2025/10/21 by Anthony Conway, Conway, Anthony, Daniel Kasprowski +1
Mathematics · #Advanced Operator Algebra Research #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2510.18796
openalex publication_date 2025/10/21 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
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 \hookrightarrow X → P2(X). Given CW pairs (X0,Y0) and (X1,Y1), as well as a map h \colon Y0 → Y1, we also prove that relative k-invariants provide a complete obstruction to constructing a map X0(3) ∪ Y0 → X1 that extends h and induces given isomorphisms on π1 and π2.