### Abstract

In this paper, we use information about the degrees of maps between manifolds to construct new functorial seminorms with interesting properties. In particular, we answer a question of Gromov by providing a functorial seminorm that takes finite positive values on homology classes of certain simply connected spaces. Our construction relies on the existence of simply connected manifolds that are inflexible in the sense that all their self-maps have degree −1, 0 or 1. The existence of such manifolds was first established by Arkowitz and Lupton; we extend their methods to produce a wide variety of such manifolds.

Original language | English |
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Pages (from-to) | 1453-1499 |

Number of pages | 48 |

Journal | Algebraic & Geometric Topology |

Volume | 15 |

Issue number | 3 |

DOIs | |

Publication status | Published - 19 Jun 2015 |

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### Keywords

- mapping degrees
- simply connected manifolds
- functorial seminorms on homology

### Cite this

*Algebraic & Geometric Topology*,

*15*(3), 1453-1499. https://doi.org/10.2140/agt.2015.15.1453

**Functorial seminorms on singular homology and (in)flexible manifolds.** / Crowley, Diarmuid; Löh, Clara.

Research output: Contribution to journal › Article

*Algebraic & Geometric Topology*, vol. 15, no. 3, pp. 1453-1499. https://doi.org/10.2140/agt.2015.15.1453

}

TY - JOUR

T1 - Functorial seminorms on singular homology and (in)flexible manifolds

AU - Crowley, Diarmuid

AU - Löh, Clara

N1 - Date of Acceptance 05/11/2014

PY - 2015/6/19

Y1 - 2015/6/19

N2 - A functorial seminorm on singular homology is a collection of seminorms on the singular homology groups of spaces such that continuous maps between spaces induce norm-decreasing maps in homology. Functorial seminorms can be used to give constraints on the possible mapping degrees of maps between oriented manifolds.In this paper, we use information about the degrees of maps between manifolds to construct new functorial seminorms with interesting properties. In particular, we answer a question of Gromov by providing a functorial seminorm that takes finite positive values on homology classes of certain simply connected spaces. Our construction relies on the existence of simply connected manifolds that are inflexible in the sense that all their self-maps have degree −1, 0 or 1. The existence of such manifolds was first established by Arkowitz and Lupton; we extend their methods to produce a wide variety of such manifolds.

AB - A functorial seminorm on singular homology is a collection of seminorms on the singular homology groups of spaces such that continuous maps between spaces induce norm-decreasing maps in homology. Functorial seminorms can be used to give constraints on the possible mapping degrees of maps between oriented manifolds.In this paper, we use information about the degrees of maps between manifolds to construct new functorial seminorms with interesting properties. In particular, we answer a question of Gromov by providing a functorial seminorm that takes finite positive values on homology classes of certain simply connected spaces. Our construction relies on the existence of simply connected manifolds that are inflexible in the sense that all their self-maps have degree −1, 0 or 1. The existence of such manifolds was first established by Arkowitz and Lupton; we extend their methods to produce a wide variety of such manifolds.

KW - mapping degrees

KW - simply connected manifolds

KW - functorial seminorms on homology

U2 - 10.2140/agt.2015.15.1453

DO - 10.2140/agt.2015.15.1453

M3 - Article

VL - 15

SP - 1453

EP - 1499

JO - Algebraic & Geometric Topology

JF - Algebraic & Geometric Topology

SN - 1472-2747

IS - 3

ER -