On characteristic classes of exotic manifold bundles

Given a closed simply connected manifold $M$ of dimension $2n\ge6$, we compare the ring of characteristic classes of smooth oriented bundles with fibre $M$ to the analogous ring resulting from replacing $M$ by the connected sum $M\sharp\Sigma$ with an exotic sphere $\Sigma$. We show that, after inverting the order of $\Sigma$ in the group of homotopy spheres, the two rings in question are isomorphic in a range of degrees. Furthermore, we construct infinite families of examples witnessing that inverting the order of $\Sigma$ is necessary.

stable normal bundle along BO n → BO. An oriented homotopy sphere Σ of dimension 2n has, up to homotopy, a unique such lift that is compatible with its orientation and hence defines a canonical class [Σ] ∈ Ω n 2n . As is common, we denote by η ∈ π 1 S the generator of the first stable homotopy group of the sphere spectrum.
If the product η ·[Σ] ∈ π 2n+1 MO n does not vanish, then the first map is nontrivial. Moreover, the converse holds for ≥ 2.
Note that H 1 (BDiff + (W ); Z) agrees with the abelianisation of the group π 0 Diff + (W ) of isotopy classes of diffeomorphisms of M. It follows from a result of Kreck [Kre79, Thm 2] that this group is finitely generated for 2n ≥ 6, so the previous theorem implies that H 1 (BDiff + (W ); Z) and H 1 (BDiff + (W ♯Σ); Z) cannot be isomorphic if the product η · [Σ] ∈ π 2n+1 MO n is nontrivial. From computations in stable homotopy theory, we derive the existence of infinite families of homotopy spheres for which this product does not vanish, hence for which the integral version of Theorem A fails in degree 1. Such examples exist already in dimension 8-the first possible dimension. Combining Theorem B with work of Kreck [Kre79], we also find Σ such that π 0 Diff + (W ) and π 0 Diff + (W ♯Σ) are nonisomorphic, but become isomorphic after abelianisation.
Remark. The final parts of Theorem B and Corollary C still hold for = 1 in most cases, but often fail for = 0 (see Remark 3.3).
By Theorem B, the first homology groups of BDiff + (W ) and BDiff + (W ♯Σ) are isomorphic after inverting 2. More generally, this holds withW replaced by any simply connected manifold, which raises the question of whether the failure for Theorem A to hold integrally is purely 2-primary. We answer this question in the negative by proving the following. It follows from a recent result of Kupers [Kup16, Cor. C] that H 3 (BDiff + (W ); Z) is finitely generated for 2n ≥ 6, so H 3 (BDiff + (W ); Z[ 1 2 ]) and H 3 (BDiff + (W ♯Σ); Z[ 1 2 ]) cannot be isomorphic if the first map in the sequence of Theorem D is nontrivial. As Θ 10 Z/6, this holds for four of the six homotopy 10-spheres by the second part of Theorem D.

Remark.
(i) As an application of parametrised smoothing theory, Dwyer-Szczarba [DS83] compared the homotopy types of the unit components Diff 0 (M) ⊂ Diff + (M) for different smooth structures of M. Their work in particular implies that for a closed manifold M of dimension d ≥ 5 and a homotopy sphere Σ ∈ Θ d , the spaces BDiff 0 (M) and BDiff 0 (M♯Σ) have the same Z[ 1 k ]-homotopy type for k the order of Σ ∈ Θ d , which results in an analogue of Theorem A for BDiff 0 (M) without assumptions on the degree. However, in general the (co)homology of BDiff + (M) and BDiff 0 (M) is very different and we do not know whether Theorem A holds outside the stable range in the sense of of Galatius-Randal-Williams. (ii) Our methods also show the analogues of Corollary E and the first part of Corollary C for homotopy instead of homology groups which implies the existence of Σ ∈ Θ 2n for which π i BDiff + (W ) and π i BDiff + (W ♯Σ) are not isomorphic for some i and all . 1. E P -T After a brief recollection on bordism theory and groups of homotopy spheres, we recall high-dimensional parametrised Pontryagin-Thom theory à la Galatius-Randal-Williams and prove Theorem A.
