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The Milnor sphere $M_{2, - 1}$ is homeomorphic but not diffeomorphic to $S^{7}$

Statement

Assume the Axiom of Choice and countable choice. The manifold $M_{2, - 1}$ is homeomorphic to $S^{7}$ but is not diffeomorphic to it. More generally, for $h + j = 1$ the constructed sphere has $λ (M_{h, j}) = (h - j)^{2} - 1 (m o d 7),$ and a nonzero value obstructs a diffeomorphism to the standard sphere even without a prescribed orientation.

Facts & Assumptions

Given: Integers $h, j$ with $h + j = 1$ , $k = h - j$ , the disk bundle $W_{h, j} = D (ξ_{h, j})$ , the sphere bundle $M_{h, j}$ , and the standard sphere $S^{7}$ .

[A1]

AC and ${A C}_{ω}$ are assumed (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )).

[L1]

The relative Pontryagin square of the disk bundle is $q (W_{h, j}) = 4 ε k^{2}$ with $ε = h + j = 1$ , and the boundary middle form is the rank-one form $[ε]$ with $σ (W_{h, j}) = ε = 1$ (Relative Pontryagin square of the Milnor disk bundle, Middle form and signature of the Milnor disk bundle).

[L2]

The invariant $λ (M) = 2 q (W) - σ (W) m o d 7$ is well defined for fillings, invariant under orientation-preserving boundary diffeomorphisms and negated by orientation reversal (The Milnor lambda invariant is well defined modulo seven).

[L3]

For $h + j = \pm 1$ the manifold $M_{h, j}$ is homeomorphic to $S^{7}$ (The Milnor homotopy seven-spheres are homeomorphic to $S^{7}$ ).

Proof

technique · direct

1.1L1L2A1

By [L1] and [L2], for $h + j = 1$ the filling $W_{h, j}$ gives $λ (M_{h, j}) = 2 \cdot 4 k^{2} - 1 = 8 k^{2} - 1 \equiv k^{2} - 1 (m o d 7)$ , because $8 \equiv 1$ ; in particular for $(h, j) = (2, - 1)$ one has $k = 3$ and $λ (M_{2, - 1}) \equiv 9 - 1 = 8 \equiv 1 (m o d 7)$ .

2.1step 1.1L2

The standard sphere $S^{7}$ bounds the disk $D^{8}$ ; for this filling $H^{4} (D^{8}, \partial D^{8}; Z) = H^{4} (D^{8}; Z) = 0$ , so $q (D^{8}) = 0$ and $σ (D^{8}) = 0$ , giving $λ (S^{7}) = 0$ .

3.1step 2.1L2

If there were an orientation-preserving diffeomorphism $M_{2, - 1} \to S^{7}$ , [L2] would give $λ (M_{2, - 1}) = λ (S^{7}) = 0$ , contradicting $λ (M_{2, - 1}) = 1$ ; if the diffeomorphism reversed orientation, [L2] would give $λ (M_{2, - 1}) = - λ (S^{7}) = 0$ , the same contradiction; hence $M_{2, - 1}$ is not diffeomorphic to $S^{7}$ in either orientation.

4.1step 3.1L3

By [L3] the manifold $M_{2, - 1}$ is homeomorphic to $S^{7}$ because $2 + (- 1) = 1$ .

5.1step 3.1step 4.1∎

Combining steps 3.1 and 4.1, $M_{2, - 1}$ is homeomorphic but not diffeomorphic to $S^{7}$ , and for general $h + j = 1$ the computation of step 1.1 gives the stated congruence, whose nonzero value is an obstruction as asserted.

Depends on

Used by

Homeomorphism type does not determine smooth structure in dimension seven Counterexample
Two Milnor spheres with distinct congruence invariants Example
Scope of the finite Milnor-sphere calculation Remark

Dependency tree · two levels

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Sources

John Milnor, On Manifolds Homeomorphic to the 7-Sphere, Annals of Mathematics 64 (1956), 399-405 (standard reference, not scraped)
John Milnor, Lectures on the h-Cobordism Theorem, section 9, printed pp. 109-110 (standard reference, not scraped)

The Milnor sphere M2,−1M2,−1​ is homeomorphic but not diffeomorphic to S7S7