# Hopf invariants, topological complexity, and LS-category of the cofiber of the diagonal map for two-cell complexes

Jesús González, Mark Grant, Lucile Vandembroucq

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Let $X$ be a two-cell complex with attaching map $\alpha\colon S^q\to S^p$, and let $C_X$ be the cofiber of the diagonal inclusion $X\to X\times X$. It is shown that the topological complexity (${\rm TC}$) of $X$ agrees with the Lusternik-Schnirelmann category (${\rm cat}$) of $C_X$ in the (almost stable) range $q\leq2p-1$. In addition, the equality ${\rm TC}(X)={\rm cat}(C_X)$ is proved in the (strict) metastable range $2p-1 Original language English Topological complexity and related topics Mark Grant, Gregory Lupton, Lucile Vandembroucq American Mathematical Society 133-150 18 702 9781470444051 9781470434366 https://doi.org/10.1090/conm/702/14109 Published - 2018 ### Publication series Name Contemporary Mathematics American Mathematical Society 1098-3627 0271-4132 ### Fingerprint Hopf Invariant Topological Complexity Cell Complex Lusternik-Schnirelmann Category Range of data Equality Inclusion ### Keywords • math.AT • 55M30, 55Q25, 55S35, 55S36, 68T40, 70B15 ### Cite this González, J., Grant, M., & Vandembroucq, L. (2018). Hopf invariants, topological complexity, and LS-category of the cofiber of the diagonal map for two-cell complexes. In M. Grant, G. Lupton, & L. Vandembroucq (Eds.), Topological complexity and related topics (Vol. 702, pp. 133-150). (Contemporary Mathematics). American Mathematical Society. https://doi.org/10.1090/conm/702/14109 Hopf invariants, topological complexity, and LS-category of the cofiber of the diagonal map for two-cell complexes. / González, Jesús; Grant, Mark; Vandembroucq, Lucile. Topological complexity and related topics. ed. / Mark Grant; Gregory Lupton; Lucile Vandembroucq. Vol. 702 American Mathematical Society, 2018. p. 133-150 (Contemporary Mathematics). Research output: Chapter in Book/Report/Conference proceedingConference contribution González, J, Grant, M & Vandembroucq, L 2018, Hopf invariants, topological complexity, and LS-category of the cofiber of the diagonal map for two-cell complexes. in M Grant, G Lupton & L Vandembroucq (eds), Topological complexity and related topics. vol. 702, Contemporary Mathematics, American Mathematical Society, pp. 133-150. https://doi.org/10.1090/conm/702/14109 González J, Grant M, Vandembroucq L. Hopf invariants, topological complexity, and LS-category of the cofiber of the diagonal map for two-cell complexes. In Grant M, Lupton G, Vandembroucq L, editors, Topological complexity and related topics. Vol. 702. American Mathematical Society. 2018. p. 133-150. (Contemporary Mathematics). https://doi.org/10.1090/conm/702/14109 González, Jesús ; Grant, Mark ; Vandembroucq, Lucile. / Hopf invariants, topological complexity, and LS-category of the cofiber of the diagonal map for two-cell complexes. Topological complexity and related topics. editor / Mark Grant ; Gregory Lupton ; Lucile Vandembroucq. Vol. 702 American Mathematical Society, 2018. pp. 133-150 (Contemporary Mathematics). @inproceedings{9fa62dfb97ff444a9ea1da49c0d1ed45, title = "Hopf invariants, topological complexity, and LS-category of the cofiber of the diagonal map for two-cell complexes", abstract = "Let$X$be a two-cell complex with attaching map$\alpha\colon S^q\to S^p$, and let$C_X$be the cofiber of the diagonal inclusion$X\to X\times X$. It is shown that the topological complexity (${\rm TC}$) of$X$agrees with the Lusternik-Schnirelmann category (${\rm cat}$) of$C_X$in the (almost stable) range$q\leq2p-1$. In addition, the equality${\rm TC}(X)={\rm cat}(C_X)$is proved in the (strict) metastable range$2p-1",
keywords = "math.AT, 55M30, 55Q25, 55S35, 55S36, 68T40, 70B15",
author = "Jes{\'u}s Gonz{\'a}lez and Mark Grant and Lucile Vandembroucq",
year = "2018",
doi = "10.1090/conm/702/14109",
language = "English",
isbn = "9781470434366",
volume = "702",
series = "Contemporary Mathematics",
publisher = "American Mathematical Society",
pages = "133--150",
editor = "Grant, {Mark } and Lupton, {Gregory } and Vandembroucq, {Lucile }",
booktitle = "Topological complexity and related topics",

}

TY - GEN

T1 - Hopf invariants, topological complexity, and LS-category of the cofiber of the diagonal map for two-cell complexes

AU - González, Jesús

AU - Grant, Mark

AU - Vandembroucq, Lucile

PY - 2018

Y1 - 2018

N2 - Let $X$ be a two-cell complex with attaching map $\alpha\colon S^q\to S^p$, and let $C_X$ be the cofiber of the diagonal inclusion $X\to X\times X$. It is shown that the topological complexity (${\rm TC}$) of $X$ agrees with the Lusternik-Schnirelmann category (${\rm cat}$) of $C_X$ in the (almost stable) range $q\leq2p-1$. In addition, the equality ${\rm TC}(X)={\rm cat}(C_X)$ is proved in the (strict) metastable range $2p-1 AB - Let$X$be a two-cell complex with attaching map$\alpha\colon S^q\to S^p$, and let$C_X$be the cofiber of the diagonal inclusion$X\to X\times X$. It is shown that the topological complexity (${\rm TC}$) of$X$agrees with the Lusternik-Schnirelmann category (${\rm cat}$) of$C_X$in the (almost stable) range$q\leq2p-1$. In addition, the equality${\rm TC}(X)={\rm cat}(C_X)$is proved in the (strict) metastable range$2p-1

KW - math.AT

KW - 55M30, 55Q25, 55S35, 55S36, 68T40, 70B15

U2 - 10.1090/conm/702/14109

DO - 10.1090/conm/702/14109

M3 - Conference contribution

SN - 9781470434366

VL - 702

T3 - Contemporary Mathematics

SP - 133

EP - 150

BT - Topological complexity and related topics

A2 - Grant, Mark

A2 - Lupton, Gregory

A2 - Vandembroucq, Lucile

PB - American Mathematical Society

ER -