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Symmetric topological complexity of projective and Lens spaces. (English) Zbl 1167.57012

The main result of the paper under review is to give a formula in terms of the symmetric topological complexity for Euclidean embeddings of the projective plane. More precisely:
Theorem 1.3. For \(r>15\) as well as for \(r\in \{ 1, 2, 4, 8, 9, 13\}\), the symmetric topological complexity of \(P^r\) satisfies \(TC^S(P^r)=E(r)+1\).
Here \(E(r)\) stands for the Euclidean embedding dimension of \(P^r\). This theorem is based on the following result which uses the the concept of level of an involution:
Theorem 1.4. For all values of \(r\), \(TC^S(P^r)= \text{level} (P^r\times P^r-\Delta_{P^r}, \;\mathbb Z/2)+1.\)
The results of this paper solve an indeterminacy left out in [A. J. Berrick, S. Feder and S. Gitler, Bol. Soc. Mat. Mex., II. Ser. 21, 39–41 (1976; Zbl 0455.57013)] when trying to identify the existence of Euclidean embeddings of the projective spaces. The last two sections are a discussion of the cases of mainly even-torsion lens spaces and complex projective spaces. In the former case the authors give an upper bound for the \(TC^S(M)\)(symmetric topological complexity), this is Corollary 5.4, in terms of some geometric numbers \(b_{n,m}^S\) (Definition 5.2). In the latter case they compute \(TC^S(\mathbb C P^n)\) (Theorem 6.1). The paper is well organized and contains several substantial informations about the development of the embedding problem.

MSC:

57R40 Embeddings in differential topology
55M30 Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects)
57R42 Immersions in differential topology

Citations:

Zbl 0455.57013
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References:

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