1.1. Bordism theory. Let θ : B → BO be a fibration. A tangential, respectively normal, θ -structure of a manifold M is a lift ℓ M : M → B of its stable tangent, respectively normal, bundle M → BO along θ , up to homotopy over BO. The collection of bordism classes of closed d-manifolds equipped with a normal θ -structure forms an abelian group Ω θ d under disjoint union (see e.g. [Sto68, Ch. 2] for details). By the classical Pontryagin-Thom theorem, this group is isomorphic to the dth homotopy group π d Mθ of the Thom spectrum Mθ associated to θ . Normal θ -structures of a manifold are in natural bijection to tangential θ ⊥ -structures, where θ ⊥ : B ⊥ → BO is the pullback of θ along the canonical involution −1 : BO → BO, so we do not distinguish between them.
1.2. Exotic spheres. The initial observation of Kervaire-Milnor's [KM63] classification of homotopy spheres is that the collection Θ d of h-cobordism classes of closed oriented d-manifolds with the homotopy type of a d-sphere forms an abelian group under taking connected sum. For d ≥ 5, this group can be described equivalently as the group of oriented d-manifolds homeomorphic to the d-sphere, modulo orientation-preserving diffeomorphisms. Kervaire-Milnor established an exact sequence of the form where bP d +1 ⊂ Θ d is the subgroup of homotopy spheres bounding a parallelisable manifold and coker( ) d = π d S/im( ) d is the cokernel of the stable -homomorphism : π d O → π d S to the stable homotopy groups of spheres. In particular, the group Θ d is finite. Noting that S ≃ M(fr : EO → BO), the ring π * S can be viewed equivalently as the ring Ω fr * of bordism classes of stably framed manifolds. The morphism Θ d → coker( ) d is induced by assigning a homotopy sphere Σ ∈ Θ d the class [Σ, ℓ] ∈ coker( ) d of its underlying manifold together with a choice of a stable framing, using that homotopy spheres are stably parallelisable, and the morphism Kerv : coker( ) d → Z/2 is the so-called Kervaire invariant. Kervaire-Milnor showed that the subgroup bP d +1 ⊂ Θ d is cyclic and computed its order in most cases: for d = 4k −1 and k ≥ 2 it is of order 2 2k−2 (2 2k−1 −1) num(|4B 2k |/k), where B 2k is the 2kth Bernoulli number (this uses Adams' [Ada66] computation of im( ) and Quillen's [Qui71] solution of the Adams conjecture) and for d = 4k + 1 the group bP d +1 has order 2 if Kerv d +1 = 0 and vanishes otherwise. Kervaire-Milnor showed that Kerv d 0 for d = 6, 14, Mahowald-Tangora [MT67] proved Kerv 30 0, Barratt-Jones-Mahowald [BJM84] established Kerv 62 0, Browder [Bro69] showed that Kerv d = 0 if d is not of the form 2 k − 2, and Hill-Hopkins-Ravenel [HHR16] proved that Kerv 2 k −2 = 0 for k ≥ 8, leaving 2 7 − 2 = 126 as the last unknown case.  Remark 1.2. Theorem 1.1 is the higher-dimensional analogue of a pioneering result for surfaces obtained by combining a classical homological stability result due to Harer [Har85] with the celebrated theorem of Madsen-Weiss [MW07].
Remark 1.3. Recent work of Friedrich [Fri17] can be used to strengthen Theorem 1.1 to manifolds that are not simply connected, but whose associated group ring Z[π 1 M] has finite unitary stable rank (cf. the remark in the introduction).
1.4. The path components of MTθ 2n . The Pontryagin-Thom correspondence identifies the group of path components π 0 MTθ 2n with the bordism group of closed smooth manifolds N of dimension 2n together with a map ℓ : N → B 2n and a stable isomorphism φ : T N s ℓ * θ * 2n γ 2n , where a weak self-equivalence [ ] ∈ π 0 hAut(θ 2n ) of B 2n over BSO(2n) acts on such a triple [N , ℓ, φ] ∈ π 0 MTθ 2n by replacing ℓ with ℓ (see e.g. [Ebe13, Thm A.2]). These bordism classes carry a canonical orientation induced from the universal orientation of γ 2n by means of φ and ℓ, and this orientation is preserved by the π 0 hAut(θ 2n )-action. Note that the factorisation (1) provides a distinguished bordism class [M, ℓ M , id] ∈ π 0 MTθ 2n of the initial manifold M defining θ . There is a cofibre sequence which in particular identifies π −1 MTθ d +1 with the bordism group Ω θ ⊥ d (see [GTMW09,Ch. 3,5]). For d = 2n and d = 2n + 1, this induces a morphism of the form whose source and target are naturally acted upon by π 0 hAut(θ 2n ); the action on the target fixes the Z-summand and changes the θ -structure on Ω θ ⊥ 2n .
