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Spaces of algebraic and continuous maps between real algebraic varieties

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Adamaszek, Michał, Kozlowski, Andrzej and Yamaguchi, Kohhei. (2011) Spaces of algebraic and continuous maps between real algebraic varieties. Quarterly Journal of Mathematics, Vol.62 (No.4). pp. 771-790. ISSN 0033-5606

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Official URL: http://dx.doi.org/10.1093/qmath/haq029

Abstract

We consider the inclusion of the space of algebraic (regular) maps between real algebraic varieties in the space of all continuous maps. For a certain class of real algebraic varieties, which include real projective spaces, it is well known that the space of real algebraic maps is a dense subset of the space of all continuous maps. Our first result shows that, for this class of varieties, the inclusion is also a homotopy equivalence. After proving this, we restrict the class of varieties to real projective spaces. In this case, the space of algebraic maps has a 'minimum degree' filtration by finite-dimensional subspaces and it is natural to expect that the homotopy types of the terms of the filtration approximate closer and closer the homotopy type of the space of continuous mappings as the degree increases. We prove this and compute the lower bounds of this approximation of these spaces. This result can be seen as a generalization of the results of Mostovoy, Vassiliev and others on the topology of the space of real rational maps and the space of real polynomials without n-fold roots. It can also be viewed as a real analogue of Mostovoy's work on the topology of the space of holomorphic maps between complex projective spaces, which generalizes Segal's work on the space of complex rational maps.

Item Type:

Journal Article

Subjects:

Q Science > QA Mathematics

Divisions:

Faculty of Science > Mathematics

Library of Congress Subject Headings (LCSH):

Holomorphic mappings, Topology, Algebraic varieties

Journal or Publication Title:

Quarterly Journal of Mathematics

Publisher:

Oxford University Press

ISSN:

0033-5606

Official Date:

December 2011

Dates:

Date	Event
December 2011	Published

Volume:

Vol.62

Number:

No.4

Page Range:

pp. 771-790

Identification Number:

10.1093/qmath/haq029

Status:

Peer Reviewed

Publication Status:

Published

Access rights to Published version:

Restricted or Subscription Access

Funder:

Engineering and Physical Sciences Research Council (EPSRC), Japan. Monbu Kagakushō [Japan. Ministry of Education, Culture, Sports, Science and Technology] (MK)

Grant number:

EP/D063191/1 (EPSRC), 19540068 (MK)

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