Milnor’s Exotic 7-Spheres

Milnor’s Exotic 7-Spheres

Jhan-Cyuan Syu (許展銓) June, 2017

0 Introduction

This article is devoted to the explicit construction of Milnor’s exotic 7-spheres, together with some prerequisites needed in the construction, such as characteristic classes, cobordisms and signatures.

There will be a nature question rising immediately as soon as we define a smooth structure on a manifold: can a topological manifold admit two different smooth structure? That is, is it true that if two smooth manifolds are homeomorphic, then they are automatically diffeomorphic? The answer is that it depends!!

If the manifold is of dimension 1 or 2, then the answers are sure, proved by Tibor Radó. The case of dimension 3 was also proved to be true by Edwin E.

Moise. However, the first counterexample given by John Milnor in 1956 states that there are at least two different smooth structures on 7-sphere. Furthermore, John Milnor and Michel Kervaire proved that there are 28 oriented differentiable structures on 7-sphere (15 if without consideration of orientation) in 1963. As time passed by, Michael Freedman gave a non-standard differentiable structure on R⁴, known as exotic R⁴, in 1982. By the way, there are no different smooth structures onRⁿ for all n̸= 4. Even though exotic R⁴ has been constructed over 30 years ago, the existence of exotic 4-sphere is still an open problem now.

1 Fiber Bundles and Characteristic Classes

Given a real vector bundle ξ : E−→ M of rank 2n, one can give each fiber a complex^π structure in the following sense: J : E → E is a continuous map which maps each fiberR-linearly to itself and satisfies J ◦J(v) = −v for all v ∈ E. With this complex structure, the real vector bundle ξ can then be viewed as a complex vector bundle.

Conversely, given a complex vector bundle ξ, we can also view it as a real vector bundle, denoted by ξ_R, by ignoring the complex structure on each fiber and think of each fiber as a real vector space.

Let ξ : E −→ M be a complex vector bundle over a manifold M. We define^π ξ : ¯¯ E −→ M the complex conjugate vector bundle of ξ by the following way:^π¯

(i) ξ_R= ¯ξ_R.

(ii) The identity map i : E → ¯E is conjugate linear, i.e., i(cv) = ¯ci(v) for each c∈ C and v∈ E.

Now we are ready for the definition of our first characteristic class: Chern classes.

Definition. (Chern Classes)

Let ξ : E −→ M be a complex vector bundle of rank n (complex dimension) over a^π manifold M . The total Chern class of this bundle is

c(ξ) =∑

i≥0

c_i(ξ)∈ H²ⁿ(M,Z) satisfying

(i) c₀(ξ) = 1, ci(ξ)∈ H²ⁱ(M,Z) for all i and ci(E) = 0 for i > n.

(ii) Naturality: for any smooth map f : M → M^′ from M to another manifold M^′, c(f^∗ξ) = f^∗c(ξ).

(iii) Whitney sum formula: c(ξ⊕ η) = c(ξ) ⌣ c(η) for any complex vector bundle η over M .

(iv) c(γ) = 1 + g, where γ is the universal line bundle of C P^∞ and g ∈ H²(C P^∞) is the generator of the cohomology.

With definition of Chern classes, we define the Euler class e(ξ_R) = c_n(ξ) ∈ H²ⁿ(M,Z), where ξ is a n-dimensional complex vector bundle over M, and the total

Pontryagin class p(η) = ∑

i∈N0pi(η) with pi(η) = (−1)ⁱc2i(η ⊗ C) ∈ H⁴ⁱ(M,Z), where η is a real vector bundle over M .

There are two facts that indicate us how to compute Pontryagin classes from Chern classes.

Fact. If ξ is a real vector bundle, then ξ⊗ C ∼=_C ξ⊗ C; if ξ is a complex vector bundle, then ξ_R⊗ C ∼=_Cξ⊕ ¯ξ.

Fact. c_i( ¯ξ) = (−1)ⁱc_i(ξ).

Now we recall some basic definitions and properties of homotopy groups, which will frequently appear throughout the whole article.

Definition. (Homotopy Groups)

Define π_n(X, x₀) = {f : Sⁿ → X | f is continuous and f(t0) = x₀}/ ∼, where t0 is some fixed point of Sⁿ and the equivalence relation∼ is the homotopic equivalence.

The composition law is [f ] + [g] := [f + g].

For two continuous maps f, g : Sⁿ → X with f(t0) = g(t₀) = x₀, we de- fine (f + g) : Sⁿ → X to be the composition of two maps Ψ ◦ ρ, where ρ : Sⁿ → Sⁿ∨

Sⁿ ∼= Sⁿ×{t0}∪{t0}×Sⁿand Ψ : Sⁿ∨

Sⁿ→ X. The space Sⁿ∨

Sⁿis obtained by choosing an equator of Sⁿ passing through t0 and then identifying the equator as one point, so the resulting space has the isomorphism Sⁿ∨

Sⁿ ∼= Sⁿ×{t0}∪{t0}×Sⁿ and ρ : Sⁿ → Sⁿ∨

Sⁿ is defined by sending the upper sphere to Sⁿ×{t0}, the lower sphere to {t0} × Sⁿ and the equator to {t0} × {t0}. The map Ψ : Sⁿ∨

Sⁿ → X is given by Ψ({·} × {t0}) = f(·) and Ψ({t0} × {·}) = g(·).