Justified by this, we identify π 0 MTθ 2n as a subgroup of Z ⊕ Ω θ ⊥ 2n henceforth. The path components of (Ω ∞ MTθ 2n )/ /hAut(θ 2n ) agree with the quotient and the path component hit by the parametrised Pontryagin-Thom map 1.5. Proof of Theorem A. By obstruction theory, the tangential n-type of a closed oriented 2n-manifold M depends only on the manifold M\ int(D 2n ) obtained by cutting out an embedded disc D 2n ⊂ M. 1 This implies that the tangential n-type of the connected sum M♯Σ of M with a homotopy sphere Σ ∈ Θ 2n has the form for the same B and θ as for M. Here ℓ M ♯Σ is the unique (up to homotopy) extension to M♯Σ of the restriction of ℓ M to M\ int(D 2n ), where D 2n ⊂ M is the disc at which the connected sum was taken. The targets of the parametrised Pontryagin-Thom maps of Theorem 1.1 for M and M♯Σ thus agree, and both maps induce an isomorphism on homology in a range of degrees onto the respective path components hit. However, different path components of the target space will usually have nonisomorphic homology; nonetheless, we have the following lemma comparing the two components in question.
Lemma 1.5. For a closed, oriented 2n-manifold M and Σ ∈ Θ 2n , there is a zig-zag between inducing homology isomorphisms with Z[ 1 k ]-coefficients, k being the order of Σ. Proof. The stable oriented tangent bundle Σ → BSO of Σ lifts to a unique tangential θstructure ℓ Σ of Σ and, by obstruction theory, the standard bordism between the connected sum M♯Σ and the disjoint union of M and Σ extends to a bordism respecting the θ -structure. This gives [M♯Σ, 2n and by the same argument, we obtain a morphism Θ 2n → Ω θ ⊥ 2n sending Σ ∈ Θ 2n to [Σ, ℓ Σ ], from which we conclude that the order of [Σ, ℓ Σ ] ∈ Ω θ ⊥ 2n divides k. As the Euler characteristics of M and M♯Σ agree, we have to the same path component, denoted Ω ∞ k ·(M, ℓ M ) MTθ 2n . As observed above, the homotopy sphere Σ has a unique θ -structure compatible with its orientation, so the class [Σ, ℓ Σ ] in Ω θ ⊥ 2n is fixed by the action of hAut(θ ), which in turn implies Stab(M, ℓ M ) = Stab(M♯Σ, ℓ M ♯Σ ) ⊂ hAut(θ 2n ). Since multiplication by k in Ω ∞ MTθ 2n is hAut(θ 2n )-equivariant, we obtain an induced zig-zag The corresponding zig-zag between the respective path components of Ω ∞ MTθ 2n before taking homotopy quotients is given by multiplication by k, so induces an isomorphism on homology with Z[ 1 k ]-coefficients. The claim now follows from a comparison of the Serre spectral sequences of the homotopy quotients, together with the equivalences (5).
By taking connected sums with Σ ∈ Θ 2n and its inverse, one sees that the genera of M and M♯Σ coincide, so Theorem A is a direct consequence of Theorem 1.1 and Lemma 1.5.
1 In fact, it only depends on the n-skeleton of the manifold.

T
In this section, we examine the homotopy fibre sequence 2.1. Triviality of the collar twist. As indicated by the following lemma, the collar twist serves as a measure for the degree of linear symmetry of the underlying manifold.
Lemma 2.2. Let M be a closed oriented d-manifold that admits a smooth orientation-preserving action of SO(k) with k ≤ d. If the action has a fixed point * ∈ M whose tangential representation is the restriction of the standard representation of SO(d) to SO(k), then the long exact sequence induced by the fibre sequence (6) reduces to split short exact sequences In particular, the collar twist is trivial on homotopy groups in this range.
Proof. On the subgroup SO(k) ⊂ SO(d), the SO(k)-action on M provides a left-inverse to the derivative map der : Diff + (M, * ) → SO(d). As the inclusion SO(k) ⊂ SO(d) is (k − 1)connected, we conclude that the derivative map is surjective on homotopy groups in degree k − 1 and split surjective in lower degrees, which implies the result.
