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The first relative k-invariant

Anthony Conway, Daniel Kasprowski

TL;DR

The paper introduces and develops the relative first $k$-invariant for pairs of spaces, providing an obstruction-theoretic framework that controls extensions of maps between Postnikov 2-types and sections of the Postnikov fibration. It builds a robust chain-level theory for augmented chain complexes and mapping cones, then defines the relative $k$-invariant for CW pairs and extends it to spaces up to homotopy equivalence with CW pairs. A central result equates the existence of a map between Postnikov 2-types with the equality of relative $k$-invariants under the induced actions on $\pi_1$ and $\pi_2$, enabling a precise, obstruction-based classification in homotopy theory and with applications to 4-manifolds with boundary. The work also proves naturality, independence from auxiliary choices, and a key vanishing property when the induced $\pi_1$-map is surjective, solidifying the relative $k$-invariant as a fundamental invariant in low-dimensional topology and beyond.

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 $X \to P_2(X)$, the relative $k$-invariant is the obstruction to the existence of a section $Bπ_1(X)\to P_2(X)$ extending $Y \hookrightarrow X \to P_2(X)$. Given CW pairs $(X_0,Y_0)$ and $(X_1,Y_1)$, as well as a map $h \colon Y_0 \to Y_1$, we also prove that relative $k$-invariants provide a complete obstruction to constructing a map $X_0^{(3)} \cup Y_0 \to X_1$ that extends $h$ and induces given isomorphisms on $π_1$ and $π_2$.

The first relative k-invariant

TL;DR

The paper introduces and develops the relative first -invariant for pairs of spaces, providing an obstruction-theoretic framework that controls extensions of maps between Postnikov 2-types and sections of the Postnikov fibration. It builds a robust chain-level theory for augmented chain complexes and mapping cones, then defines the relative -invariant for CW pairs and extends it to spaces up to homotopy equivalence with CW pairs. A central result equates the existence of a map between Postnikov 2-types with the equality of relative -invariants under the induced actions on and , enabling a precise, obstruction-based classification in homotopy theory and with applications to 4-manifolds with boundary. The work also proves naturality, independence from auxiliary choices, and a key vanishing property when the induced -map is surjective, solidifying the relative -invariant as a fundamental invariant in low-dimensional topology and beyond.

Abstract

Motivated by work on the homotopy classification of -manifolds with boundary, we define a relative -invariant for pairs of spaces that are homotopy equivalent to CW pairs. We show that for such a pair with Postnikov -type , the relative -invariant is the obstruction to the existence of a section extending . Given CW pairs and , as well as a map , we also prove that relative -invariants provide a complete obstruction to constructing a map that extends and induces given isomorphisms on and .
Paper Structure (8 sections, 20 theorems, 65 equations)

This paper contains 8 sections, 20 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Preparatory lemmas
  3. Relative $k$-invariants of chain complexes.
  4. Relative $k$-invariants of CW pairs.
  5. Set-up and lemmas
  6. The $k$-invariant of a CW pair
  7. Relative $k$-invariants of spaces
  8. Additional properties of relative $k$-invariants.

Key Result

Theorem 1.1

Let $(X_0,Y_0)$ and $(X_1,Y_1)$ be pairs of spaces that are homotopy equivalent to CW pairs, and let $c_1\colon X_1\to P_2(X_1)$ be the Postnikov $2$-type of $X_1$. For a map $h \colon Y_0 \to Y_1$, an isomorphism $u\colon \pi_1(X_0)\to \pi_1(X_1)$ with $u\circ(\iota_0)_*= (\iota_1)_* \circ h$, and

Theorems & Definitions (52)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • proof
  • Lemma 2.1
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 42 more