We will next introduce two important examples of vector bundles, which both play crucial roles in the following story.

Example. (Hopf Quaternion Bundles)

Let H := {a + bi + cj + dk | a, b, c, d ∈ R} the quaternions. Let {q0,· · · , qn} be the standard basis of Hⁿ⁺¹. We now consider S⁴ⁿ⁺³. Indeed, S⁴ⁿ⁺³ can be embedded into Hⁿ⁺¹ ∼= R⁴ⁿ⁺⁴, i.e., S⁴ⁿ⁺³ ,→ R⁴ⁿ⁺⁴ ∼= Hⁿ⁺¹, so we can represent S⁴ⁿ⁺³ in quaternion coordinate:

S⁴ⁿ⁺³ :={(q0,· · · , qn)∈ Hⁿ⁺¹| |(q0,· · · , qn)|² =

∑n α=0

|qα|² = 1}.

Then there is a natural action of SU(2) ∼= S³ ,→ H on S⁴ⁿ⁺³ given by q· (q0,· · · , qn) := (qq₀,· · · , qqn)∈ S⁴ⁿ⁺³ ,→ Hⁿ⁺¹,

the left multiplication in quaternions by q ∈ H with |q| = 1. Hence we obtain a principal SU(2)-bundle with fiber SU(2) ∼= S³:

SU(2) S⁴ⁿ⁺³

S⁴ⁿ⁺³/ SU(2)

action

π

where S⁴ⁿ⁺³/ SU(2) ∼= S⁴ⁿ⁺³/ S3 ∼=H Pⁿ.

Let γ_n be the universal line bundle of H Pⁿ and define un:= c2(γnC) = e(γnR)∈ H⁴(H Pⁿ,Z).

By CW-decomposition ofH Pⁿ, one can conclude:

(i) H⁴ⁱ(H Pⁿ,Z) ∼= uⁱ_nZ.

(ii) Hⁱ(H Pⁿ,Z) = 0 if 4 - i.

So c(γ_nC) = 1 + c₁(γ_nC) + c₂(γ_nC) = 1 + u_n.

For p(γ_nR), notice that c₀(γ_nC) = 1, c₂(γ_nC) = u and c_i(γ_nC) = 0 for i ̸= 0, 2 and compute

p(γ_nR) = ∑

i∈N0

p_i(γ_nR) = ∑

i∈N0

(−1)ⁱc_2i(γ_nR⊗ C) = ∑

i∈N0

(−1)ⁱc_2i(γ_nC⊕ ¯γnC)

= c0(γnC)c0(γnC)− c0(γnC)c2(γnC)− c2(γnC)c0(γnC) + c2(γnC)c2(γnC)

= 1− un− un+ u²_n = 1− 2 un+ u²_n.

Example. (SO(4)-Vector Bundles over S⁴ of Rank 4) We first recall a useful lemma:

Lemma. The G-bundles over S^k are determined by πk−1(G).

There is an intuitive explanation of this lemma: View S^k as the union of the upper sphere and the lower sphere. Each half sphere is contractible and thus their vector bundles are both trivial. Therefore, to obtain a vector bundle over S^k, we should glue the two trivial bundles along the fiber of the equator, and each distinct approach to gluing up two bundles gives distinct vector bundles over S^k.

To study SO(4)-vector bundles over S⁴, we should look at π3(SO(4)). Since

π₃(SO(4)) ∼= π3(SO(3))⊕ π3(S³) ∼=Z ⊕ Z,

we know that SO(4)-vector bundles over S⁴ are determined by integer pairs (h, j)∈ Z².

Define f_hj : S³ → SO(4) by fhj(v)w := v^hwv^j, where v, w ∈ H and v ∈ S³ ,→ H.

Let ξ_hj be the vector bundle defined by f_hj with fiber R⁴ and S_hj the sphere bundle of ξ_hj.

Define σ : S³ → SO(4) and σ^′ : S³ → SO(4) by σ(v)w := vw and σ^′(vw) := wv, the left and right multiplication in H. Since H P¹ ∼= S⁴, so this is a special case of Hopf quaternion bundles with n = 1. Let γ := γ₁ be the left Hopf bundle corresponding to σ discussing in last example and γ^′ the vector bundle corresponding to σ^′. Then σ and σ^′ generate π₃(SO(4)).

Proposition 1. e(ξ_hj) = (h + j) u and p₁(ξ_hj) = 2(h− j) u, where u := u1.

Proof. As noted above, this is a special case of Hopf quaternion bundles with n = 1.