The action of SO(d) on the standard sphere S d by rotation along an axis satisfies the assumption of the lemma, so the collar twist of S d is trivial on homotopy groups up to degree d − 1. In fact, in this case it is even nullhomotopic. Another family of manifolds that admit a smooth action of SO(k) as in the lemma is given by the -fold connected sums an alternative description, kindly pointed out to us by Jens Reinhold: consider the action of SO(n) on S n × S n by rotating the first factor around the vertical axis. Both the product of the two upper hemispheres of the two factors and the product of the two lower ones are preserved by the action and are, after smoothing corners, diffeomorphic to a disc D 2n , acted upon via the inclusion SO(n) ⊂ SO(2n) followed by the standard action of SO(2n) on D 2n . Taking the -fold equivariant connected sum of S n × S n using these discs results in an action of SO(n) on W as in Lemma 2.2 and thus has the following as a consequence.

Detecting the collar twist in bordism.
Recall that a manifold M is called (stably) nparallelisable if it admits a tangential n -structure for the n-connected cover BO n → BO. This map factors for n ≥ 1 over BSO and obstruction theory shows that there is a unique (up to homotopy) equivalence BO n ⊥ ≃ BO n over BSO, so tangential and normal nstructures of a manifold M are naturally equivalent. For oriented manifolds M, we require that n -structures on M are compatible with the orientation; that is, they lift the oriented stable tangent bundle M → BSO of M. Another application of obstruction theory shows that the map BSO × BSO → BSO classifying the external sum of oriented stable vector bundles is up to homotopy uniquely covered by a map BO n × BO n → BO n turning MO n into a homotopy commutative ring spectrum.
In the following, we describe a method to detect the nontriviality of certain collar twists. For this, we restrict our attention to oriented manifolds M that are (n − 1)-connected, nparallelisable, and 2n-dimensional. These manifolds have by obstruction theory a unique For an oriented (n − 1)-connected n-parallelisable 2n-manifold M, define morphisms as follows: a class [φ] ∈ π i BDiff(M, D 2n ) classifies a smooth fibre bundle together with the choice of a trivialised D 2n -subbundle S i × D 2n ⊂ E φ . The latter induces a trivialisation of the normal bundle of S i = S i × {0} in E φ . Using this, a given n -structure on E φ induces an n -structure on the embedded S i and conversely, obstruction theory shows that every n -structure on S i is induced by a unique n -structure on E φ in this manner. We can hence define Φ i by sending a homotopy class [φ] to the total space E φ of the associated bundle, together with the unique n -structure extending the canonical n -structure on S i induced by the standard stable framing of S i . Lemma 2.6. The following diagram is commutative the left vertical map being the -homomorphism followed by the map from framed to nbordism, and the bottom map the multiplication by [M, ℓ M ] ∈ Ω n 2n .
Proof. Recall from the beginning of the chapter that the map t is given by twisting a collar [0, 1] × S 2n−1 ⊂ M\ int(D 2n ). The composition of t * with Φ i maps a class in π i SO(2n), represented by a smooth map φ : S i → SO(2n), to the bordism class of a certain manifold E t (φ) , equipped with a particular n -structure. The manifold E t (φ) is constructed using a clutching function that twists the collar using φ and is constant outside of it. Its associated n -structure is the unique one that extends the canonical one on S i via the given trivialisation of its normal bundle. Untwisting the collar using the standard SO(2n)-action on the disc D 2n yields a diffeomorphism E t (φ) S i × M, which coincides with the twist Indeed, if there is a nontrivial element in the subgroup im( ) i · [M, ℓ M ] of π 2n+i MO n , then Lemma 2.6 implies that the collar twist of M is nontrivial on homotopy groups in degree i. The -fold connected sum W is the boundary of the parallelisable handlebody ♮ (D n+1 × S n ) and is thus trivial in framed bordism and so also in n -bordism, giving in Ω n 2n for all Σ ∈ Θ 2n . This proves the first part of the following proposition.
Proposition 2.7. Let Σ ∈ Θ 2n be a homotopy sphere and ≥ 0. If the subgroup is nontrivial for some i ≥ 1, then the abelianisation of the morphism is nontrivial. Assuming 2n ≥ 6, the converse holds for i = 1 and ≥ 2.
