A FIBRATION FOR DIFFΣ^{n}

ERIK KJÆR PEDERSEN AND NIGEL RAY

0. Introduction

Let M^{n} be an oriented smooth manifold, and let Diff M^{n} be its group of orientation preserving diffeomorphisms. Let Diff(N^{n}, M_{0}^{n}) and Diff_{∗} M^{n} be the two subgroups (with the C^{∞} topology ) which consist of diffeomorphisms fixing M_{0}^{n} = M^{n} − D^{n} ( for some embedded disc D^{n}) and the base point ∗ ∈ D^{n} ⊂ M^{n} respectively.

It is a common procedure when studying π_{∗}(Diff M^{n}) to restrict attention to either one of the above subgroups; it is therefore of some interest to study the homotopy braid of the triple (Diff M^{n}, Diff_{∗} M^{n}, Diff(M^{n}, M_{0}^{n})).

In fact this is equivalent to considering the following ‘comparison diagram’ of principal fibrations, constructed by the usual techniques.

(0.1). Diagram.

\xymatrix{ & Diff_{∗} M^{n} \ar[rr]^{d} \ar[dd]^{j} &  & SO(n) \ar[dd]_{j_{P}} \ar[dr] \\ Diff(M^{n}, M_{0}^{n}) \ar@{-->}[ur]^{i′} \ar@{-->}[dr]_{i} &  &  &  & B Diff(M^{n}, M_{0}^{n}) \\  & Diff M^{n} \ar[dr]_{π} \ar[rr]^{e} &  & P τ(M^{n}) \ar@{-->}[ur] \ar[dl]^{π} \\  &  & M^{n}}

Here d is the derivative map at ∗, P τ(M^{n}) is the total space of the oriented smooth principal tangent bundle to M^{n}, e maps f ∈ Diff M^{n} to the derivative at ∗ of the restriction f|D^{n}, and π is evaluation at ∗.

The only fibration on display above which is not well documented is the lower horizontal one, i. e.

Diff(M^{n}, M_{0}^{n}) ⟶_{i} Diff M^{n} ⟶_{e} P τ(M^{n})  (F)

Its existence follows from the fibration

Diff(M^{n}, M_{0}^{n}) ⟶_{i} Diff M^{n} ⟶_{e′} E(D^{n}, M^{n}),

where E( , ) denotes the space of orientation preserving smooth embeddings on D^{n}. But E(D^{n}, M^{n}) may be identified with P τ(M^{n}) , since

E_{∗}(D^{n}, M^{n}) → E(D^{n}, M^{n}) → M^{n}

is a model for the principal tangent bundle of M^{n} via the derivative map.

The homotopy braid of (0.1) gives rise to two interlaced problems. Firstly, to what extent does the ‘linearization’ j_{P} determine the map j; and secondly, to what extent does d determine e?

We shall discuss the second of these questions in the special case of M^{n} an exotic sphere Σ^{n}. In this case, (F) generalizes an exact sequence described by R. Schultz [6]. In particular we study the difference between the boundary maps associated to d and to e, and reduce the detection of a certain class of ‘stable’ homotopy element so arising to an interesting, but apparently unsolved, problem in the homotopy groups of spheres.

Throughout, we shall write S_{β}^{n} for the exotic sphere given by an element β ∈ π_{n}(Top /O), n ≥ 7. Such β arises from an isotopy class of diffeomorphisms β ∈ Diff S^{n−1}, so any S_{β}^{n} can be represented as D_{0}^{n} ∪_{β} D_{c}^{n}, where 0 ∈ D_{0}^{n} is the basepoint, and D_{c}^{n} is the complementary disc.

We are grateful to Dick Lashof for a helpful letter.

1. The fibration

Our fibration (F) of §0 can be further simplified when M^{n} = S_{β}^{n}. For it has long been known that, whatever β, P τ(S^{n}) = SO(n + 1). It is most convenient to describe this fact by means of

(1.1). Lemma. There is a homeomorphism of degree 1, say h : S_{β}^{n} → S^{n}, such that the diagram

\xymatrix{S^{n} \ar[dr]^{τ(S^{n})} \\  & B SO(n) \\ S_{β}^{n} \ar[uu]^{h} \ar[ur]^{τ(S_{β}^{n})}}

homotopy commutes. Thus P τ(S_{β}^{n}) is homeomorphic to SO(n + 1) via an SO(n) equivariant map,

Proof. Since S_{β}^{n} is stably parallelizable, τ(S_{β}^{n}) lifts to S^{n}. We can choose the lift h to have degree 1 by appealing to the euler characteristic if n is even, and the Kervaire semicharacteristic if n is odd □ (1.2). Note. The resulting homeomorphism h̄ : P τ(S_{β}^{n}) → SO(n + 1) is defined only up to alteration by any map S_{β}^{n} → SO(n).