In this case, e(γ_R) = u and

p₁(γ_R) = −c2(γ_C⊗ C) = −c2(γ_C⊕ ¯γ_C) = −[c0(γ_C)c₂(¯γ_C) + c₂(γ_C)c₀(¯γ_C)] =−2 u . Similarly, e(¯γ_R) = u and p₁(¯γ_R) = 2 u. Then

e(ξhj) = (h + j) u and p1(ξhj) = 2(h− j) u follow from definition.

Example. (Oriented Canonical Vector Bundles of Grassmannians)

Let G_k(R^k+p) be the Grassmannian consisting of all k-dimensional subspaces of R^k+p. Further, define ˜Gk(R^k+p) be the oriented Grassmannian, i.e., its elements are those oriented k-dimensional subspaces of R^k+p. Given G_k(R^k+p), there is a canonical vector bundle

γ_p^k ≡ γ^k(R^k+p) :={(X, v) | X ∈ Gk(R^k+p) and v ∈ X}.

Similarly, define oriented canonical vector bundle of ˜G_k(R^k+p) by

˜

γ_p^k≡ ˜γ^k(R^k+p) :={( ˜X, ˜v)| ˜X ∈ ˜G_k(R^k+p) and ˜v ∈ ˜X}.

We now introduce two notations about partition numbers:

P(n) := {(i1,· · · , ir)∈ N^r | r ∈ N, i1 ≤ · · · ≤ ir, i₁+· · · + ir= n} p(n) :=|P(n)|, the cardinality of P(n).

Theorem 1. Let R be a ring with 2 being invertible. ˜ξ^m denotes the universal bundle over ˜G_2n−1(R^∞) of rank m. Then

H^∗( ˜G2n+1(R^∞); R) = R[p1( ˜ξ²ⁿ⁺¹),· · · , pn( ˜ξ²ⁿ⁺¹)]

H^∗( ˜G2n(R^∞); R) = R[p1( ˜ξ²ⁿ),· · · , pn−1( ˜ξ²ⁿ), e( ˜ξ²ⁿ)].

2 Thom Spaces and Cobordisms

Let M be an oriented manifold. We denote by−M the same manifold with opposite orientation. For another oriented manifold M^′, M + M^′ denotes the disjoint union of M and M^′.

Definition. (Cobordism Groups)

We say that two smooth oriented compact manifolds M₁, M₂ both of dimension n are (oriented) codordant if M₁+ (−M2) = ∂W for some (n + 1)-dimensional smooth oriented compact manifold-with-boundary W . In this case, we say that M, M^′ are in the same cobordiam class, denoted by [M ] = [M^′].

The n-th (oriented) cobordism (group) Ω_n is the set of all (oriented) cobordism classes of dimension n.

The above definition indeed make sense because the relation of cobordism classes is clearly an equivalence relation. Another fact is that Ω_n is an abelian group. Our attention will turn to the direct sum of all cobordisms: Ω :=⊕

n∈N0Ω_n. Recall that a graded ring is a direct sum of abelian groups {Gα} such that G_α × Gβ ⊂ Gα+β. We check that Ω is definitely a graded ring. The only thing we have to verify is that the map Ω_i × Ωj → Ωi+j given by ([M ], [N ]) 7→ [M × N] is well-defined. Let [M ] = [M^′]∈ Ωi and [N ] = [N^′]∈ Ωj. Write M − M^′ = ∂W and N− N^′ = ∂V . Then

M × N − M^′× N^′ = (M^′+ ∂W )× N − M^′× (N − ∂V )

= ∂W × N + M^′× ∂V

= ∂(W × N − M^′× V ) Hence [M ]× [N] = [M^′]× [N^′].

Definition. (Transversality)

Let M, N be two smooth manifolds and N^′ a submanifold of N . f : M → N is a smooth map. For a subset A of M we say f is transverse to N^′ over A if for each p∈ A ∩ f⁻¹(N^′) the following condition holds:

df (T_pM ) + T_{f (p)}N^′ = T_{f (p)}N.

In this case, we denote it by f tAN^′. In particular, we write f t N^′ if A = M . If f t {y}, then we called y ∈ N is a regular value of f.

In other words, f tAN^′ if and only if the composition of maps

T_xM −→ T^dfx f (x)N → Tf (x)N /T_{f (x)}N^′

is surjective for all x ∈ A ∩ f⁻¹(N^′). Specifically, if N^′ = {y} is a set containing only one point, then df_x is surjective for all x∈ A ∩ f⁻¹(y).

Fact. Let f : M → N be a smooth map and y ∈ N a point. Assume dim M = m and dim N = n. If f t {y}, then f⁻¹(y) is a smooth manifold of dimension m− n.

Proof. Let x ∈ f⁻¹(y). Since the map df_x : T_xM → T yN is surjective, we then have N := ker df_x is a smooth manifold of dimension of m−n. Embed M into R^kfor some k. Choose a linear map L :R^k → R^m−nto be non-singular on N⊂ TxM ,→ R^k. Define

F : M → N × R^m−n x7→ (f(x), L(x)).

