Algebraic & Geometric Topology

Symmetric topological complexity of projective and lens spaces

Jesús González and Peter Landweber

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For real projective spaces, (a) the Euclidean immersion dimension, (b) the existence of axial maps and (c) the topological complexity are known to be three facets of the same problem. But when it comes to embedding dimension, the classical work of Berrick, Feder and Gitler leaves a small indeterminacy when trying to identify the existence of Euclidean embeddings of these manifolds with the existence of symmetric axial maps. As an alternative we show that the symmetrized version of (c) captures, in a sharp way, the embedding problem. Extensions to the case of even-torsion lens spaces and complex projective spaces are discussed.

Article information

Algebr. Geom. Topol., Volume 9, Number 1 (2009), 473-494.

Received: 15 January 2009
Revised: 18 February 2009
Accepted: 18 February 2009
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55M30: Ljusternik-Schnirelman (Lyusternik-Shnirelʹman) category of a space 57R40: Embeddings

symmetric topological complexity Euclidean embedding biequivariant map symmetric axial map projective space lens space


González, Jesús; Landweber, Peter. Symmetric topological complexity of projective and lens spaces. Algebr. Geom. Topol. 9 (2009), no. 1, 473--494. doi:10.2140/agt.2009.9.473.

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