The derivative map for diffeomorphism of disks: An example

We prove that the derivative map $d \colon \mathrm{Diff}_\partial(D^k) \to \Omega^kSO_k$, defined by taking the derivative of a diffeomorphism, can induce a nontrivial map on homotopy groups. Specifically, for $k = 11$ we prove that the following homomorphism is non-zero: $$ d_* \colon \pi_5\mathrm{Diff}_\partial(D^{11}) \to \pi_{5}\Omega^{11}SO_{11} \cong \pi_{16}SO_{11} $$ As a consequence we give a counter-example to a conjecture of Burghelea and Lashof and so give an example of a non-trivial vector bundle $E$ over a sphere which is trivial as a topological $\mathbb{R}^k$-bundle (the rank of $E$ is $k=11$ and the base sphere is $S^{17}$.) The proof relies on a recent result of Burklund and Senger which determines those homotopy 17-spheres bounding $8$-connected manifolds, the plumbing approach to the Gromoll filtration due to Antonelli, Burghelea and Kahn, and an explicit construction of low-codimension embeddings of certain homotopy spheres.


Geometry & Topology 1 Introduction
The derivative map is a basic invariant of the diffeomorphism group of the k-disk; in fact the first-order approximation in the embedding calculus approach to the diffeomorphism group.While d Q W Diff @ .D k / Q ! .k SO k / Q , the rationalisation of d, is nullhomotopic, as we explain in Section 3, much less is known about the derivative map d integrally.For example, to the best of our knowledge, it was not yet known whether the map induced by d on homotopy groups, was ever nontrivial.Burghelea and Lashof showed that d vanishes for i D 0; 1.At odd primes p, they also showed that d D 0 provided i < k 3 and they made a conjecture equivalent to the claim that this holds for p D 2 as well [6,Conjecture,page 40].Burghelea and Lashof also report A'Campo informing them about a proof that d D 0 for i D 2 (however, a written proof has not appeared).
Using smoothing theory, or an explicit geometric construction we introduce here, the map d admits an interpretation as describing the normal bundle of certain homotopy spheres embedded in euclidean space.Combining this interpretation with recent results of Burklund and Senger and the refined plumbing construction of Antonelli, Burghelea and Kahn, we obtain a counterexample to the conjecture of Burghelea and Lashof.
In more detail, in [3; 4] Antonelli, Burghelea and Kahn constructed families of diffeomorphisms of the disk using a pairing W p SO q a ˝ q SO p b !aCbC1 Diff @ .D pCq a b 1 / for 0 Ä a Ä q and 0 Ä b Ä p, refining Milnor's plumbing pairing; see below.Now 8 SO 6 Š Z=24 (see [12, page 162]) and we have: Using Morlet's smoothing theory isomorphism, the derivative map d on k is identified with the boundary map To the best of our knowledge, this is the first example of a nontrivial vector bundle over a sphere which is known to be trivial as a topological R n -bundle.
where the map on the left is the tensor product of the canonical stabilisations.We now consider the homotopy where the last map is induced by the classifying map of the tangent bundle of the 11-sphere.
It remains to prove Lemmas 2.1 and 2.2.
.x; y/ 7 !.x;.x;y//; and A.OE / is represented by the homotopy sphere † nC1 ‰ obtained by gluing two copies of D nC1 along the boundary using the diffeomorphism ‰.Note also that the image of OE under the derivative map is represented by For technical reasons, we actually assume without loss of generality that the maps are the identity maps in a neighbourhood of the boundaries.

We construct an explicit embedding
‰ , compute the normal bundle of this embedding and show explicitly that it is obtained by clutching with d .
As might be expected, given that all our data is on disks (and trivial near the boundary of the disks), we actually produce an interesting embedding The desired embedding Ã is explicitly given by .x; y; t/ 7 !.x;˛.t/y; ˇ.t/ x .y/;t/: Here, ˛; ˇW OE0; 1 !OE0; 1 are smooth maps such that ˛.t/ D 1 for t < 0:6 and ˛.t/ D 0 for t > 0:9, and ˇ.t/ D ˛.1 t/.This is evidently a smooth embedding whose image we denote by S , and we let @S D Ã .@DnC1 /.Then @S @D nCkC1 : to see this, observe that if either the xor the t-coordinate is in the boundary, then the first or fourth coordinate of the image point is so, too.For each t 2 OE0; 1, then ˛.t/ D 1 or ˇ.t/ D 1.If y 2 @D k , then therefore either the second or the third component of the image point is in the boundary (or both).As @.D n k D k OE0; 1/ is the union of those points with at least one component in the boundary, this proves the claim.