Its differential is given by dFx(v) = (dfx(v), L(v)), so dFx is surjective. Hence F maps some neighborhood U of x diffeomorphically to some neighborhood V of (y, L(x)). Therefore, F maps f⁻¹(y)∩ U diffeomorphically to ({y} × R^m−n)∩ V . That is, f⁻¹(y) is a smooth manifold of dimension m− n.

Fact. (Brown’s Theorem)

Let f : M → N be a smooth map. Then the set of regular values of f is dense in N , i.e., the set {y ∈ N | f t {y}} is dense in N.

Proof. This is a corollary of Sard’s theorem. We first recall the statement of ssard’s theorem.

Sard’s Theorem. Let f : M → N be a smooth map between manifolds M, N.

Then the set

C :={x ∈ M | dfx is not surjective.}

has (Lebesgue) measure 0 in N.

Then Brown’s theorem follows from N − f(C) = {y ∈ N | f t {y}} and N− f(C) is dense in N.

The task for us is to approximate a map transverse to {0} over a closed subset by a map transverse to{0} over a larger closed subset.

Lemma 1. Suppose M is an open subset of R^m. Let X ⊂ M be a closed subset in M and K a compact subset of M . Suppose f : M → Rⁿ is a smooth map and f tX {0}. Fix a compact K^′ ⊂ M with K ⊂ (K^′ − ∂K^′). Given ε > 0 then there exists a smooth map g : M → Rⁿ satisfying:

(i) gtX∪K {0}.

(ii) f |^cK^′= g |^cK^′, where ^cK^′ := M − K^′. (iii) |f(x) − g(x)| < ε for all x ∈ M.

Proof. Let λ : M → [0, 1] be a smooth cut-off function such that λ |K≡ 1 and λ|^cK^′≡ 0. According to the fact mentioned above, we can take y ∈ Rⁿ with |y| < ε such that f t y. Define g(x) = f(x) − λ(x)y and check that

(a) f |^cK^′= g |^cK^′.

(b) |f(x) − g(x)| < ε for all x ∈ M.

(c) gtK {0}.

(a) and (b) are obvious. For (c), if x₀ ∈ g⁻¹(0) ∩ K, then 0 = g(x0) = f (x₀)− λ(x₀)y = f (x₀)− y =⇒ x0 ∈ f⁻¹(y). By our assumption, f t y, i.e.,

df (T_x0M ) = T_yRⁿ = T₀Rⁿ. In addition,

dg(T_x0M ) = d(f + λy)(T_x0M ) = df (T_x0M ) = T₀Rⁿ. Hence we verify that (c) holds.

Claim: If y is chosen to be sufficiently close to 0, then g tK^′∩X {0}.

With this claim, together with (b) and (c), we then prove g tX∪K {0}.

Now we prove the claim. f tX∩K^′ {0} implies Df(x) is of full rank for all x ∈ X ∩ K^′ ∩ f⁻¹(0), where Df = (∂f_i/∂x_j)_ij. Since X ∩ K^′ is compact, one can

find K^′′ compact such that (X∩ K^′)⊂ (K^′′− ∂K^′′) and Df is of full rank on K^′′. Let U := X∩ K^′ ∩ g⁻¹(0). By taking|y| sufficiently small, one has U ⊂ K^′′. Then for each x∈ U, (

∂g_i

∂x_j )

ij

= (∂f_i

∂x_j − yi· ∂λ_i

∂x_j )

ij

is of full rank by choosing |y| small enough again (view det(Df) as a continuous function). Hence the claim.

Remark. In the above proof, the verification of g tK {0} can also be carried out as we do in proving g tK^′∩X {0}. In this case, it is much easier because of K∩ g⁻¹(0) = f⁻¹(y). By our assumption, det(Df ) ̸= 0 on f⁻¹(y). In fact, these two proofs are the same but write in different ways.

Definition. (Thom Spaces)

Let ξ be a vector bundle over a smooth manifold M . Define the Thom space of ξ to be T(ξ) := D(ξ)/ S(ξ), where D(ξ) contains all elements in ξ with length ≤ 1, and S(ξ) contains all elements in ξ with length = 1. Let t₀ be the point of T(ξ) identified by S(ξ).

Fact. Suppose a smooth manifold M is also a CW-complex. If ξ is a k-dimensional vector bundle over M , then T(ξ) is a (k− 1)-connected CW-complex.

Theorem 2. Let ξ be a vector bundle of a closed manifold M of rank k. Then there is a group homomorphism τ : π_n+k(T(ξ))→ Ωn.

Proof. There are several steps.

For convenience, denote T(ξ) by T. Given a map f : S^n+k → T, one can ap- proximate f by a map f₀ : S^n+k → T on f0⁻¹(T−t0) = f⁻¹(T−t0). Choose an open covering{W1,· · · , Wr} of compact set f0⁻¹(M ) such that f₀(W_i) is contained in some π⁻¹(U_i) ∼= Ui × R^k, where U_i are some open subsets of M . Let ρ_i : π⁻¹(U_i) → R^k be the projection. Pick compact K_i ⊂ Wi such that f₀⁻¹(M ) ⊂ (K1 ∪ · · · ∪ Kr).

