### Abstract

We show that in every codimension greater than 1, there exists a mod 2 homology class in some closed manifold (of sufficiently high dimension) that cannot be realized by an immersion of closed manifolds. The proof gives explicit obstructions (in terms of cohomology operations) for realizability of mod 2 homology classes by immersions. We also prove the corresponding result in which the word 'immersion' is replaced by 'map with some restricted set of multi-singularities'.

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

Pages (from-to) | 329-340 |

Number of pages | 12 |

Journal | Bulletin of the London Mathematical Society |

Volume | 45 |

Issue number | 2 |

Early online date | 14 Nov 2012 |

DOIs | |

Publication status | Published - Apr 2013 |

### Keywords

- self-intersections
- singularities
- immersions
- spaces

### Cite this

*Bulletin of the London Mathematical Society*,

*45*(2), 329-340. https://doi.org/10.1112/blms/bds090

**On realizing homology classes by maps of restricted complexity.** / Grant, Mark; Szucs, Andras.

Research output: Contribution to journal › Article

*Bulletin of the London Mathematical Society*, vol. 45, no. 2, pp. 329-340. https://doi.org/10.1112/blms/bds090

}

TY - JOUR

T1 - On realizing homology classes by maps of restricted complexity

AU - Grant, Mark

AU - Szucs, Andras

PY - 2013/4

Y1 - 2013/4

N2 - We show that in every codimension greater than 1, there exists a mod 2 homology class in some closed manifold (of sufficiently high dimension) that cannot be realized by an immersion of closed manifolds. The proof gives explicit obstructions (in terms of cohomology operations) for realizability of mod 2 homology classes by immersions. We also prove the corresponding result in which the word 'immersion' is replaced by 'map with some restricted set of multi-singularities'.

AB - We show that in every codimension greater than 1, there exists a mod 2 homology class in some closed manifold (of sufficiently high dimension) that cannot be realized by an immersion of closed manifolds. The proof gives explicit obstructions (in terms of cohomology operations) for realizability of mod 2 homology classes by immersions. We also prove the corresponding result in which the word 'immersion' is replaced by 'map with some restricted set of multi-singularities'.

KW - self-intersections

KW - singularities

KW - immersions

KW - spaces

U2 - 10.1112/blms/bds090

DO - 10.1112/blms/bds090

M3 - Article

VL - 45

SP - 329

EP - 340

JO - Bulletin of the London Mathematical Society

JF - Bulletin of the London Mathematical Society

SN - 0024-6093

IS - 2

ER -