We also note that the subset @S @D nCkC1 is in fact independent of (as is fixed to be the identity map near the boundary).Let us identify this image set @S with , where it is given by ‰.Therefore, we can glue two copies of D n k D k D k OE0; 1 along the boundary by the identity map to obtain S nCkC1 , and the embeddings Ã in one copy and Ã Id in the other glue together to form the desired embedding of Strictly speaking, one has to round the corners off to get an actual smooth embedding.This can easily be achieved, as is the identity in a neighbourhood of the boundaries.We omit spelling out the somewhat cumbersome details.
It remains to compute the normal bundle of the embedding.To do this, we first compute the differential to be given in each fibre by We obtain an explicit trivialisation of the normal bundle of this embedding via the fibrewise linear map covering Ã , For the other half-disk which produces the embedding of † nC1 ‰ into S nCkC1 , we obtain a trivialisation of the normal bundle by the same recipe, replacing by Id.We observe then that we obtain the global normal bundle by gluing these two explicitly chosen normal subbundles of TD nCkC1 along the boundary, where they coincide.The trivialisations differ precisely on the half-disk Ã.D n k D k f1g/, and there they differ by the derivative map d .On the other half-disk, the two trivialisations coincide.
Consequently, the normal bundle of the embedding Ã is obtained by clutching with d , precisely as claimed, and the lemma is proved.
Remark 2.3 It is tempting to hope that the explicit geometric construction of d as the normal bundle of the embedding Ã can be used to get some new information about d .On the other hand, the information obtained by the formulas in the proof given above seems rather limited.At least in the case where lies in the image of , we present Conjecture 3.1 below on d ı .
Proof of Lemma 2.2 Let A 18  8 denote the group of bordism classes relative boundary of 8-connected 18-manifolds with boundary a homotopy sphere, which are defined in [23,Section 17].Specifically, elements of A 18  8 are represented by compact oriented 8-connected 18-manifolds W with boundary a homotopy sphere, and W 1 is bordant to W 2 if there is an h-cobordism Y between their boundaries such that the closed manifold Furthermore, S 3 ./ stabilises to a generator of 8 SO, by Lemma 2.4 below.Thus, S˛W maps both x and y to a generator and so we may take y W D x C y.We now compute that where 2 denotes reduction mod 2.
Now, taking the homotopy sphere on the boundary defines a homomorphism (3) @W A 18 8 !‚ 17 : From the short exact sequence 0 !bP 18 .D Z=2/ !‚ 17 !coker.J 17 / !0 and [7, Theorem 1.4], we see that the image of the map @ from (3) consists of precisely 4 different elements, so the map @ is injective.Each of the bP -spheres is the boundary of a manifold P which satisfies S˛P D 0 and therefore ' P .yP / D 0 and it follows that the element W from above must map under @ to a non-bP -sphere, which then represents OEÁÁ 4 in view of [7,Theorem 1.4].Lemma 2.4 The map Z=24 Š 8 SO 6 ! 8 SO Š Z=2 is surjective.
Proof By [14, Theorem 1.4], .Z=2/ 3 Š 8 SO 8 ! 8 SO Š Z=2 is onto and therefore has a kernel of 4 elements.(We refer to [12] for the computation of the relevant homotopy groups.)On the other hand, .Z=2/2 Š 8 SO 7 !8 SO 8 is injective (its cokernel injects into 8 .S 7 / Š Z=2) and so has an image of 4 elements.These two subgroups do not coincide: Since the maximal number of pointwise linearly independent vector fields on S 9 is 1 [1, Theorem 1.1], the tangent bundle of S 9 defines an element in 8 SO 8 that is not in the image of 8 SO 7 but maps to 0 2 8 SO.Therefore, .Z=2/ 2 Š 8 SO 7 !8 SO is surjective and has a kernel of precisely two elements; similarly the image of Z=24 Š 8 SO 6 ! 8 SO 7 Š .Z=2/ 2 consists of precisely two elements (its cokernel injects into 8 S 6 Š Z=2), and we are left to show that these two subgroups do not agree.To see this, we consider the element a WD .2/ 7 Á 7 where .2/ 7 is a generator of Z Š 7 SO 7 and Á 7 W S 8 !S 7 is the nontrivial class: By [14, Theorem 1.4], .2/ 7 stabilises to an element divisible by 2 and so a is in the kernel of the stabilisation; and it does not lift to 8 SO 6 by the commutativity of the following diagram with exact rows: We conclude this section by giving the promised second proof of Lemma 2.1.To this end we recall from [18, Section 6] that a smoothing of S nC1 in S nCkC1 consists of a smooth manifold W and a PL homeomorphism H W W ! S nCkC1 , such that † WD H 1 .S nC1 / W is a smooth submanifold, and such that H is concordant to the identity smoothing of S nCkC1 ; and recall the group d k nC1 of concordance classes of such smoothings. 2We note that up to diffeomorphism, W is a standard sphere mapping to S nCkC1 by a PL homeomorphism concordant to the identity, so that elements of d k nC1 are represented by PL homeomorphisms H W S nCkC1 !S nCkC1 which are concordant to the identity (ie orientation-preserving).Note also that † is a homotopy .nC1/-sphere,oriented through the PL homeomorphism h WD H j † , which is smoothly embedded into S nCkC1 .