Our strategy is to modify f₀ within one W_i after another, and then define the last one to be our desired function. We want to construct maps f₁,· · · , frsatisfying:

(a) Every f_i is smooth in f_i⁻¹(T−t0) and f_i |Wi−Ki= f_i−1 |Wi−Ki. (b) f_i tK1∪···∪Kr M for i = 1,· · · , r.

(c) π(fi(x))∈ M equals π(f0(x)) for all x∈ f0⁻¹(T−t0).

We start from f₀ and construct f_i inductively. Assume f_i−1 has already been con- structed. Condition (c) implies that fi−1(Wi)⊂ π⁻¹(Ui) ∼= Ui× R^k. Also, it follows from condition (b) that the map ρ_i◦ fi−1 : W_i → R^k has

ρ_i◦ fi−1 t(K1∪···∪Ki)∩Wi {0}.

By lemma 1, we can approximate ρ_i◦ fi−1 : W_i → R^k by a map ρ_i◦ fi : such that (a^′) ρ_i◦ fi |Wi−Ki= ρ_i◦ fi−1 |Wi−Ki.

(b^′) f_i tK1∪···∪Kr {0} N := g⁻¹(T(ξ)− t0).

So we can define f_i : W_i → π⁻¹(U_i) ∼= Ui × R^k whose first coordinate π(f_i(x)) is determined by condition (c) and the second coordinate ρ_i◦ fi(x) is determined by condition (a), (a^′) and (b^′). It is then clear that condition (b) holds. Hence we define f₁, f₂,· · · , fr inductively.

Now let g := f_r. We must prove the following claim.

Claim: g⁻¹(M )⊂ (K1 ∪ · · · ∪ Kr).

Indeed, g tK1∪···∪Kr M . If we can prove the claim, then we will conclude g t M.

Now we prove the claim. Since K₁∪· · · Kris a compact neighborhood of f₀⁻¹(M ) in the compact manifold S^n+k, one can find c ∈ (0, 1) such that |f0(x)| < c for all x /∈ (K1∪ · · · ∪ Kr). Let f_i is chosen to satisfy

|fi(x)− fi−1(x)| < c

r for all x ∈ S^n+k.

Consequently, |g(x) − f0(x)| < c for all x ∈ S^n+k and thus |g(x)| ̸= 0 for any x /∈ (K1 ∪ · · · ∪ Kr). That is, g⁻¹(M ) ⊂ (K1 ∪ · · · ∪ Kr). Hence g t M. So we naturally define τ ([f ]) = [g⁻¹(M )], where g⁻¹(M ) is a compact manifold of dimension n by our construction.

Next we have to check that this map is well defined, i.e., [f₀] = [f₁]∈ πn+k(T(ξ)) such that f₀ t M and f1 t M implies f0⁻¹(M ) = f₁⁻¹(M ). We take a smooth map F : S^n+k×[0, 1] → T(ξ) such that

F (x, [0,1

3]) = f₀(x) F (x, [2

3, 1]) = f₁(x).

Since f₀ t M and f1 t M, we have

F tS^n+k×(0,¹₃]∪N×[²₃,1) M.

By lemma 1 and similar process of above construction, we can approximate F by F^′ : S^n+k×[0, 1] → T(ξ) such that

F^′ tS^n+k×(0,1) and F^′(x, [0, δ)) = f₀(x), F^′(x, (1− δ, 1]) = f1(x),

where δ is some positive number less than 1/3. Then ∂F^′(M ) = f₁⁻¹(M )− f0⁻¹(M ).

Hence [f₀⁻¹(M )] = [f₁⁻¹(M )]∈ Ωn.

The final thing is to verify τ is definitely a group homomorphism. It is obvious that the addition in homotopy group corresponds to the disjoint union in cobordism group. Hence we construct the homomorphism.

Recall in section 1 we have defined the oriented canonical vector bundle

˜

γ_p^k:= ˜γ^k(R^k+p) over ˜G_k(R^k+p).

Lemma 2. If k ≥ n and p ≥ n, then the homomorphism τ : π(T(˜γp^k)) → Ωn is surjective.

Proof. Let Mⁿ be a compact smooth manifold of dimension n. By Whitney em- bedding theorem, one can embed Mⁿ intoR^n+k for some k. Let T N^k be the normal vector bundle of Mⁿ in R^n+k (the superscript indicates that T N is a k-dimensional vector space). By the existence of tubular neighborhood of Mⁿ, there exists a neighborhood U of Mⁿ inR^n+k diffeomorphic to T N^k. Thus

U ∼= T N^k −−−−−−−→ ˜γ^{Gauss map} _n^k,→ ˜γ_p^k canonical map

−−−−−−−−−→ T(˜γ_p^k).

