Exotic Smooth Structures and Milnor Spheres

Empty manifolds have the empty CW model. A compact zero-dimensional manifold is finite, since its singleton open cover has a finite subcover, and has a finite discrete CW model. Hence assume $\dim W>0$. Use [L1] to collar the boundary. For the absolute model take $(W;\mathrm{\varnothing },\mathrm{\partial }W)$ and choose $f_0:W\to [0,1]$ equal to $1-t$ near the outgoing face, with all other values in $(0,1)$; cut off this collar formula to the constant $1\mathrm{/}2$ in the interior. When the boundary is empty use $f_0=1\mathrm{/}2$. For the relative model instead take $(W;\mathrm{\partial }W,\mathrm{\varnothing })$ and use $f_0=t$ near its incoming face. These functions have no critical point on a fixed boundary strip and the required boundary level sets.

Let $K$ be the compact complement of a smaller boundary strip, contained in the interior. Take finitely many coordinate charts with compact cores covering $K$ and bumps ${\rho }_i$ equal to one near those cores, supported in the interior, by [L2]. The smooth functions ${\rho }_i{x}_i^j$, extended by zero, have differentials spanning $T_p^{\ast }W$ on an open neighbourhood $O$ of $K$.

Put $f_t=f_0+\sum t_{ij}{\rho }_i{x}_i^j$ with parameter space ${\mathbb{R}}^N$. The section $(p,t)\mapsto d{f}_t(p)$ over $O\times {\mathbb{R}}^N$ is transverse to the zero section because its parameter derivatives span the fibre. Its zero set $Z$ is therefore a smooth $N$-manifold by [L3]. At a zero its tangent equation in local coordinates is $H_{p}v+B_p\tau =0$, where $H_p$ is the Hessian and $B_p:{\mathbb{R}}^N\to T_p^{\ast }W$ is onto. Thus the projection $Z\to {\mathbb{R}}^N$ is regular at $(p,t)$ exactly when $H_p$ is onto, equivalently invertible.

Apply Euclidean Sard in countably many charts of $Z$. Under [A1] their critical-value sets have null union (choose covers with budgets $ϵ2^{-i}$), so that union contains no open parameter ball. Take a regular parameter $t$ arbitrarily near zero. On the remaining compact boundary strip $d{f}_0$ is bounded away from zero in a metric built by a finite chart partition; a sufficiently small $t$ preserves this property. Its support misses a neighbourhood of the boundary, and a small perturbation keeps the interior values strictly between zero and one. Hence $f_t$ is adapted and Morse throughout $W$.

The critical set is finite by [L4]. Choose disjoint small critical-point neighbourhoods and bumps constant one near each critical point. Adding sufficiently small independent constants times those bumps preserves the critical points and their Hessians; on the compact transition annuli the differential was bounded away from zero, so it remains nonzero. Choose the constants to make the finitely many critical values distinct, retaining the boundary formulas and range. This gives an excellent adapted function $f$.

By [L7], choose Morse charts at the finitely many critical points and their prescribed negative Euclidean gradient fields. Away from those charts choose local fields with $df(X)<0$, and on the boundary collars take the descending collar direction. A partition as in [L1], equal to one near the critical points, glues these fields: strict negativity is preserved by convex combination. Append exterior collars, extend the collar fields, and cut off outside a compact neighbourhood of $W$. The resulting ambient field is complete by [L7] and has the adapted local models and boundary signs. Thus $(f,X)$ meets the pair hypotheses of [L5].

Apply [L5] to obtain a finite handle decomposition relative to the chosen incoming face. In the absolute construction this face is empty, and [L6] gives a finite CW model homotopy equivalent to $W$. In the relative construction the incoming face is the supplied finite CW boundary; [L6] gives a finite relative CW pair and an equivalence fixing that face, with its initial collar compressed onto it.

These models prove both assertions. Only finite selections and the countable chart, null-cover, collar, partition and completeness suppliers used ${\mathrm{A}\mathrm{C}}_{\omega }$.

By [L1] the map $H_n({\mathrm{\Sigma }}_i)\to H_n({\mathrm{\Sigma }}_i,P_i)$ is an isomorphism, since it sends the fundamental generator to the local disk generator. Exactness gives ${\tilde{H}}_{\ast }(P_i)=0$. Removing a disk leaves a path-connected manifold: paths entering the disk can be diverted along its connected boundary collar. Van Kampen for $P_i$ and the disk, thickened to an open cover, has simply connected overlap $S^{n-1}$ for $n\ge 3$, and gives ${\pi }_1(P_i)={\pi }_1({\mathrm{\Sigma }}_i)=0$. Thus [L2], applied on finite models to $P_i\to \{\ast \}$, makes $P_i$ contractible.