Diarmuid Crowley, Thomas Schick and Wolfgang Steimle
There are two obvious homomorphisms out of d k nC1 , the left one mapping the class of H to the diffeomorphism class of †, and the right one to the classifying map of the normal bundle of † S nCkC1 (where, as usual, we identify a homotopy sphere up to homotopy equivalence with a standard sphere using the given orientation).Then, Lemma 2.1 is clearly implied by the following result: Lemma 2.5 There exists a group homomorphism B W n k Diff @ .D k / !d k nC1 such that the following diagram commutes up to possible signs: Proof We recall the homotopy equivalence of Morlet (see [5,Theorem 4.4]) and consider the diagram (4) Here ‰ is the map which sends a homotopy sphere † to the element represented by the tangent PL microbundle of the mapping cylinder cyl.hW † !S nC1 / of an orientationpreserving PL homeomorphism h, along with its linear structure induced by the smooth structure of † on the † end of the cylinder and its canonical trivialisation at the S nC1 end.The map ‰ is an isomorphism by surgery theory; see eg [15,Theorem 6.48].The map d k nC1 !nC1 f PL k =O k is an isomorphism by [18,Corollary 6.7]: it is defined by sending the class of .H; h/W .S nCkC1 ; † nC1 / !.S nCkC1 ; S nC1 / to the normal block bundle cyl of cyl.h/ inside cyl.H / along with its linear reduction at the † nC1 end of the cylinder and its canonical trivialisation at the other end.Finally, the map z S is obtained from the fact that the inclusion PL !f PL is an equivalence [19, Corollary 5.5(ii)]; that is, there is no essential difference between stable PL (micro)bundles and stable block bundles.
We claim that the lower left square of ( 4) is commutative up to sign.To see this, we may assume, increasing k if necessary, that the normal block bundle cyl is given by a PL microbundle.Then, the sum of the two composites, applied to OE.H; h/, is represented by the direct sum microbundle T cyl.h/˚ cyl over cyl.h/ along with its linear reduction at the front end and its canonical trivialisation at the other end.But now, we have an isomorphism T cyl.h/ ˚ cyl Š T cyl.H /j cyl.h/ of microbundles which extends isomorphisms T † ˚ † S nCkC1 Š T S nCkC1 j † and T S nC1 ˚ S nC1 S nCkC1 Š T S nCkC1 j S nC1 of vector bundles.
Since H is PL isotopic to the identity (being an orientation-preserving PL homeomorphism of the sphere), we conclude that the sum of the two composite maps, applied to OE.H; h/, represents the zero element.
All other parts of this diagram commute up to possible signs: the commutativity of the squares on the right and of the triangle in the middle follows from the definitions.That M n ı D S ı M k follows from [8,Lemma 2.5], and that M n D ‰ ı C is proven in [8,Lemma 2.7].The lemma now follows by a diagram chase.

Concluding remarks
In this section we discuss some of the background to our results and state a conjecture about the map d ı .
(1) The homotopy fibre of d W Diff @ .D k / !k SO k is the H -space Diff fr @ .D k / of framing-preserving diffeomorphisms.It is the loop space of the classifying space BDiff fr @ .D k /, which features in the recent work of Kupers and Randal-Williams [13] on the rational homotopy groups of Diff @ .D k /.We see that d is rationally trivial because the Alexander trick implies that d becomes nullhomotopic after composition with the natural map k SO k !k SPL k .It is well known that .SO k / Q is Eilenberg-Mac Lane, detected by the suspensions of the rational Pontryagin classes and rational Euler class.Since these classes are defined on .