Proof. We are left to prove the second claim. The group π 1 BDiff(W ♯Σ, D 2n ) is the group of isotopy classes of orientation-preserving diffeomorphisms of W ♯Σ that fix the embedded disc D 2n pointwise. Letting such diffeomorphisms act on the middle-dimensional homology group H n (W ♯Σ; Z) provides a morphism π 1 BDiff(W ♯Σ, D 2n ) → Aut(Q W ♯Σ ) to the subgroup Aut(Q W ♯Σ ) ⊂ GL(H n (W ♯Σ; Z)) of automorphisms that preserve Wall's [Wal62b] quadratic form associated to the (n − 1)-connected 2n-manifold W ♯Σ. Together with the morphism (7) in degree 1, this combines to a map which induces an isomorphism on abelianisations by [Kra19, Cor. 2.4, Cor. 4.3 (i), Thm 4.4], assuming ≥ 2 and 2n ≥ 6. 3 The isotopy classes in the image of π 1 SO(2n) → π 1 BDiff(W ♯Σ, D 2n ) are supported in a disc and hence act trivially on homology. Consequently, the collar twist is for ≥ 2 nontrivial on abelianised fundamental groups if and only if it is nontrivial after composition with π 1 BDiff(W ♯Σ, D 2n ) → Ω n 2n+1 , so the claim is a consequence of Lemma 2.6 and the discussion prior to this proposition.
Remark 2.8. For = 1, the second part of Proposition 2.7 remains valid for n even and for n = 3, 7, since (8) still induces the abelianisation in these cases by [Kra19, Thm 4.4, Lem. 3.9]. The cited results moreover show that (8) no longer induces the abelianisation for = 1 and n 1, 3, 7 odd, but they can be used in combination with [Kre79, Lem. 4, Thm 3 c)] to show that the conclusion of the previous proposition is valid in these cases nevertheless, at least when assuming a conjecture of Galatius-Randal-Williams [GRW16, Conj. A]. For = 0, the second part of the proposition fails in many cases, which can be seen from Proposition 2.12 and its proof (cf. Remark 2.13).
(i) Since the image of the stable -homomorphism is cyclic, the condition in the previous is equivalent to the nonvanishing of a single class, e.g. η · [Σ, ℓ Σ ] ∈ π 2n+1 MO n for i = 1 or ν · [Σ, ℓ Σ ] ∈ π 2n+3 MO n for i = 3. (ii) By obstruction theory, the sphere S i has a unique n -structure for i ≥ n + 1, so the left vertical morphism in the diagram of Lemma 2.6 is trivial in this range. Hence, the morphism π i BDiff(M, D 2n ) → Ω n 2n+i can only detect the possible nontriviality of the collar twist in low degrees relative to the dimension. Another consequence of this uniqueness is that the image im( ) i of the -homomorphism is contained in the kernel of the canonical morphism from framed to n -bordism for i ≥ n + 1. In what follows, we attempt to detect nontrivial elements in the subgroup for homotopy spheres Σ ∈ Θ 2n by mapping MO n to ring spectra whose rings of homotopy groups are better understood. It is known to contain many periodic families of nontrivial products with η ∈ π 1 S and ν ∈ π 3 S which provide infinite families of homotopy spheres Σ for which (9) is nontrivial. The following proposition carries out this strategy in the lowest dimensions possible.
Proposition 2.11. In the following cases, there exists Σ ∈ Θ 2n for which the subgroup im( ) i · [Σ, ℓ Σ ] of π 2n+i MO n is nontrivial at the prime p: is nontrivial at the prime p for all ≥ 0.
Proof. During the proof, we make use of various well-known elements in π * S, referring to [Rav86] for their definition (c.f. Thm 1.1.14 loc. cit.). From the discussion above, we conclude in particular that all elements in π 2n S for 2n as in one of the three cases of the claim can be represented by homotopy spheres. The elements ε = ν − ησ ∈ π 8 S and κ ∈ π 14 S, respectively, generate coker( ) in these dimensions and give rise to 192-periodic families in π * S at the prime 2 whose elements are nontrivial when multiplied with η ∈ im( ) 1 and ν ∈ im( ) 3 , respectively (see e.g. [HM14,Ch. 11]). All of these products are detected in tmf and hence are also nontrivial in MO n . This proves the first and the second case. To prove the third one, we make use of a 72-periodic family in π * S at the prime 3 generated by the generator β 1 ∈ π 10 S of the 3-torsion of coker( ) 10 . The products of the elements in this family with ν ∈ π 3 S are nontrivial and are detected in tmf (see e.g. [DFHH14,Ch. 13]). This proves the first part of the proposition. The second part is implied by the first by means of Proposition 2.7.