Use the fixed orientation-reversing linear reflection between the disk-coordinate boundary spheres to glue $P_1$ and $P_2$. This is the oriented connected sum; no arbitrary boundary diffeomorphism is substituted for that coordinate identification. Collar thickenings of the two pieces form an open cover whose overlap retracts to $S^{n-1}$.

Reduced Mayer–Vietoris for that cover gives ${\tilde{H}}_k({\mathrm{\Sigma }}_1\mathrm{\#}{\mathrm{\Sigma }}_2)\cong {\tilde{H}}_{k-1}(S^{n-1})$ for $k\ge 1$, because both pieces are contractible. In degree zero the union is connected. Hence its integral homology is that of $S^n$.

Van Kampen for the same open cover, with simply connected pieces and overlap, gives ${\pi }_1({\mathrm{\Sigma }}_1\mathrm{\#}{\mathrm{\Sigma }}_2)=0$.

Choose an oriented coordinate disk $D$ in the connected sum $X$ and collapse $X\setminus \mathrm{int}D$ to a point. The target is $D\mathrm{/}\mathrm{\partial }D\cong S^n$; the map has degree one because it carries the local oriented disk generator to the sphere generator. By step 3.1 it is a homology equivalence. Transport it to the finite CW model of $X$ and use [L2] to obtain $X\simeq S^n$.

The oriented connected sum is therefore an oriented smooth homotopy $n$-sphere.

An oriented connected sum uses small coordinate disks and the fixed orientation-reversing linear reflection of their boundary coordinates, as in [L1]. Changes of small coordinate disks are joined by isotopies: shrink each disk within its chart, move its centre along a finite chain of charts on a path in the connected manifold, and move its oriented frame through $G{L}_n^{+}(\mathbb{R})$: Gram–Schmidt deforms the positive triangular factor to identity and plane rotations deform the orthogonal factor to identity. On a sufficiently small disk each chart change is isotopic to its derivative by rescaling, retaining positive determinant; the finitely many stages yield the required isotopy. Reparameterize it to be stationary near its ends. Applying [L3] to a $1$-handle joining the outgoing faces of two disjoint product collars transfers this isotopy to an orientation-preserving diffeomorphism of their connected-sum boundary. Thus coordinate-disk choices do not affect the sum. This argument asserts no independence under an arbitrary nonextendable boundary twist.

By [L2] the h-cobordisms $W,W^{\mathrm{\prime }}$ are connected and their faces are simply connected homotopy spheres. Their dimension is $n+1\ge 6$, so [L4] gives orientation-preserving diffeomorphisms $f:{\mathrm{\Sigma }}_0\to {\mathrm{\Sigma }}_1$ and $f^{\mathrm{\prime }}:{\mathrm{\Sigma }}_0^{\mathrm{\prime }}\to {\mathrm{\Sigma }}_1^{\mathrm{\prime }}$: a product diffeomorphism relative to the incoming face preserves its orientation, and hence the orientation throughout the connected cobordism. Choose outgoing disk charts by transporting the incoming ones through $f,f^{\mathrm{\prime }}$. The restrictions of $f,f^{\mathrm{\prime }}$ then agree with the coordinate reflection used at the neck and glue to an orientation-preserving diffeomorphism ${\mathrm{\Sigma }}_0\mathrm{\#}{\mathrm{\Sigma }}_0^{\mathrm{\prime }}\to {\mathrm{\Sigma }}_1\mathrm{\#}{\mathrm{\Sigma }}_1^{\mathrm{\prime }}$. Its product cylinder is an oriented h-cobordism. Step 1.1 removes dependence on the disk choices, proving descent to classes.

For three summands choose two disjoint small disks in the middle one and one disk in each outer one. Both groupings are the same quotient of the three punctured manifolds with the same two neck identifications; regrouping the quotient gives an orientation-preserving diffeomorphism. Interchanging the summands reverses the neck parameter and applies the fixed reflection on its sphere factor, so the two orientation signs cancel and give an orientation-preserving diffeomorphism. Together with step 1.1 these prove associativity and commutativity.

The complement of a standard coordinate disk in $S^n$ is a standard disk. The coordinate reflection extends linearly over it, so gluing this complement into the removed disk of $\mathrm{\Sigma }$ restores $\mathrm{\Sigma }$ orientation-preservingly. Thus $S^n$ is an identity; [L1] ensures that every sum remains a homotopy sphere.

