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Corrigendum to “Topological complexity of real Grassmannians”

Published online by Cambridge University Press:  19 April 2022

Petar Pavešić*
Affiliation:
Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 21, 1000 Ljubljana, Slovenija ([email protected])
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Abstract

There is a problem with the proofs of [1], Lemma 4.4 and the related Theorems 4.5, 4.8 and 4.12 regarding the computation of zero-divisor cup-length of real Grassmann manifolds ${G_k({{\mathbb {R}}}^{n})}$. The correct statements and improved estimates of the topological complexity of ${G_k({{\mathbb {R}}}^{n})}$ will appear in a separate paper by M. Radovanović [2].

Type
Corrigendum
Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

In a recent article [Reference Pavešić1] we described new estimates of Farber's topological complexity of real Grassmann manifolds $G_k({{\mathbb {R}}}^{n})$. In particular, in [Reference Pavešić1, Theorems 4.5, 4.8, 4.12] we computed the zero-divisor cup-lengths $\mathop {\rm zcl}\nolimits (G_2({{\mathbb {R}}}^{n}))$, $\mathop {\rm zcl}\nolimits (G_3({{\mathbb {R}}}^{n}))$ and $\mathop {\rm zcl}\nolimits (G_4({{\mathbb {R}}}^{n}))$ together with the corresponding lower bounds for topological complexity. This was achieved by finding new non-trivial products in the cohomology of Grassmannians, similar in spirit to those used by Stong [Reference Stong3] to estimate Lusternik-Schnirelmann category of Grassmannians, but chosen in a way that is better suited for the computation of topological complexity. Soon after the publication, we were informed by Prof. M. Radovanović that by using similar methods one can find non-trivial products in the cohomology of Grassmannians that in most cases yield even longer non-trivial products of zero-divisors. As a consequence, the values of zero-divisor cup-lengths stated in the previously mentioned theorems are incorrect, and the resulting lower bounds for topological complexity of real Grassmannians are further improved by the new estimates. For instance, the estimate $\mathop {\rm zcl}\nolimits (G_2({{\mathbb {R}}}^{2^{s}+1}))\ge 2^{s+1}-1$ in [Reference Pavešić1, Theorem 4.5] is improved by Radovanović to $\mathop {\rm zcl}\nolimits (G_2({{\mathbb {R}}}^{2^{s}+1}))\ge 2^{s+1}+2^{s}-4$, and the estimate $\mathop {\rm zcl}\nolimits (G_3({{\mathbb {R}}}^{2^{s}+1}))\ge 3\cdot 2^{s}-2$ in [Reference Pavešić1, Theorem 4.8] is improved to $\mathop {\rm zcl}\nolimits (G_2({{\mathbb {R}}}^{2^{s}+1}))\ge 3\cdot 2^{s}+2^{s-2}-7$. The precise formulations include several cases and sub-cases and will be presented in detail in a forthcoming paper by M. Radovanović [Reference Radovanović2]. We are grateful to Prof. Radovanović who discovered the error and provided corrected and improved estimates.

Footnotes

Date: March 9, 2022.

*

Supported by the Slovenian Research Agency program P1-0292 and grants N1-0083, N1-0064.

References

Pavešić, P., Topological complexity of real Grassmannians, Proc. Royal Society of Edinburgh Section A: Mathematics, 151 (2021), 20132029.CrossRefGoogle Scholar
Radovanović, M., On the topological complexity and zero-divisor cup-length of real Grassmannians, submitted.Google Scholar
Stong, R., Cup products in Grassmannians, Top. Appl. 13 (1982), 103113.CrossRefGoogle Scholar