Let g : U → T(˜γp^k) be the resulting map. There is no doubt that g t M and g⁻¹( ˜Gk(R^k+p)) = M . Extend g to a map ˆg : S^n+k → T(˜γp^k) by viewing S^n+k ∼= R^n+k∪{∞} and sending S^n+k−U to t0. Hence

[ˆg]∈ π(T(˜γp^k)) and [Mⁿ] = [ˆg⁻¹( ˜G_k(R^k+p))]

Corollary 1. The manifolds

C P²ⁱ¹× · · · × C P^2ir,

where (i₁,· · · , ir) ∈ P(m), are independent, i.e., free of relations, in Ω4m. Hence Ω_4m has rank ≥ p(m).

Theorem 3. Let X be a finite CW complex which is r-connected with r ≥ 1. Then we have the isomorphism π_n(X)⊗ Q ∼= Hn(X;Q) for n ≤ 2r.

Theorem 4. (Thom)

Ω⊗ Q ∼=Q[C P²,C P⁴,C P⁶,· · · ].

Proof. By lemma 2, we know that Ω_n is a homomorphic image of π_n+k(T(˜γ_p^k)). By theorem 3, we have

π_n+k(T(˜γ_p^k))⊗ Q ∼= Hn+k(T(˜γ_p^k);Q).

Note that we have the natural isomorphism H_n+k(T(˜γ_p^k);Q) ∼= H^n+k(T(˜γ_p^k);Q). By theorem 1, we have

rank Ω_n≤ p(m) if n = 4m rank Ω_n= 0 if 4- n.

However, corollary 1 tells us that rank Ω_n ≥ p(m) when n = 4m. As a result, rank Ω_n= p(m) when n = 4m. In fact, corollary 1 implies more:

C P²ⁱ¹× · · · × C P^2ir (i₁,· · · , ir)∈ P(m)

is a set of basis of Ω4m⊗ Q. Hence the Thom’s theorem.

3 Hirzebruch’s Signature Formula

We first recall some properties about homology and cohomology of manifolds.

Fact. Let F be a field. If M is a n-dimensional compact oriented manifold, then

Hⁿ(M ; F ) ∼= F

H^m(M ; F ) = 0 for all m > n.

Theorem 5. (Poincaré Duality)

If M is a n-dimensional oriented compact manifold, then

H^k(M ; R) = Hn−k(M ; R) for k = 0, 1,· · · , n, where R can be any coefficient ring.

We consider the pairing

H_i(M ; F )× Hn−i(M ; F )→ F.

Since dim Hⁱ(M ; F ) = dim H_n−i(M ; F ) = dim Hⁿ⁻ⁱ(M ; F ) by Poincaré duality and natural isomorphism of vector spaces, we can view Hⁿ⁻ⁱ(M ; F ) as the dual space of Hⁱ(M ; F ). In particular, we have

H_i(M ;Z) × Hn−i(M ;Z) → Z .

For n being even, we can define a non-degenerate bilinear form on H_n/2(M ;Z) by

⟨x, y⟩ := ˜x(y),

where x, y ∈ Hn/2(M ;Z) and ˜x ∈ H^n/2(M ;Z) is the isomorphic image of x. Note that ⟨ ·, · ⟩ is symmetric if n/2 is even an is alternating if n/2 is odd.

Definition. (Fundamental Classes)

Let M be a n-dimensional compact oriented manifold. The fundamental (homology) class of M , denoted by µ_M, is the generator of H(M ;Z) ∼=Z, which is compatible with the orientation of M .

Now given a compact oriented manifold M of dimension 4n. For any two α, β ∈ H²ⁿ(M,R), we define the pairing

⟨α, β⟩ := (α ⌣ β)(µM).

This pairing is non-degenerate. Recall that deRham theorem tells us that the deR- ham cohomology is isomorphic singular cohomology, i.e.,

H_dR^p (M ;R) ∼= H^p(M ;R).

In some situations, the viewpoint of deRham cohomology is easier to compute than singular cohomology.

Definition. (Signatures of Compact Oriented Smooth Manifolds)

Let M be a compact oriented smooth manifold of dimension n. If 4 - n, then its signature, denoted by σ(M ), is defined to be zero. If n = 4k, then σ(M ) is defined to be the signature of the symmetric bilinear form ⟨ ·, · ⟩ on H^2k(M ;R).

Remark.

(a) Recall that the signature of a symmetric real bilinear form is the difference of the number of positive eigenvalues and the number of negative eigenvalues.

(b) We now have two bilinear forms: one on homology, the other on cohomology.

However, we use the same notation⟨ ·, · ⟩ because the signature of a compact oriented smooth manifold can be defined to be the signature of ⟨ ·, · ⟩ on homology or ⟨ ·, · ⟩ on cohomology.

For short, σ : Ω → Z is a map between rings. As we expected, σ is actually a ring homomorphism. We state and prove this result in the following theorem, which was first presented in Thom’s paper.

Theorem 6. (Thom)

σ : Ω→ Z is a ring homomorphism, i.e., σ satisfies (a) σ(M + N ) = σ(M ) + σ(N ).

(b) σ(M × N) = σ(M) × σ(N).

(c) σ(M ) = 0 if M = ∂W .