Connected sum consequently defines the asserted associative, commutative operation on oriented h-cobordism classes with identity $[S^n]$.

By [L1] the punctured homotopy sphere $P=\mathrm{\Sigma }\setminus \mathrm{int}{D}^n$ is a compact contractible smooth $n$-manifold with boundary $\mathrm{\partial }P=S^{n-1}$.

Round $P\times [0,1]$ explicitly in the collars of [L2]. Near either corner, let $s,t\ge 0$ be the two inward coordinates and put $w=s-t$, $z=s+t$, so the quadrant is $z\ge \mathrm{\mid }w\mathrm{\mid }$. Choose a smooth convex function $g(w)\ge \mathrm{\mid }w\mathrm{\mid }$ equal to $\mathrm{\mid }w\mathrm{\mid }$ outside a small interval (convolve $\mathrm{\mid }w\mathrm{\mid }$ with a nonnegative even smooth kernel of integral one and small compact support). Replace $z\ge \mathrm{\mid }w\mathrm{\mid }$ by $z\ge g(w)$, using the same profile over $\mathrm{\partial }P$ and disjoint neighborhoods at the two ends. The graph is smooth and agrees with the faces away from the corner, so the retained region $V$ is a compact smooth manifold with boundary. The homotopy $z\mapsto z+u\max \{0,g(w)-z\}$, $0\le u\le 1$, extended by the identity, retracts the original product onto $V$; its support stays inside the chosen collar. Thus $V$ is contractible. Its boundary joins two shortened copies of $P$ by the boundary collar cylinder, hence is the double of $P$, namely $\mathrm{\Sigma }\mathrm{\#}(-\mathrm{\Sigma })$; collar reparametrizations identify the shortened copies with $P$. Choose the orientation of $V$ so the first copy has the orientation of $\mathrm{\Sigma }$; the other copy then has the opposite orientation.

Remove from $V$ the interior of a small smoothly embedded $(n+1)$-disk meeting only the interior, obtaining the compact oriented manifold $C$ whose boundary has the face $\mathrm{\Sigma }\mathrm{\#}(-\mathrm{\Sigma })$ and a standard sphere face $S^n$.

Using collar thickenings, excision identifies $H_{\ast }(C,S^n)$ with $H_{\ast }(V,D^{n+1})=0$ because $V$ and the disk are contractible, so the inclusion $S^n\to C$ is an integral homology isomorphism; the boundary relation of [L3] gives $[\mathrm{\Sigma }\mathrm{\#}(-\mathrm{\Sigma })]+[S^n]=0$ in $H_n(C)\cong \mathbb{Z}$, so the other face inclusion also induces an isomorphism on $H_n$ and hence on all reduced homology, since $C$ has the homology of a point in intermediate degrees.

Van Kampen applied after reattaching the removed disk shows ${\pi }_1(C)={\pi }_1(V)=0$, and both faces are simply connected: $\mathrm{\Sigma }\mathrm{\#}(-\mathrm{\Sigma })$ by the connected-sum lemma and $S^n$ for $n\ge 5$.

The compact smooth manifold $C$ has a finite CW model by [L1]'s finite-CW clause, so the simply connected homology Whitehead criterion upgrades both face inclusions to homotopy equivalences; hence $C$ is an h-cobordism from $S^n$ to $\mathrm{\Sigma }\mathrm{\#}(-\mathrm{\Sigma })$, proving that orientation reversal is the inverse.

If $f:\mathrm{\Sigma }\to {\mathrm{\Sigma }}^{\mathrm{\prime }}$ is an orientation-preserving diffeomorphism, the product $\mathrm{\Sigma }\times [0,1]$ with the two boundary identifications given by the identity and by $f$ is an oriented h-cobordism from $\mathrm{\Sigma }$ to ${\mathrm{\Sigma }}^{\mathrm{\prime }}$, so oriented diffeomorphism implies oriented h-cobordism.

Conversely, suppose $\mathrm{\Sigma },{\mathrm{\Sigma }}^{\mathrm{\prime }}$ are oriented h-cobordant and let $C$ be a compact oriented h-cobordism between them; then $C$ has dimension $n+1\ge 6$, and its boundary faces are the closed simply connected manifolds $\mathrm{\Sigma },{\mathrm{\Sigma }}^{\mathrm{\prime }}$, since both are homotopy spheres.

