Exotic spheres and the Whitehead space

Translation of: Sphères exotiques et l’espace de Whitehead by Michael Weiss; C.R. Acad. Sc. Paris, vol. 303 (1986). Some serious misprints have been corrected.

I. Review: An obstruction theory. — Let Θ_{n} be the group of diffeomorphism classes of oriented smooth homotopy spheres of dimension n. Let Diff(𝕊^{n−1}) be the simplicial group of orientation preserving diffeomorphisms 𝕊^{n−1} → 𝕊^{n−1}. For n > 5, the natural homomorphism from π_{0}Diff(𝕊^{n−1}) to Θ_{n} is surjective by Smale’s h–cobordism theorem, and injective by Cerf’s pseudo–isotopy theorem. It is easily seen that π_{0}Diff(𝕊^{n−1}) ≅ π_{0}Diff^{∂}(I^{n−1}), where Diff^{∂}(I^{n−1}) is the simplicial group of diffeomorphisms I^{n−1} → I^{n−1} which restrict to the identity near the boundary ∂I^{n−1}. Summarizing, there is an isomorphism Θ_{n} ≅ π_{0}Diff^{∂}(I^{n−1}) for n > 5.

Question. — Given a class in π_{0}Diff^{∂}(I^{n−1}) and given an integer q with 0 ≤ q ≤ n − 1, is there a representative f of that class such that

(⋆) \begin{matrix}I^{n−1} & ⟶^{f} & I^{n−1} \\ ↓ & & ↓ \\ I^{q} & ⟶^{=} & I^{q} \end{matrix}

commutes ? (The vertical arrows are the standard projections.)

Let V_{q} ⊂ Diff^{∂}(I^{n−1}) be the simplicial group of all those f for which (⋆) commutes. Then

{1} = V_{n−1} ⊂ V_{n−2} ⊂ ... ⊂ V_{1} ⊂ V_{0} = Diff^{∂}(I^{n−1}) .

To describe the simplicial set V_{j}/V_{j+1} of left cosets of V_{j+1} in V_{j} we will use the following notation. For a smooth compact manifold M, let C(M) be the (simplicial) group of pseudoisotopies alias concordances of M; that is, diffeomorphisms M ×I → M ×I which are the identity near M ×{0} and ∂M × I. There are (almost obvious) homotopy equivalences

V_{j} ≃ Ω^{j}Diff^{∂}(I^{n−1−j}) , V_{j}/V_{j+1} ≃ Ω^{j}C(I^{n−2−j}) .

They lead to exact sequences

π_{j+1}C(I^{n−2−j}) ⟶ π_{0}V_{j+1} ⟶ π_{0}V_{j} ⟶ π_{j}C(I^{n−2−j})

and therefore to an obstruction theory for the study of π_{0}V_{0} ≅ Θ_{n}. For more details, see [2]. Now it is well known that the spaces C(I^{q}) are closely related to the algebraic K–theory of ℤ. More precisely, there are stabilisation maps

⋯ → C(I^{q−1}) → C(I^{q}) → C(I^{q+1} → ⋯

and a map

C(I^{∞}) := lim_{q} C(I^{q}) ⟶ Ω^{2}(BGL(∞, ℤ)^{+});

see [3], [4], [5]. Hence there is a connection between homotopy spheres and algebraic K–theory.

II. Tying the obstructions together. — To elucidate this connection, we will construct:

(i) a fibration p: E → 𝕊^{n−2} whose fibers have the homotopy type of C(I^{n−2}); (ii) an involution τ: E → E sucht that pτ = αp, where α is the antipodal involution on 𝕊^{n−2}; (iii) for each oriented smooth homotopy sphere, a homotopy class of equivariant sections of p.

Here are the details:

(i) The fibration. — Let 𝔻^{n−1} ⊂ ℝ^{n−1} be the unit disk. For each x ∈ 𝕊^{n−2} = ∂𝔻^{n−1} let

W (x) = {y ∈ 𝕊^{n−2} | ⟨x, y⟩ ≥ 0} ,

where ⟨ , ⟩ is the usual inner product. Let E_{x} be the group of diffeomorphisms f from 𝔻^{n−1} to 𝔻^{n−1} for which f(z) = z whenever z ∈ W (x). Then clearly E_{x} ≃ C(I^{n−2}).

(ii) The involution. — For each x ∈ 𝕊^{n−2} there is a surjection

pr_{x}: W (x) × I → 𝔻^{n−1}
(v, t) ↦ v − 2t⟨v, x⟩ · x.