Proof. Let dim M = m, dim N = n and dim W = m + 1.

(a) This is obvious.

(b) Let V := M× N. If 4 - dim V , then 4 - m or 4 - n. Thus σ(M × N) = 0 and σ(M )× σ(N) = 0.

Now suppose dim V = 4k. By Künneth theorem, H^2k(V ;R) ∼= ⊕

s+t=2k

H^s(M ;R) ⊗RH^t(N ;R).

Two elements x, y ∈ H^2k(V ;R) are said to be orthogonal if xy(µV) := x ⌣ y(µ_V) = 0. Let{v^si}, {wj^t} be basis of H^s(M ;R), H^t(N ;R) such that vi^sv^m_j ^−s= δ_ij, w^t_iw_j^n−t = δ_ij for s ̸= m/2, t ̸= n/2. Let A = H^m/2(M ;R) ⊕ H^n/2(N ;R) if m, n are both even and A = 0 for other cases. Define B := A^⊥ in H^2k(V ;R). So

{v^si ⊗ wj^t | s + t = 2k, s ̸= m

2, t̸= n 2} is an orthogonal basis of B.

(c) The coefficient ring of the following diagram isR.

· · · H_2k+1(W^4k+1, M^4k) H_2k(M^4k) H_2k(W^4k+1) · · ·

· · · H_2k(W^4k+1) H^2k(M^4k) H^2k+1(W^4k+1, M^4k) · · ·

∂_∗

≀

i_∗

≀ ≀

i^∗ δ^∗

Note that imi^∗ is of half dimension of H^2k(M ). For any two cocycles x, y in M^4k are obtained by restricting cocycles x^′, y^′ in W^4k+1, i.e., i^∗(x^′) = x, i^∗(y^′) = y. Then

⟨ x, y ⟩ = (x ⌣ y)(µM) = i^∗(x^′ ⌣ y^′)(µM) = (x^′ ⌣ y^′)i_∗(µM).

Thus the number of positive and negative eigenvalues are the same.

Let A = ⊕

n∈N⁰Aⁿ be a graded ring. Define A^Π :={a0+ a₁+ a₂+· · · | ai ∈ Aⁱ}.

Particularly, we are interested in A^Π₁ :={1 + a1 + a₂ +· · · | ai ∈ Aⁱ} ⊂ A^Π. Definition. (Multiplicative Sequences)

Let x∈ A^Π. We say that {Kn}n∈N0 is a multiplicative sequence if (i) each K_n(x₁,· · · , xn) is a homogeneous polynomial of degree n.

(ii) K(ab) = K(a)K(b) for any a, b∈ A^Π1, where

K(x) := 1 + K₁(x₁) + K₂(x₁, x₂) +· · · .

Proposition 2. Let A be the graded polynomial ring R[t] where t is a variable of degree 1. Given f (t)∈ 1 + λ1t + λ₂t²+· · · ∈ R[[t]], there is an unique multiplicative sequence {Kn} such that K(1 + t) = f(t).

Proof. By definition, we want to find {Kn} satisfying

K(1 + t) = 1 + K₁(t) + K₂(t, 0) + K₃(t, 0, 0) +· · · = 1 + λ1t + λ₂t²+ λ₃t³· · · . That is, we want to find K_n(x₁,· · · , xn) whose coefficient of xⁿ₁-term is λ_n for each n.

We prove the existence of {Kn} at first. Fix n ∈ N. Let {t1,· · · , tn} be algebraically independent and all of degree 1. For I = (i₁,· · · , ir) ∈ P(n), de- fine λ_I := λ_i1· · · λir. Let s₁,· · · , sn be the elementary symmetric polynomials of {t1,· · · , tn}. Note that {s1,· · · , sn} is also algebraically independent. We claim that

K_n(s₁,· · · , sn) := ∑

I∈P(n)

λ_Ig_I(s₁,· · · , sn)

is the desired multiplicative sequence. The definition of g_I is as following:

g_I(s₁,· · · , sn) = ∑ tⁱ_j¹

1· · · tⁱj^rr

with 1 ≤ j1,· · · , jr ≤ n all distinct and no ”repeated terms”. ¹ It is clear that we have the formula

g_I(ab) = ∑

HJ =I

g_H(a)g_J(b).

Hence K(ab) = K(a)K(b).

Remark. In the case of proposition 2, we call the multiplicative sequence {Kn} belongs to the formal power series f (t).

Now we define the action of multiplicative sequence {Kn} on m-dimensional compact oriented smooth manifold M^m. If 4 - m, define K(M^m) = 0. If m = 4k, define

K(M⁴k) := K_k(p₁,· · · , pk)(µ_M).

1See chapter 16 of [MS] for the complete definition of gI. Notice that the notations of [MS] are different from this article.

Theorem 7. (Hirzebruch’s Signature Formula) Let {Ln} be the multiplicative sequence belonging to

√t tanh√

t = 1 + 1 3t− 1

45t²+· · · +(−1)ⁿ⁻¹2²ⁿB_n

(2n)! tⁿ+· · · , where Bk denotes the k-th Bernoulli number. Then σ(M^4k) = L(M^4k).

