The degrees of maps between (n − 1)-connected (2n + 1)-dimensional manifolds and Poincar ́e complexes and their applications
The degrees of maps between (n − 1)-connected (2n + 1)-dimensional manifolds and Poincar ́e complexes and their applications
In this paper using homotopy theoretical methods, we study degrees of maps between (n − 1)-connected (2n + 1)-dimensional Poincare complexes. Necessary and sufficient algebraic conditions for the existence of mapping degrees between such Poincare complexes are established. These conditions allow us to construct explicitly, up to homotopy, all maps with a given degree.
As an application of mapping degrees, we consider maps between (n − 1)-connected (2n+1)-Poincare complexes with degree ±1, and give a sufficient condition when those are homotopy equivalences. This resolves a homotopy theoretical analogue of Novikov’s question when a map between manifolds of degree 1 is a homeomorphism. For low n, we classify, up to homotopy, torsion free (n − 1)-connected (2n + 1)-dimensional Poincare complexes.
mapping degree, highly connected manifolds and Poincar ́e complexes, homotopy theory, classification of Poincar ́e complexes
Grbic, Jelena
daaea124-d4cc-4818-803a-2b0cb4362175
Vučić, Aleksandar
8c274ce2-5721-4637-ac29-7948665317bf
Grbic, Jelena
daaea124-d4cc-4818-803a-2b0cb4362175
Vučić, Aleksandar
8c274ce2-5721-4637-ac29-7948665317bf
Grbic, Jelena and Vučić, Aleksandar
(2021)
The degrees of maps between (n − 1)-connected (2n + 1)-dimensional manifolds and Poincar ́e complexes and their applications.
Matematicheskii Sbornik.
(doi:10.1070/SM9436).
Abstract
In this paper using homotopy theoretical methods, we study degrees of maps between (n − 1)-connected (2n + 1)-dimensional Poincare complexes. Necessary and sufficient algebraic conditions for the existence of mapping degrees between such Poincare complexes are established. These conditions allow us to construct explicitly, up to homotopy, all maps with a given degree.
As an application of mapping degrees, we consider maps between (n − 1)-connected (2n+1)-Poincare complexes with degree ±1, and give a sufficient condition when those are homotopy equivalences. This resolves a homotopy theoretical analogue of Novikov’s question when a map between manifolds of degree 1 is a homeomorphism. For low n, we classify, up to homotopy, torsion free (n − 1)-connected (2n + 1)-dimensional Poincare complexes.
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Accepted/In Press date: 20 January 2021
e-pub ahead of print date: 21 October 2021
Keywords:
mapping degree, highly connected manifolds and Poincar ́e complexes, homotopy theory, classification of Poincar ́e complexes
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Local EPrints ID: 447011
URI: http://eprints.soton.ac.uk/id/eprint/447011
PURE UUID: c70dae56-f702-4e02-b4ab-6a39fa50a966
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Date deposited: 01 Mar 2021 17:34
Last modified: 06 Jun 2024 01:51
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Author:
Aleksandar Vučić
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