For each f ∈ E_{α(x)} (where α(x) is the antipode of x) let ∂f: W (x) → W (x) be the restriction of f to W (x). Define ∂f × I: 𝔻^{n−1} → 𝔻^{n−1} by

(∂f × I)(pr_{x}(v, t)) := pr_{x}(∂f(v), t).

The map

E_{α(x)} ⟶ E_{x} ; f ↦ (∂f × I)^{−1} · f

is a homeomorphism. Letting x vary in 𝕊^{n−2}, one obtains an involution τ: E → E for which pτ = αp.

(iii) The equivariant section ψ(Σ^{n}) associated with a smooth homotopy sphere Σ^{n}. — We represent Σ^{n} by a diffeomorphism f: 𝔻^{n−1} → 𝔻^{n−1} which fixes ∂𝔻^{n−1} = 𝕊^{n−2} pointwise. For each x ∈ 𝕊^{n−2} we can view f as an element of the fiber E_{x} , by the very definition of E_{x}. The section ψ(Σ_{n}) of p: E → 𝕊^{n−2} obtained in this way is indeed equivariant. (It can be shown that the section is non–equivariantly nullhomotopic; hence it is a 2–primary invariant.)

Now let Wh(∗) = Wh^{DIFF}(∗) be the spectrum associated with the infinite loop space B^{2}C(I^{∞}). This comes with a canonical involution tw, independent of n, which we will regard as an operation of ℤ/2 = π_{1}(ℝP^{∞}) on Wh(∗).

Remark. — One can view ψ(Σ^{n}) as an element of

H_{n}(ℝP^{∞}; Wh(∗)^{tw}) = π_{n}(𝕊_{+}^{∞} ∧_{ℤ/2} Wh(∗)).

Sketch proof. — Let p̂: Ê → ℝP^{n−2} be the fibration obtained from p: E → 𝕊^{n−2} by passage to the orbit spaces Ê = E_{ℤ/2} and ℝP^{n−2} = (𝕊^{n−2})_{ℤ/2}. Then ψ(Σ^{n}) is a section of p̂, in other words a ‘twisted map’ from ℝP^{n−2} to C(I^{n−2}). This leads to a twisted map from ℝP^{n−2} to C(I^{∞}) by means of a stabilisation of the fibers of p̂. But C(I^{∞}) is a representing space for the representable cofunctor H^{0}( ; Σ^{−2}Wh(∗)). The twisted map ℝP^{n−2} → C(I^{∞}) therefore represents an element of H^{0}(ℝP^{n−2}; Σ^{−2}Wh(∗)^{tw′} where tw′: Wh(∗) → Wh(∗) is an appropriate involution (which depends on n). To complete the proof, one uses the homomorphisms

H^{0}(ℝP^{n−2}; Σ^{−2}Wh(∗)^{tw′}) ⟶^{≅} H_{n−2}(ℝP^{n−2}; Σ^{−2}Wh(∗)^{tw})
⟶^{≅} H_{n}(ℝP^{n−2}; Wh(∗)^{tw}) ⟶ H_{n}(ℝP^{∞}; Wh(∗)^{tw}) ;

the first of these is a Poincaré duality isomorphism, the second is a suspension isomorphism, and the third is induced by the inclusion ℝP^{n−2} ⊂ ℝP^{∞}.

For any positive integer q and sufficiently large n (larger than an integer n(q) depending on q), the following conditions are equivalent:

(i) the smooth homotopy sphere Σ^{n} can be obtained from a diffeomorphism f ∈ Diff^{∂}(I^{n−1}) making the square (⋆) above commutative;

(ii) the invariant ψ(Σ^{n}) ∈ H_{n}(ℝP^{∞}; Wh(∗)^{tw}) belongs to the image of the homomorphism induced by inclusion,

H_{n}(ℝP^{n−q−2}; Wh(∗)^{tw}) ⟶ H_{n}(ℝP^{∞}; Wh(∗)^{tw}).

Moreover (i) implies (ii) for all n > q.

The existence of an invariant having roughly the properties of ψ was conjectured by Bruce Williams (Notre Dame) in a more general setting, along with the theorem just below. A joint article is in preparation.

III. Calculations. — Recall that for n ≥ 4, the surgery obstruction group L_{n+1}(ℤ) is isomorphic to the bordism group of smooth compact stably framed manifolds M^{n+1} whose boundary ∂M is a homotopy sphere. The isomorphism is obtained by associating to such an M its signature divided by 8 if n + 1 = 4k, and its Kervaire invariant if n + 1 = 4k + 2. This description of L_{n+1}(ℤ) gives us a homomorphism

\begin{matrix}ψ∂: L_{n+1}(ℤ) & ⟶ & H_{n}(ℝP^{∞}; Wh(∗)^{tw}) \\ [M] & ↦ & ψ(∂M) . \end{matrix}

Waldhausen has obtained a map of spectra

λ: Wh(∗) ⟶ K(ℤ).