Proof. By Thom’s cobordism theorem, we only need to check that σ(C P^2k) = L(C P^2k) for each k ∈ N. We have already computed that σ(C P^2k) = 1. To compute L(C P^2k), we recall that p(C P^2k) = (1 + a²)^2k+1, where a :=−c1(γ¹) with γ₁ the canonical line bundle of C P^2k. By definition,

L(1 + a²+ 0 +· · · ) =

√a² tanh√

a² =⇒ L(p(C P^2k)) =

( a

tanh a )2k+1

. Now we replace a by a complex variable z. We want to compute the coefficient of z²k in the Taylor expansion of

( z tanh z

)2k+1

. The substitution u = tanh z with

dz = du 1− u² gives

1 2πi

∫ dz

(tanh z)^2k+1 = 1 2πi

∫ 1 + u²+ u⁴+· · ·

u^2k+1 du = 1.

Hence L(C P²k) = 1.

4 Construction of Exotic 7-Spheres

We recall some definitions and properties in Morse theory.

Definition. (Morse Function)

A Morse function f on a manifold M is a real-valued function whose critical points (the points where the first derivative of f vanishes) are all non-degenerate, i.e., its Hessian matrix is non-singular.

Note that the index of a non-degenerate critical point y of f is the dimension of the largest subspace of the tangent space of M at y on which the Hessian is negative definite.

Lemma 3. (Morse Lemma)

Let y be a non-degenerate critical point of f : Mⁿ → R with index α. Then there exists a chart (x₁, x₂,· · · , xn) in a neighborhood U of y such that

x_i(y) = 0 ∀1 ≤ i ≤ n and

f (x) = f (b)− x²1− · · · − x²α+ x²_α+1+· · · + x²n ∀x ∈ U \ {y}.

Recall that we have introduced the SO(4)-vector bundles over S⁴ of rank 4. Let ξhj be the vector bundle of rank 4 defined by fhj : S³ → SO(4) where fhj(v)w :=

v^hwv^j (By viewing v ∈ S³ ,→ H, the multiplication v^hwv^j is doing in H). Let Shj be the sphere bundle of ξhj. Now we are ready to construct the exotic seven spheres.

Idea: Suppose h + j = 1.We will show that M_k⁷ := S_hj is a topological 7-sphere by constructing a Morse function on M_k⁷, where k is an odd number and assume h− j = k. Finally, if we can show that Mk⁷ is not diffeomorphic to standard S⁷, then we complete the construction of exotic 7-spheres by explicitly constructing a exotic 7-sphere, M_k⁷. The final step will be carried out by computing characteristic classes.

Step 1. If f : M_k⁷ → R is a Morse function with two critical points. Let y0, y₁ be the two critical points. Since M_k⁷ is compact, the two critical points are actually the

maximun and minimun of f . By rescaling the function f , we may assume f (y0) = 0 and f (y₁) = 1. Now consider the gradient flow:

d x

dt =∇f(x).

Note that this flow is orthogonal to each level set f⁻¹(a) for a ∈ (0, 1). Hence we have f⁻¹([0, a]) ∼= f⁻¹([0, b]) for each a, b∈ (0, 1). For a sufficiently small, it follows from Morse lemma that there exists a chart (x₁,· · · , x7) of neighborhood f⁻¹([0, a]) of y₀ such that

f (x) = x²₁+· · · + x²7.

That is, f⁻¹([0, a]) ∼= D⁷. Therefore, f⁻¹([0, 1)) = M_k⁷− {y1} ∼= D⁷. Hence 7_k ∼= S⁷ topologically.

Step 2. Now our mission is to construct a Morse function and apply step 1 to conclude that 7_k ∼= S⁷ topologically. To construct a Morse function on M_k⁷, we have to realize M_k⁷ as a more understandable structure. We will check in this step that M_k⁷ can be realized as gluing two copies of R⁴× S³ along (R⁴−{0}) × S³ via a diffeomorphism g of (R⁴−{0}) × S³ given by

g : (u, v)→ (u^′, v^′) ( u

|u|²,u^hvu^j

|u|

) .

(Notice that the operation are done inH for R⁴−{0} ,→ H and S³ ,→ H.) This map is well-defined because of h + j = 1. Now take the case u = u^′ into consideration.

In this case, we get a restricted map of g on S³:

˜

g := S³ → SO(4)

˜

g(u)v := u^hvu^j.

This is exactly the map that we define M_k⁷. Hence the result.

Step 3. From step 2, we have two coordinate charts (u, v) and (u^′′, v^′). In this step, we will verify that

f (u, v) = Re(v)

√1 +|u|² = Re(u^′′)

√1 +|u^′′|² where u^′′ := u^′(v^′)⁻¹ = u

|u| · u^hvu^j is our desired Morse function on M_k⁷, i.e., it has two non-degenerate critical points.

First of all, direct computation shows that the second equality holds. It is clear that