Proof. Since S_{β}^{n} is stably parallelizable, τ(S_{β}^{n}) lifts to S^{n}. We can choose the lift h to have degree 1 by appealing to the euler characteristic if n is even, and the Kervaire semicharacteristic if n is odd. □

We can now construct our special version of (0.1) as follows:

(1.3). Diagram.

\xymatrix{ & Diff_{∗} S_{β}^{n} \ar[rr]^{d} \ar[dd]^{j} &  & SO(n) \ar[dd]_{j_{P}} \ar[dr]^{∂′} \\ Diff(D^{n}, ∂) \ar@{-->}[dr]_{i} \ar[ur]^{i′} &  &  &  & B Diff(D^{n}, ∂) \\  & Diff S_{β}^{n} \ar[rr]^{ε} \ar[dr]_{π} &  & SO(n + 1) \ar[dl]^{π} \ar@{-->}[ur]_{g(β)} \\  &  & S^{n}}

Note that we have labeled h̄ ∘ e as ε, and the classifying map of ε as g(β). Of course j_{P} is precisely the standard inclusion, and the homotopy commutativity of the central square is assured by construction.

(1.4). Definitions. (i) Let W_{∗}(S^{n}) ⊂ π_{∗}(Diff S^{n}) be the graded subgroup Im i′_{∗}∩ker j_{∗}.

(ii) Let X_{∗}(S^{n}) ⊂ π_{∗}(SO(m + 1)) be the graded set of elements x with the property that π_{∗}(x) ≠ 0 ≠ g(β)_{∗}(x).

Thus W_{∗}(S_{β}^{n}) is a measure of the extent to which j_{P} fails to determine j, and X_{∗}(S_{β}^{n}) is a measure of the extent to which ∂′ fails to determine g(β). Also 0 ≠ x ∈ W_{∗}(S_{β}^{n}) yields (i′)^{−1}(w) ∈ X_{∗}(S_{β}^{n}).

We investigate X_{∗}(S_{β}^{n}) below. Note that, if k < n, then W_{k}(S_{β}^{n}) = 0 and X_{k}(S_{β}^{n}) = ∅. Also, W_{∗}( ) and X_{∗}( ) are defined for arbitrary M^{n}.

If S_{β}^{n} is the standard sphere S^{n}, then (1.3) ‘collapses’. For the symmetry of S^{n} allows splitting Diff S^{n} ⟵ SO(n + 1) of (F), which restricts to a splitting of the upper fibration. Thus W_{∗}(S^{n}) = 0 and X_{∗}(S^{n}) = ∅.

Hence the cardinalities of W_{∗}() and X_{∗}() in some sense reflect the asymmetry of S_{β}^{n}. We develop below a detection procedure for ‘stable’ elements in X_{∗}( ).

2. Detecting elements in X_{∗}()

We first summarize some information from [6] concerning the map ∂′ of (1.3). Before so doing, however, it is convenient to recall some familiar notation which will also be useful for the remainder of this section.

We shall write Top S^{n} for the group of orientation preserving homeomorphisms of S^{n}, so that we have the following commutative diagram:

\xymatrix{SO \ar[r] & S Top \ar[rr] &  & Top /O \ar[dddd] \\  &  & S Top(n) \ar@{-->}[ul] \ar[dl]_{k} \\ SO(n + 1) \ar[uu] \ar[dd] \ar[r]^{i} & Top S^{n} \ar[dr]_{j} \\  &  & Ω_{1}^{n+1}S^{n+1} \ar[dl] \\ SO \ar[r]^{J} & Ω_{1}^{∞}S^{∞} = SG \ar[rr] &  & G/O}

Here i is the standard inclusion, j is the suspension, and k is one-point compactification.