(2) The proof of Theorem 1.1 relies on the fact that the normal bundle of any embedding † ; ,! S 28 is nontrivial.Despite the elementary argument we give for this in Section 2, computing the normal bundle of an embedding of a homotopy sphere g W † nC1 ,! S nCkC1 is a subtle problem.Provided one is in the metastable range n < 2k 4, Haefliger [10] proved that the isotopy class of g depends only on the diffeomorphism type of †, so that, in particular, the normal bundle is independent of the choice of embedding.Hsiang, Levine and Sczarba [11] proved that the latter statement holds even for n < 2k 2, defined the homomorphism where n < 2k 2; and proved that 13  16 ¤ 0; ie the exotic 16-sphere embeds into S 29 with nontrivial normal bundle.Then Antonelli [2] made a systematic study of normal bundles of homotopy spheres in the metastable range, which includes the statement that 11  17 ¤ 0.
(3) Concerning A'Campo's claim that d vanishes for i D 2, we note that, since 13 16 ¤ 0, Lemma 2.1 entails that if A'Campo's claim holds, then the exotic 16-sphere does not lie in the image of the map AW 2 Diff @ .D 13 / !‚ 16 D Z=2.This is consistent with computations we have made for the refined plumbing pairing W 8 SO 6 ˝ 7 SO 8 ! 2 Diff @ .D 13 /; which show that A ı D 0, even though M W 8 SO 7 ˝ 7 SO 8 !‚ 16 is nontrivial, a statement which can be deduced from [20,Satz 12.1].
(4) Finally, we present a conjectural description of the homomorphism d ı W p SO q a ˝ q SO p b !pCq SO pCq a b 1 in purely homotopy-theoretic terms.
Let hW i SO j !i .S j 1 / be the map induced by the canonical projection SO j !S j 1 .For maps f W W ! X and f W Y !Z let f g W W Y !X Z be their join.Let @W mC1 .S k / !m SO k denote the boundary map in the homotopy long exact sequence of the fibration SO k !SO kC1 !S k .For compactness, we use the notation p 0 WD p b and q 0 WD q a and let 1 2 p SO q 0 and 2 2 q SO p 0 .Then we have h. 1 / 2 p .S q 0 1 /; h. 2 / 2 q S p 0 1 and h. 1 / h. 2 / 2 pCqC1 .S p 0 Cq 0 1 /; so that @ h. 1 / h. 2 / 2 pCq SO p 0 Cq 0 1 .
In addition, we have the J -homomorphisms J p;q 0 W p SO q 0 !pCq 0 S q 0 and J q;p 0 W q SO p 0 !qCp 0 S p 0 ; and we can suspend in the target of each of these to get the homomorphisms † a ı J p;q 0 W p SO q 0 !pCq S q and † b ı J q;p 0 W q SO p 0 !pCq S p : We then take compositions with the maps induced by i for i D 1; 2 and the inclusions i p 0 W SO p 0 !SO p 0 Cq 0 1 and i q 0 W SO q 0 !SO p 0 Cq 0 1 .Hence we have homomorphisms 2 W p SO q 0 † a ıJ p;q 0 !pCq S q 2 !pCq SO p 0 i p 0 !pCq SO p 0 Cq 0 1 ; 1 W q SO p 0 † b ıJ q;p 0 !pCq S p 1 !pCq SO q 0 i q 0 !pCq SO p 0 Cq 0 1 : Conjecture 3.1 Up to sign, the homomorphism d ı W p SO q 0 ˝ q SO p 0 !pCq SO p 0 Cq 0 1 is given by d . . 1 ; 2 // D @ h.
which has standard form near the boundary, and then obtain an embedding of † nC1 ‰ by gluing with a standard embedding of D nC1 into D nCkC1 in the appropriate way.
[6, this allows for our interpretation of[6, Conjecture, page 40]in terms of the derivative map).We conclude that the map SO 11 !SPL 11 is not injective on 16 .More specifically, if 11 W S 11 !BSO 11 represents the tangent bundle of the 11-sphere and f W S 17! S 11 represents the unique nontrivial homotopy class (see [21, Proposition 5.11]), we have:Corollary 1.2The pullback f 11 is a nontrivial vector bundle which becomes trivial as an R 11 -bundle, even when considered as a bundle with structure group SPL 11 .
