### Abstract

To clarify the method behind (Iwase, Bull. Lond. Math. Soc. 30 (1998), 623-634), a generalisation of Berstein-Hilton Hopf invariants is defined as 'higher Hopf invariants'. They detect the higher homotopy associativity of Hopf spaces and are studied as obstructions not to increase the LS category by one by attaching a cone. Under a condition between dimension and LS category, a criterion for Ganea's conjecture on LS category is obtained using the generalised higher Hopf invariants, which yields the main result of (Iwase, Bull. Lond. Math. Soc. 30 (1998), 623-634) for all the cases except the case when p = 2. As an application, conditions in terms of homotopy invariants of the characteristic maps are given to determine the LS category of sphere-bundles-over-spheres. Consequently, a closed manifold M is found not to satisfy Ganea's conjecture on LS category and another closed manifold N is found to have the same LS category as its 'punctured submanifold' N - {P}, P ε N. But all examples obtained here support the conjecture in (Iwase, Bull. Lond. Math. Soc. 30 (1998), 623-634).

Original language | English |
---|---|

Pages (from-to) | 695-723 |

Number of pages | 29 |

Journal | Topology |

Volume | 41 |

Issue number | 4 |

DOIs | |

Publication status | Published - Apr 3 2002 |

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### All Science Journal Classification (ASJC) codes

- Geometry and Topology

### Cite this

_{∞}-method in Lusternik-Schnirelmann category.

*Topology*,

*41*(4), 695-723. https://doi.org/10.1016/S0040-9383(00)00045-8

**A _{∞}-method in Lusternik-Schnirelmann category.** / Iwase, Norio.

Research output: Contribution to journal › Article

_{∞}-method in Lusternik-Schnirelmann category',

*Topology*, vol. 41, no. 4, pp. 695-723. https://doi.org/10.1016/S0040-9383(00)00045-8

_{∞}-method in Lusternik-Schnirelmann category. Topology. 2002 Apr 3;41(4):695-723. https://doi.org/10.1016/S0040-9383(00)00045-8

}

TY - JOUR

T1 - A∞-method in Lusternik-Schnirelmann category

AU - Iwase, Norio

PY - 2002/4/3

Y1 - 2002/4/3

N2 - To clarify the method behind (Iwase, Bull. Lond. Math. Soc. 30 (1998), 623-634), a generalisation of Berstein-Hilton Hopf invariants is defined as 'higher Hopf invariants'. They detect the higher homotopy associativity of Hopf spaces and are studied as obstructions not to increase the LS category by one by attaching a cone. Under a condition between dimension and LS category, a criterion for Ganea's conjecture on LS category is obtained using the generalised higher Hopf invariants, which yields the main result of (Iwase, Bull. Lond. Math. Soc. 30 (1998), 623-634) for all the cases except the case when p = 2. As an application, conditions in terms of homotopy invariants of the characteristic maps are given to determine the LS category of sphere-bundles-over-spheres. Consequently, a closed manifold M is found not to satisfy Ganea's conjecture on LS category and another closed manifold N is found to have the same LS category as its 'punctured submanifold' N - {P}, P ε N. But all examples obtained here support the conjecture in (Iwase, Bull. Lond. Math. Soc. 30 (1998), 623-634).

AB - To clarify the method behind (Iwase, Bull. Lond. Math. Soc. 30 (1998), 623-634), a generalisation of Berstein-Hilton Hopf invariants is defined as 'higher Hopf invariants'. They detect the higher homotopy associativity of Hopf spaces and are studied as obstructions not to increase the LS category by one by attaching a cone. Under a condition between dimension and LS category, a criterion for Ganea's conjecture on LS category is obtained using the generalised higher Hopf invariants, which yields the main result of (Iwase, Bull. Lond. Math. Soc. 30 (1998), 623-634) for all the cases except the case when p = 2. As an application, conditions in terms of homotopy invariants of the characteristic maps are given to determine the LS category of sphere-bundles-over-spheres. Consequently, a closed manifold M is found not to satisfy Ganea's conjecture on LS category and another closed manifold N is found to have the same LS category as its 'punctured submanifold' N - {P}, P ε N. But all examples obtained here support the conjecture in (Iwase, Bull. Lond. Math. Soc. 30 (1998), 623-634).

UR - http://www.scopus.com/inward/record.url?scp=0036120054&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036120054&partnerID=8YFLogxK

U2 - 10.1016/S0040-9383(00)00045-8

DO - 10.1016/S0040-9383(00)00045-8

M3 - Article

AN - SCOPUS:0036120054

VL - 41

SP - 695

EP - 723

JO - Topology

JF - Topology

SN - 0040-9383

IS - 4

ER -