [6]. Chaperon’s well-chosen terminology was soon adopted in the following papers on the subject (among others [1, 3]).

  3. Statements (ii) and (iii), as well as a different proof of (iv), appear in [8]. Case (v) for \(s=0\) with some restrictions was proved in [6].

  4. These operations can be defined for intersections of any number \(k\) homogeneous quadrics and the unit sphere, but are only used in this article for \(k=2\) (a and b) and \(k=1\) (c).

  5. It can be shown that any simple polytope can be realized as the associated polytope of an intersection of diagonal quadrics, in general more than two, but here the construction will be explicit.

  6. This idea was used in [3] to find the topology of some intersections of more than two quadrics.

  7. See Sect. 2 for the definition of \(d_i\). Observe that when \(s=0,\ell =1\), \(Y\) is not even homotopy equivalent to the connected sum in the theorem, but it does have the same homology groups.

  8. Sometimes this sum is called the complexity of the space.

  9. This theorem formalizes an idea already used in [6].

References

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Authors and Affiliations

  1. Facultad de Ciencias, Universidad Nacional Autónoma de México, Mexico City, Mexico

    Vinicio Gómez Gutiérrez

  2. Instituto de Matemáticas, Universidad Nacional Autónoma de México, Mexico City, Mexico

    Santiago López de Medrano

Authors
  1. Vinicio Gómez Gutiérrez
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  2. Santiago López de Medrano
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Correspondence to Santiago López de Medrano.

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Dedicated to Fico González Acuña on his 70th birthday.

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Gómez Gutiérrez, V., López de Medrano, S. Topology of the intersections of quadrics II. Bol. Soc. Mat. Mex. 20, 237–255 (2014). https://doi.org/10.1007/s40590-014-0038-2

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  • DOI: https://doi.org/10.1007/s40590-014-0038-2

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