This map commutes with the standard involutions; note that K(ℤ) has a standard involution coming from the involutions GL(m, ℤ) → GL(m, ℤ) ; A ↦ (A^{t})^{−1}. Hence there is an induced homomorphism

λ: H_{n}(ℝP^{∞}; Wh(∗)^{tw}) ⟶ H_{n}(ℝP^{∞}; K(ℤ)^{tw}) .

Our goal is to describe the composite homomorphism

λψ∂: L_{n+1}(ℤ) ⟶ H_{n}(ℝP^{∞}; K(ℤ)^{tw}).

On needs to know that, for n = 4k − 1, there is a Poincaré duality isomorphism

H_{k}(ℝP^{n}; K(ℤ)^{tw}) ≅ H^{n−k}(ℝP^{n}; K(ℤ)^{tw}).

The element corresponding to 1 ∈ H^{0}(ℝP^{n}; K(ℤ)^{tw}) under this duality will be called the fundamental class [ℝP^{n}] ∈ H_{n}(ℝP^{n}; K(ℤ)^{tw}).

Theorem. — For x ∈ L_{n+1}(ℤ) one has

λψ∂(x) = \{ \begin{matrix}(signature of x) · [ℝP^{n}] & if n + 1 = 4k \\ 0 & otherwise \end{matrix}

in H_{n}(ℝP^{∞}; K(ℤ)^{tw}).

Suppose now that n = 4k − 1 (where k > 1) and let Σ^{n} be the Milnor exotic sphere. It can be regarded as the boundary of the generator of L_{4k}(ℤ) ≅ ℤ. Using the above theorem and easy calculations with real topological K–theory, one can estimate the order of ψ(Σ^{n}) in H_{n}(ℝP^{∞}; Wh(∗)^{tw}). It is divisible by 2^{2k−3} if k is even, and by 2^{2k−2} if k is odd. Using the Atiyah–Hirzebruch spectral sequence, one can deduce:

Corollary. — Let β(k) be the number of indices i such that 0 ≤ i ≤ 4k − 1 and π_{i}(Wh(∗)) has odd order (note that 1 is odd). Then β(k) ≤ 2k + 3 if k is even, and β(k) ≤ 2k + 2 if k is odd.

This result has also been obtained by Bökstedt and Waldhausen, see [1], with completely different methods.

A more detailed investigation of the Atiyah–Hirzebruch spectral sequence gives the following. Let f ∈ Diff^{∂}(I^{4k−2}) represent Milnor’s exotic sphere.

Corollary. — It is impossible to choose f in such a way that the diagram

I^{4k−2} ⟶^{f} I^{4k−2}
↓  ↓
I^{2}  ⟶^{=} I^{2}

(where the vertical arrows are the standard projections) commutes.

In fact one encounters a nonzero obstruction in π_{3}(Wh(∗)) ≅ ℤ/2. (There is no obstruction in π_{2}(Wh(∗)) because that group is zero by Cerf’s pseudoisotopy theorem.) It is surprising that such a simple description of the generator of π_{3}(Wh(∗)) ≅ ℤ/2 exists; until 1983, the structure of π_{3}(Wh(∗)) was unknown. See also [1].

References

[1] M. Bökstedt and F. Waldhausen, The map BG→A(⋆) →Q𝕊^{0}, in Proc 1983 Princeton conf. John Moore’s birthday, Ann. Math. Studies 113, Princeton University Press, 1987, pp. 418–431.

[2] A. Hatcher, Concordance spaces, higher simple homotopy theory, and applications, in Algebraic and geometric topology, (Proc. Symp. Pure Math., Stanford University 1976), Proc. Symp Pure Math. vol XXXII, Amer. Math. Soc., Providence RI, 1978, pp. 3–21.

[3] J.-L. Loday, Homotopie des espaces de concordances (d’après Waldhausen), in Séminaire Bourbaki 1977/1978, Springer Lect. Notes vol. 710, 1979, pp. 187–205.

[4] F. Waldhausen, Algebraic K-theory of spaces – a manifold approach, in Current trends in algebraic topology, 1981 London Ontario conf., Canadian Math. Soc. Conf. Proceedings 2, Part 1, Amer. Math. Soc., 1982, pp. 141–184.

[5] , Algebraic K-theory of spaces, in Proc. of 1983 Rutgers conf. on algebraic topology, Springer Lecture Notes vol. 1126, 1984, pp. 318–419.