Now according to [1], the composite π_{∗}(S Top(n)) → π_{∗}(Top /O) is epic, so we may suppose that S_{β}^{n} is represented by a map S^{n} → S Top(n) (tantamount to choosing a framing for S_{β}^{n}); or, via k_{∗}, a map β̄ : S^{n} → Top S^{n}. Continuing around the diagram, we also obtain β̃ ∈ π_{n}(G/O) as the image of both β̄ and β. Moreover β̃ lies in the summand π_{n}(G)/ Im J.

We can now state

(2.1). Theorem. (R. Schultz). Given α ∈ π_{k}(SO(n)), then ∂′_{∗}α ≠ 0 in π_{k}(B Diff(D^{n}, ∂)) whenever β̃J(α) ≠ 0 in π_{n+k}^{S}/ Im J.

Our result concerning X_{∗}( ) is in the same spirit, and can be stated thus:

(2.2). Theorem. Given x ∈ π(SO(n + 1)), then x ∈ X_{k}(S_{β}^{n}) whenever β̃^{2} · H(x) ≠ 0 in π_{n+k}^{S}/ Im J.

Here H(x) ∈ π_{k}(S^{n}) is represented by the composite S^{k} ⟶^{x} SO(n + 1) ⟶^{π} S^{n} (π being the usual projection), which is proven in [2] to be the Hopf invariant (up to sign and suspension) of the element J(x) ∈ π_{k+n+1}(S^{n+1}).

Elements so detectable constitute a stable subset SX_{∗}(S_{β}^{n}) ⊂ X_{∗}(S_{β}^{n}). Unfortunately, we know of no non-zero classes of the form y^{2} ·H(x) in any stem mod Im J. The experts seem to regard this as a potentially accessible, but unsolved , problem of homotopy theory. Certainly, all x ∈ π_{n+t}(SO(n + 1)) give y^{2} · H(x) = 0 for all y when t is small.

There are two main steps involved in establishing (2.2): these follow below as (2.3) and (2.4).

The first includes generalizing a diagram of [6, p. 240]. Since in (1.3) we have set up a map ε : Diff S^{n} → SO(n + 1) (which depends on the choice of h̄ in (1.2)), it is important to relate ε with the standard inclusion of both Diff S^{n} and SO(n + 1) in Top S^{n}. This is done by

(2.3). Lemma. The following diagram is homotopy commutative:

\xymatrix{Diff S_{β}^{n} \ar@{^{(}->}[r] \ar[d]_{e} \ar[dd]_{ε} & Top S_{β}^{n} \ar[d]_{χ_{h}} \\ P τ(S^{n}) \ar[d]_{h̄} & Top S^{n} \ar[d]_{c_{β}} \\ SO(n + 1) \ar@{^{(}->}[r]^{i} & Top S^{n}}

Here χ_{h} is conjugation by the homeomorphism h, whereas c_{β} is the composition

Top S^{n} ⟶_{π×1} S^{n} × Top S^{n} ⟶_{β̄×1} Top S^{n} × Top S^{n} ⟶_{μ} Top S^{n}

where π projects a homeomorphism onto its value at 0 ∈ S^{n} = ℝ^{n}∪{∞}, and μ is composition of functions. Note that π ∘_{β} (f) = π(f) ∈ S^{n} for all f, and that c_{β} is a homeomorphism with inverse c_{−β}.

The proof of (2.3) proceeds by passing between three equivalent versions of P^{Top}τ(S^{n}), the oriented principal topological tangent bundle of S^{n}. These may be displayed by the commutative diagram of (principal) fibrations

\xymatrix{S Top(n) \ar[r]^{σ_{1}} \ar[d] & Top_{∗} S^{n} \ar[d] \ar[r]^{σ_{2}} & E_{∗}^{Top}(D^{n}, S^{n}) \ar[d] \\ P^{Top}τ(S^{n}) \ar[d]_{π} \ar[r]^{φ_{1}} & Top S^{n} \ar[d]_{π} \ar[r]^{φ_{2}} & E^{Top}(D^{n}, S^{n}) \ar[d]^{π′} \\ S^{n} \ar[r]^{1} & S^{n} \ar[r]^{1} & S^{n}}

The maps σ_{1} and φ_{1} are induced by compactification, and σ_{2} and φ_{2} be restricting a homeomorphism of ℝ^{n} ∪ {∞} to D^{n}.