Milnor famously gave examples of nontrivial vector bundles over Moore spaces, for example the Moore space M.Z=7; 7/ D S 7 [ 7 D 8 , which are trivial as R n -bundles [17, Lemma 9.1].These examples are stable bundles over 4-connected spaces and so the vector bundles are trivial as piecewise linear bundles too.
where ‚ nC1 is the group of homotopy .nC1/-spheres.The first map includes fibrewise diffeomorphisms of D n k D k into all diffeomorphisms, and the second C uses a diffeomorphism of D n S n as a datum to clutch two .nC1/-disksandmakeahomotopysphere.1Lemma2.1Forany OE 2 n k Diff @ .D k /, the homotopy sphere A.OE / 2 ‚ nC1 admits an embedding into R nCkC1 whose normal bundle is classified (up to possible sign) by d .OE / 2 n SO k Š nC1 BSO k .We will offer two proofs of this result; one by an explicit geometric construction and a more abstract one by the classification of smoothings through Rourke and Sanderson's theory of block bundles.M W p SO q ˝ q SO p ! ‚ pCqC1 :
To observe that this really describes the normal bundle, for dimension reasons we just have to check that the image of intersects the tangent bundle of S , ie the image of DÃ , trivially.It is clear that .v/can only be equal to a tangent vector of the form .0; ˛.t/w; ˇ.t/ d .w/;0/ for w 2 R k .This implies ˛.t/v D ˇ.t/ d .w/and ˇ.t/v D ˛.t/ d .w/; the two equations imply ˛.t/ 2 v D ˇ.t/ 2 v and finally (as ˛.t/ 2 C ˇ.t/ 2 > 0) then v D 0 and then also w D 0. It follows that the image of represents the normal bundle of S in D nCkC1 .
an 8-connected 19-manifold.Since the stabilisation map S W 8 SO 9 !8 SO D Z=2 is split surjective with kernel Z=2, from ˛W we obtain a quadratic map ' W W H 9 .W I Z/ !Z=2 with values in Z=2 D ker.S / by fixing a splitting of 8 SO 9 .The first component of (2) is the Arf invariant of ' W and we next define the second component.Let S˛W W H 9 .W I Z/ !Z=2 D 8 .SO/ be the composition of ˛W with the stabilisation map S above.Using [22, Lemma 2] again, we see that S˛W is a homomorphism.Define W 2 H 9 .W I Z=2/ to be the Poincaré dual of S˛W 2 Hom.H 9 .W I Z/; Z=2/ D H 9 .W I Z=2/ Š H 9 .W; @W I Z=2/:The second component of (2) is given by evaluating ' W on any integral lift y W of W .Let S 3 ./ 2 8 SO 9 be the image of 2 8 SO 6 under the inclusion SO 6 !SO 9 .By the commutativity of (1), † ; is the boundary of the Milnor plumbing W of S 3 ./ 2 8 SO 9 with itself, and we compute ' [22, ' W .y W //:Here ' W W H 9 .W I Z/ !Z=2 is a quadratic refinement of the mod 2 intersection form defined as follows.By[22, Lemma 2], representing an integral homology class by an embedded sphere and taking its normal bundle gives rise to a quadratic map ˛W W H 9 .W I Z/ ! 8 SO 9 : W .y W / as follows: with H 9 .W I Z/ D Z.x/ ˚Z.y/ the normal bundles obtained from representing x and y by embeddings are both given by S 3 ./.We conclude that ' W .x/ D ' W .y/.Moreover, we may use that in this basis the intersection form W of W has matrix 1 / h. 2 / C 1 .2 / C 2 . 1 /: We briefly discuss Conjecture 3.1 in light of Theorem 1.1 and Corollary 1.2.For 2 8 SO 6 a generator, h./ 2 8 S 5 Š s 3 is again a generator and we choose so that h./ D 5 .Hence Conjecture 3.1 gives d . .; // D @. 5 5 / C 2 ./. Now 16 S 8 Š .Z=2/ 4 , which entails that 2 ./ D 0 and the proof of Corollary 1.2 shows that d . .; // D @. 2 11 /.Since 5 5 D 2 11 , Conjecture 3.1 is consistent with Theorem 1.1 and Corollary 1.2, with both giving the same nonzero expression for d ı W 8 SO 6 ˝ 8 SO 6 !16 SO 11 .