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The first relative k-invariant

The first relative k-invariant
The first relative k-invariant
Motivated by work on the homotopy classification of 4-manifolds with boundary, we define a relative k-invariant for pairs of spaces that are homotopy equivalent to CW pairs. We show that for such a pair (X,Y) with Postnikov 2-type X → P2(X), the relative k-invariant is the obstruction to the existence of a section Bπ1(X) → P2(X) extending Y → X → P2(X).Given CW pairs (X0,Y0) and (X1,Y1), as well as a map h: Y0 → Y1, we also prove that relative k-invariants provide a complete obstruction to constructing a map X(3)extends h and induces given isomorphisms on π1 and π2.1.
Conway, Anthony
8522231c-1da6-4cbe-926f-c064441bb872
Kasprowski, Daniel
44af11b9-4d22-49f2-a6a3-04009f45b075
Conway, Anthony
8522231c-1da6-4cbe-926f-c064441bb872
Kasprowski, Daniel
44af11b9-4d22-49f2-a6a3-04009f45b075

Conway, Anthony and Kasprowski, Daniel (2026) The first relative k-invariant. Homology, Homotopy and Applications. (In Press)

Record type: Article

Abstract

Motivated by work on the homotopy classification of 4-manifolds with boundary, we define a relative k-invariant for pairs of spaces that are homotopy equivalent to CW pairs. We show that for such a pair (X,Y) with Postnikov 2-type X → P2(X), the relative k-invariant is the obstruction to the existence of a section Bπ1(X) → P2(X) extending Y → X → P2(X).Given CW pairs (X0,Y0) and (X1,Y1), as well as a map h: Y0 → Y1, we also prove that relative k-invariants provide a complete obstruction to constructing a map X(3)extends h and induces given isomorphisms on π1 and π2.1.

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The first relative k-invariant - Accepted Manuscript
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Accepted/In Press date: 6 February 2026

Identifiers

Local EPrints ID: 510828
URI: http://eprints.soton.ac.uk/id/eprint/510828
PURE UUID: 5a0b81f5-271d-41b1-9788-82d4c8c6edac
ORCID for Daniel Kasprowski: ORCID iD orcid.org/0000-0001-5926-2206

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Date deposited: 22 Apr 2026 16:52
Last modified: 23 Apr 2026 02:15

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Contributors

Author: Anthony Conway
Author: Daniel Kasprowski ORCID iD

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