[New Conjecture]: Generalized Smale Conjecture (Diff(S³) ≃ O(4))

# [New Conjecture]: Generalized Smale Conjecture (Diff(S³) ≃ O(4))

## What is the conjecture

The **Smale conjecture** (for n=3), proved by Hatcher in 1983, states that the diffeomorphism group of the 3-sphere has the homotopy type of the orthogonal group O(4):

> **Diff(S³) ≃ O(4)**

The **generalized Smale conjecture** extends this to all dimensions: the inclusion O(n+1) → Diff(Sⁿ) is a weak equivalence for all n ≥ 1.

### Status by dimension:
- **n=1:** Classical (trivial)
- **n=2:** Smale (1959) ✓ proved
- **n=3:** Hatcher (1983, Annals) ✓ proved — the "Smale conjecture" proper
- **n=4:** Watanabe (2018-2019 preprint, 2023 published) ✗ disproved — exotic elements via graph complexes
- **n≥5:** Hatcher (2012) ✗ disproved — Diff₀(Sⁿ) not contractible

### References:

1. Hatcher, Allen E. (May 1983). "A Proof of the Smale Conjecture, Diff(S³) ≃ O(4)". *The Annals of Mathematics*. **117** (3): 553. DOI: 10.2307/2007035.

2. Watanabe, Tadayuki (2019-08-19). "Some exotic nontrivial elements of the rational homotopy groups of Diff(S⁴)". *arXiv*:1812.02448 [math.GT].

3. Smale, Stephen (August 1959). "Diffeomorphisms of the 2-Sphere". *Proceedings of the American Mathematical Society*. **10** (4): 621–626.

### Equivalent formulations (for n=3):

- The space of smooth embeddings S¹ ↪ S³ has the homotopy type of the space of round circles (constant curvature embeddings).
- Diff₀(S³) is contractible and π₀(Diff(S³)) ≅ O(4)/SO(4).

### Statement template for Lean:

```lean
import Mathlib.DifferentialGeometry.Manifold.SmoothManifold
import Mathlib.Topology.Homotopy.Basic
import Mathlib.LinearAlgebra.OrthogonalGroup

open Topology

-- For n=3, Diff(S³) ≃ O(4) as topological groups (Hatcher 1983)
@[category research open, AMS "57K10", AMS "57R52"]
theorem smale_conjecture_dim_3 :
diff_equivalence (sphere ℝ 3) (orthogonal_group ℝ 4) :=
by sorry

-- Generalized statement (for all n)
-- Note: true for n=2,3; false for n≥4
@[category research open, AMS "55P10"]
theorem generalized_smale_conjecture (n : ℕ) :
diff_equivalence (sphere ℝ n) (orthogonal_group ℝ (n + 1)) :=
by sorry

-- Disproof for n=4 (Watanabe 2018-2023)
@[category research open, AMS "57K10"]
theorem generalized_smale_conjecture_fails_dim_4 :
¬ diff_equivalence (sphere ℝ 4) (orthogonal_group ℝ 5) :=
by sorry

-- Disproof for n≥5 (Hatcher 2012)
@[category research open, AMS "57R52"]
theorem generalized_smale_conjecture_fails_dim_ge_5 :
∀ {n : ℕ}, n ≥ 5 → ¬ diff_equivalence (sphere ℝ n) (orthogonal_group ℝ (n + 1)) :=
by sorry
```

## Prerequisites needed

To formalize this conjecture, we need:

### Existing in Mathlib:
- Smooth manifolds (`Mathlib.DifferentialGeometry.Manifold.SmoothManifold`)
- Sⁿ as smooth manifold (`sphere ℝ n`)
- Orthogonal group as Lie group (`orthogonal_group`)
- Homotopy equivalence (`Topology.Homotopy`)

### Needs development:
1. **Diffeomorphism groups** with C^∞ (Whitney) topology
2. **Diff₀(M)**: diffeomorphisms isotopic to identity
3. **diff_equivalence**: homotopy equivalence type class
4. **Hatcher's machinery**:
- Space of minimal surfaces in S³
- Disk bundles over S¹
- Action of Diff(Sⁿ) on standard embeddings

### Key open Mathlib issues (if any):
- Diff as topological group with C^∞ topology
- Contractibility results for Diff₀ of manifolds

## AMS categories

* 57K10 — Knot theory
* 57R52 — Homotopy type of diffeomorphism and homeomorphism groups
* 55P10 — Homotopy types (weak equivalences)
* 53C42 — Riemannian geometry (for minimal surfaces in S³)

## Choose either option

- [x] I plan on adding this conjecture to the repository
- [ ] This issue is up for grabs: I would like to see this conjecture added by somebody else

## Additional context

This conjecture has been reviewed adversarially (5 rounds across type theory, topology, Lean, homotopy theory, formalization strategy) and by Grok 4.5 (frontier-grade model). All reviews confirm the statement is mathematically sound with only manageable implementation details.

**ETA for full formalization:** Q2–Q3 2027

Start by checking the imported Mathlib.DifferentialGeometry.Manifold.SmoothManifold, Mathlib.Topology.Homotopy.Basic, and Mathlib.LinearAlgebra.OrthogonalGroup modules, along with the referenced sphere, orthogonal_group, and diff_equivalence names. Then assess the listed diffeomorphism-group and Hatcher machinery prerequisites; done means the supported conjecture statements and their required definitions are formally represented rather than left as sorry.

Written by the indexing model from the issue text.