To introduce the second ingredient in the proof of (2.2), let us return to our fibration of (1.3). Suppose that α : Y → SO(n + 1) is a map of some reasonable space into the base. If α does not factor through ε : Diff S^{n} → SO(n + 1) and does not lift to SO(n), then it represents a class with the properties we are seeking for X_{∗}(S_{β}^{n}) (in case Y is a sphere). We thus wish to discuss the obstruction to lifting α to Diff S_{β}^{n}.

We may assume without loss of generality that the suspension S^{1} ∧ Y is given as an open subset of some euclidean space ℝ^{n}; in other words as an open smooth manifold. Now let α^{∗}γ_{β} be the topological S^{n} bundle over S^{1} ∧ Y which arises by adjoining the composite

\xymatrix{Y \ar[r]^{α} & SO(n + 1) \ar[r]^{i} \ar@{-->}[dr]_{γ_{β}} & Top S^{n} \ar[d]^{c_{−β}} \\  &  & Top S^{n}}

Then if α lifts to Diff S_{β}^{n}, (2.3) tells us that the total space E(α^{∗}γ_{β}) admits a smoothing which restricts to β on each fibre.

In fact it is most useful to work universally, and to consider the case of Y = SO(n + 1) and α the identity map. Then E(γ_{β} can be constructed by first choosing the ‘core’ plus a single fibre, i. e. S^{n} ∪ C SO(n + 1), where the attaching map is π ∘ γ_{β} = π. To this we must further attach a cone on the join S^{n−1} ∗ SO(n + 1) by a suitable map η. We therefore have a cofibre sequence

S^{n} ∧ SO(n + 1) ⟶^{η} S^{n} ∪_{π} C SO(n + 1) ⟶^{θ} E(γ_{β})  (C)

But E(γ_{β}) is a topological manifold, fibered by S^{n}’s and over a smooth base. As such it admits a S Top(n) bundle of tangents along the fibres, say

Top_{τ_{F}} : E(γ_{β}) → BS Top(n).

Our aim is to determine the extent to which E(γ_{β}) admits a smoothing fibred by S_{β}^{n}’s, or equivalently to which it carries an n-plane bundle τ_{F}, agreeing with Top_{τ_{F}} topologically and restricting to τ(S_{β}^{n}) on each fibre.

Now from §1, τ_{β} : S_{β}^{n} → B SO(n) extends to some bundle τ̄_{β} over S_{β}^{n} ∪_{π} C SO(n+1). Thus we may construct τ_{F} at least over S^{n} ∪_{π} C SO(n + 1), by composing τ̄_{β} with h̄^{−1}.

Returning to our cofibration (C), we can consider η^{∗}τ_{F} over S^{n} ∧ SO(n + 1). This is topologically trivialized by the existence of Top_{τ_{F}} , so we have a map σ(β) : S^{n}∧SO(n+1) → Top(n)/O(n) which fits into the following homotopy commutative diagram

\xymatrix{S^{n} ∧ SO(n + 1) \ar[d]^{σ(β)} \ar[r]^{η} & S^{n} ∪_{π} C SO(n + 1) \ar[d]^{τ_{F}} \ar[r]^{θ} & E(γ_{β}) \ar[d]^{_{Top_{τ_{F}}}} \\ Top(n)/O(n) \ar[r] & B SO(n) \ar[r] & BS Top(n)}

So σ(β). which we shall confuse with its adjoint SO(n + 1) → Ω(Top(n)/O(n)), is the obstruction to extending τ_{F} over the whole of E(γ_{β}).

Thus in terms of our original α^{∗}γ_{β} we deduce that if α : Y → SO(n + 1) lifts to Diff S_{β}^{n} then the composite

Y ⟶^{α} SO(n + 1) ⟶^{σ(β)} Ω(Top(n)/O(n))

is null-homotopic.

We have shown only that this map is a necessary obstruction to lifting α. In the light of the celebrated Morlet equivalence B Diff(D^{n}, ∂) ≃ Ω(Top(n)/O(n)) (e. g. see [3]) it seems highly likely that σ(β) and g(β) of (1.3) are the same map. Note that on homotopy groups σ(β) induces a non-bilinear extension to π_{k}(SO(n + 1)) of the Milnor pairing ( ; β) : π_{k}(SO(n)) → π_{k+n}(Top /O).

