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Connected sum preserves oriented homotopy spheres

Statement

Assume ${A C}_{ω}$ . For $n \geq 3$ , the oriented connected sum of two oriented smooth homotopy $n$ -spheres is again an oriented smooth homotopy $n$ -sphere.

Facts & Assumptions

Given: Oriented smooth homotopy $n$ -spheres $Σ_{1}, Σ_{2}$ with $n \geq 3$ , smoothly embedded oriented disks $D_{i} \subseteq Σ_{i}$ , and the oriented connected sum $Σ_{1} # Σ_{2}$ .

[A1]

Countable choice ${A C}_{ω}$ is assumed (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[L1]

The punctured manifold $P_{i} = Σ_{i} ∖ int D_{i}$ has boundary $S^{n - 1}$ . Excision and the pair sequence identify $H_{k} (Σ_{i}, P_{i})$ with $H_{k} (D_{i}, \partial D_{i})$ ; the oriented fundamental class maps to the relative disk generator. Van Kampen applies after enlarging the pieces by collars (Smooth homotopy sphere, Excision for singular homology, Long exact sequence of a pair, Seifert–van Kampen identifies the fundamental group with a group pushout).

[L2]

Under ${A C}_{ω}$ compact smooth manifolds have finite CW homotopy models (Compact smooth manifolds have finite CW models under countable choice). For a homology equivalence $f : X \to Y$ between simply connected finite CW complexes, cellular approximation and its finite mapping cylinder give a simply connected finite pair $(M_{f}, X)$ with zero relative homology (Cellular approximation for maps of CW pairs, Cellular mapping cylinders and relative cylinders are CW complexes, Long exact sequence of a pair). The pair is $1$ -connected; if it is $(r - 1)$ -connected, the choice-free relative Hurewicz comparison makes $π_{r} (M_{f}, X) = H_{r} (M_{f}, X) = 0$ (Relative Hurewicz comparison through a choice-free weak model). Induction and the relative homotopy sequence show that $f$ is weak, and finite Whitehead makes it a homotopy equivalence (Long exact sequence of relative homotopy groups, Whitehead theorem). This criterion uses no full choice.

[L3]

Mayer-Vietoris computes the homology of a union of two subspaces from the homology of the pieces and their intersection (Mayer–Vietoris sequence in singular homology), and van Kampen computes the fundamental group of a union with connected intersection (Seifert–van Kampen identifies the fundamental group with a group pushout).

Proof

technique · direct

1.1L1L2A1given

By [L1] the map $H_{n} (Σ_{i}) \to H_{n} (Σ_{i}, P_{i})$ is an isomorphism, since it sends the fundamental generator to the local disk generator. Exactness gives ${\tilde{H}}_{*} (P_{i}) = 0$ . Removing a disk leaves a path-connected manifold: paths entering the disk can be diverted along its connected boundary collar. Van Kampen for $P_{i}$ and the disk, thickened to an open cover, has simply connected overlap $S^{n - 1}$ for $n \geq 3$ , and gives $π_{1} (P_{i}) = π_{1} (Σ_{i}) = 0$ . Thus [L2], applied on finite models to $P_{i} \to {*}$ , makes $P_{i}$ contractible.

2.1step 1.1givenconstruct

Use the fixed orientation-reversing linear reflection between the disk-coordinate boundary spheres to glue $P_{1}$ and $P_{2}$ . This is the oriented connected sum; no arbitrary boundary diffeomorphism is substituted for that coordinate identification. Collar thickenings of the two pieces form an open cover whose overlap retracts to $S^{n - 1}$ .

3.1step 1.1step 2.1L3

Reduced Mayer–Vietoris for that cover gives ${\tilde{H}}_{k} (Σ_{1} # Σ_{2}) ≅ {\tilde{H}}_{k - 1} (S^{n - 1})$ for $k \geq 1$ , because both pieces are contractible. In degree zero the union is connected. Hence its integral homology is that of $S^{n}$ .

3.2step 1.1step 2.1L3

Van Kampen for the same open cover, with simply connected pieces and overlap, gives $π_{1} (Σ_{1} # Σ_{2}) = 0$ .

4.1step 3.1step 3.2L1L2construct

Choose an oriented coordinate disk $D$ in the connected sum $X$ and collapse $X ∖ int D$ to a point. The target is $D / \partial D ≅ S^{n}$ ; the map has degree one because it carries the local oriented disk generator to the sphere generator. By step 3.1 it is a homology equivalence. Transport it to the finite CW model of $X$ and use [L2] to obtain $X ≃ S^{n}$ .

5.1step 4.1∎

The oriented connected sum is therefore an oriented smooth homotopy $n$ -sphere.

Depends on

Used by

Dependency tree · two levels

68 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Michel Kervaire and John Milnor, Groups of Homotopy Spheres I, Annals of Mathematics 77 (1963), 504-537 (standard reference, not scraped)
Allen Hatcher, Algebraic Topology, Cambridge University Press 2002 (complete book) (standard reference, not scraped)