(i) For = 1, the conclusion of the second part of Proposition 2.12 still holds in most cases (see Remark 2.8), but has to fail for = 0 since π 1 BDiff(Σ, D 2n ) is abelian. (ii) Recall that π 1 BDiff(W ♯Σ, D 2n ) is the group of isotopy classes of diffeomorphisms of W ♯Σ that fix an embedded disc D 2n pointwise. The image of t * : π 1 SO(2n) → π 1 BDiff(W ♯Σ, D 2n ) is generated by a single diffeomorphism t(η), which is the higherdimensional analogue of a Dehn twist for surfaces. It twists a collar around D 2n by a generator in π 1 SO(2n) and is the identity elsewhere. Consequently, the morphism t * is trivial if and only if t(η) is isotopic to the identity, and it is trivial in the abelianisation if and only t(η) is isotopic to a product of commutators.

D
Evaluating diffeomorphisms of a closed oriented d-manifold M at a basepoint * ∈ M induces a homotopy fibre sequence whose base space is simply connected. Evidently, the rightmost space in this sequence is not affected by replacing M by M♯Σ for homotopy spheres Σ, and the following lemma shows that the same holds for the leftmost space as well. Proof. The group Diff(M♯Σ, D d ) can be described equivalently as the group of diffeomorphisms of M♯Σ\ int(D d ) that extend over the disc D d ⊂ M♯Σ by the identity. Manifolds of dimension d ≥ 5 with nonempty boundary are insensitive to taking the connected sum with a homotopy sphere, so there is a diffeomorphism M♯Σ\int(D d ) M\ int(D d ). This diffeomorphism does not necessarily preserve the boundary, but conjugation with it nevertheless induces an isomorphism as claimed.
The first map is trivial in the case of the standard sphere Σ = S 2n by Corollary 2.3, a fact which, combined with Lemma 3.1, gives isomorphisms π 1 BDiff(W ♯Σ, D 2n ) π 1 BDiff(W , D 2n ) π 1 BDiff + (W ) whose composition we denote by ψ . We arrive at an exact sequence Z/2 ψ t * −→ π 1 BDiff(W ) ψ −1 ι * −→ π 1 BDiff + (W ♯Σ) −→ 0, whose first map agrees up to isomorphism with the collar twist of W ♯Σ on fundamental groups. Abelianising this exact sequence implies Theorem B by virtue of Proposition 2.7. From work of Sullivan [Sul77,Thm 13.3], we know that the mapping class group π 0 Diff + (W ) is commensurable to an arithmetic group and hence finitely generated and residually finite. By a classical result of Malcev [Mal40], this implies that it is Hopfian; that is, every surjective endomorphism is an isomorphism. Therefore, π 1 BDiff(W ) and π 1 BDiff(W ♯Σ) are isomorphic if and only if π 1 SO(2n) → π 1 BDiff(W ♯Σ, D 2n ) is trivial, and the two respective abelianisations are isomorphic if and only if this morphism is trivial after abelianisation. Thus, combining Proposition 2.11 and 2.12 proves Corollary C.
Remark 3.3. The preceding proof also shows that the final parts of Theorem B and Corollary C are valid for = 0 and = 1 if Propositions 2.7 and 2.12 hold in these cases. This is the case for = 1 and most n, but fails for = 0, as discussed in Remarks 2.8 and 2.13. Proof of Theorem D. Since Θ 6 vanishes by [KM63], we may assume 2n ≥ 8. Employing the universal coefficient theorem and the fact that BSO(2n) has the homotopy type of a CWcomplex of finite type, the well-known calculation of H * (BSO(2n); Z[ 1 2 ]) (see e.g. [Sto68, p. 87]) implies that H * (BSO(2n); Z[ 1 2 ]) vanishes in degrees * ≤ 3 and is isomorphic to Z[ 1 2 ] in degree 4. From (10) and the Serre exact sequence with Z[ 1 2 ]-coefficients of the homotopy fibre sequence (6), one sees that the morphism nontrivial away from the prime 2. In the case 2n = 10, the group Θ 10 is isomorphic to [Wal62a] C. T. C. Wall, The action of Γ 2n on (n − 1)-connected 2n-manifolds, Proc. Amer. Math. Soc. 13 (1962