For calculational purposes, and given the current state of the art, unstable results such as we have obtained are not especially helpful. We must therefore show

(2.4). Lemma. The stabilization

sσ(β) : SO(n + 1) ⟶^{σ(β)} Ω^{n}(Top(n)/O(n) → Ω^{n}(Top /O)

may be described as

SO(n + 1) ⟶_{γ_{β}} Top S^{n} ⟶_{J} Ω^{n+1}S^{n+1} ⟶_{Ω^{n+1}β} Ω^{n+1}B(Top /O)

This formula follows simply from stabilizing the bundles in our discussion above.

To complete the proof of (2.2), we must choose Y = S^{k} and α to represent a class x ∈ π_{k}(SO(n + 1)) such that π_{∗}(x) ≠ 0 in π_{k}(S^{n}). Then by (2.4), x ∈ X_{k}(S^{n}) if

S^{k} ⟶_{x} SO(n + 1) ⟶_{σ(β)} Ω^{n}(Top /O)

is not null-homotopic. The usual detection procedure for such a map is then to pass to Ω^{n}(G/O), and to compute its value in the summand π_{n+k}^{S}/ Im J ⊂ π_{k+n}(G/O).

In our case the maps involved can be unraveled to give sσ(β)_{∗}x module Im J as

S^{k} ⟶^{x} SO(n + 1) ⟶^{π×J} S^{n} × G ⟶^{−β̃×1} G × G ⟶^{∘} G ⟶^{∘β̃} Ω^{n}G.

This represents β̃ ∘ (J(x) − β̃(π ∘ x)) in π_{n+k}^{S}/ Im J. But π ∘ x = H(x), whilst Novikov [5] and Kosinski [4] have shown that β̃ ∘ J(x) ∈ Im J whenever k > \frac{1}{2}n + 1 (which is certainly the case here).

We can now deduce our detection formula (2.2) in the form

sσ(β)_{∗}x = ±β̃^{2} · H(x) in π_{n+k}^{S}/ Im J.

Note that if S_{β}^{n} bounds a parallelizable manifold, then β̃ = 0 by definition. So SX_{∗}(S_{β}^{n}) = ∅. We conclude with a result which is a more subtle version of this same fact

(2.5). Proposition. Let S′X_{∗}(S_{β}^{n}) be the intermediate set SX_{∗}(S_{β}^{n}) ⊂ S′X_{∗}(S_{β}^{n}) ⊂ X_{∗}(S_{β}^{n}) of elements detected by σ(β)_{∗}x ∈ π_{n+k}(Top /O). Then S′X_{∗}(S_{β}^{n}) = ∅ if S_{β}^{n} bounds a parallelizable manifold.

Proof. By choice, β ∈ π_{n+k}(B Top /O)) lifts to π_{n}(G/ Top). But localized at 2, G/ Top is a product of Eilenberg-MacLane spaces, and at odd primes is equivalent to BO. In either case β ∘ f = 0 for any f ∈ π_{k+n+1}(S^{n+1}). □

This may be one more way of saying that such S_{β}^{n}’s are the most symmetric of exotic spheres.

References

1. D. Burghelea and R. Lashof, The homotopy type of the space of diffeomorphisms II, Trans. Amer. Math. Soc. 196 (1974), 37–50.

2. M. Kervaire, An interpretation of G. Whitehead’s generalization of H. Hopf’s invariant, Ann. of Math. (2) 69 (1959), 345–365.

3. R. Kirby and L. Siebenmann, Foundational Essays on Topological Manifolds, Smoothings and Triangulations, Annals of Mathematics Studies, vol. 88, Princeton Univ. Press, 1977.

4. A. Kosinski, On the inertia group of π-manifolds, Amer. J. Math. 89 (1967), 227–248.

5. S. P. Novikov, Differentiable sphere bundles, Izv. Akad. Nauk SSSR, Ser. Mat. 29 (1965), 1–96.

6. R. Schultz, Improved estimates for the degree of symmetry of certain homotopy spheres, Topology 10 (1971), 227–235.

Matematisk Institut, Odense Universitet, 5000 Odense, Denmark

Mathematics Department, The University, Manchester M13 9PL, England