By [L2] and [A1] the cobordism $C$ is diffeomorphic to a product relative to the face $\mathrm{\Sigma }$; hence its other face ${\mathrm{\Sigma }}^{\mathrm{\prime }}$ is orientation-preservingly diffeomorphic to $\mathrm{\Sigma }$.

Combining steps 1.1 and 3.1, oriented h-cobordism and orientation-preserving diffeomorphism define the same equivalence relation on oriented smooth homotopy $n$-spheres, so the classes of ${\mathrm{\Theta }}_n$ are oriented diffeomorphism classes.

Representative independence: if oriented homotopy spheres $\mathrm{\Sigma },{\mathrm{\Sigma }}^{\mathrm{\prime }}$ represent the same class of ${\mathrm{\Theta }}_n$ and $\mathrm{\Sigma }=\mathrm{\partial }V$ for a compact oriented parallelizable filling $V$, then [L1] gives an orientation-preserving diffeomorphism $f:\mathrm{\Sigma }\to {\mathrm{\Sigma }}^{\mathrm{\prime }}$; transporting the tangent trivialization along a collar and pulling $V$ back across $f$ produces a compact oriented parallelizable filling of ${\mathrm{\Sigma }}^{\mathrm{\prime }}$.

Closure under addition: if $\mathrm{\Sigma }=\mathrm{\partial }V$ and ${\mathrm{\Sigma }}^{\mathrm{\prime }}=\mathrm{\partial }{V}^{\mathrm{\prime }}$ with $V,V^{\mathrm{\prime }}$ compact oriented parallelizable, choose small boundary coordinate $n$-disks whose parametrizations extend to slightly larger disks. Attach the $1$-handle $[-1,1]\times D^n$ to $V\bigsqcup V^{\mathrm{\prime }}$ at its two feet, using product collars and rounding as in [L2]. Choose the disk identifications so the handle orientation extends both filling orientations. The attaching disks disappear from the boundary and are replaced by $[-1,1]\times S^{n-1}$, joining the two punctured boundaries by the standard orientation-reversing disk-coordinate identification. Absorbing the intervening collars therefore gives boundary $\mathrm{\Sigma }\mathrm{\#}{\mathrm{\Sigma }}^{\mathrm{\prime }}$. The attachment is compact.

Take positively oriented tangent frames on the fillings. In collar coordinates near each attaching disk, each frame is a smooth map to $G{L}_{n+1}^{+}(\mathbb{R})$. Shrink the disk and use radial contraction on a slightly larger coordinate neighborhood to deform that map to its value at the centre, keeping the frame unchanged off that neighborhood. Any positive frame is joined to the coordinate frame: Gram–Schmidt deforms its positive upper-triangular factor to the identity, and a product of plane rotations deforms its orthogonal factor to the identity. Make these deformations constant near their ends and use smooth collar cutoffs to match both frames to the product frame on the $1$-handle. They then agree on neighborhoods of the attaching regions, including their edges, and extend in the ambient product coordinates used for rounding. Restricting those frames to the rounded region gives a smooth tangent trivialization. Thus the new filling is parallelizable.

Inverses and identity: $-V$ is parallelizable with $\mathrm{\partial }(-V)=-\mathrm{\Sigma }$, so the inverse class stays in the set, and the standard sphere $S^n=\mathrm{\partial }{D}^{n+1}$ bounds the parallelizable disk, giving the zero class.

Steps 1.1-4.1 prove representative independence, closure under addition, closure under inverse and membership of the zero class, hence the classes represented by parallelizable boundaries form a subgroup of ${\mathrm{\Theta }}_n$.

Put $s_{+}=v_0$ on the upper hemisphere. By the upper-to-lower transition, the lower boundary value must be $s_{-}(a)=g(a)v_0$. Extend this value to a smooth map $F:D_{-}^4\to {\mathbb{R}}^4$, constant in the radial coordinate near the boundary and zero near the centre, by a smooth radial cutoff. It agrees with the constant upper section through the equatorial product charts. The base orientation on $D_{-}^4$ is its standard coordinate orientation, and its coordinate boundary orientation is the source orientation fixed in the statement.

The map $F$ is nonzero on a boundary strip. Choose a smooth cutoff $\rho$ equal to one on the compact complement of that strip, supported away from the boundary, and with its transition region inside the strip. By [L6] choose a sufficiently small regular value $t$ of $F$ on the open disk. Then $F_t=F-\rho t$ has no zeros in the transition strip, while every remaining zero lies where $\rho =1$ and has invertible derivative $DF$. Thus the section with $s_{+}=v_0$, $s_{-}=F_t$ is smooth and transverse to zero, with a finite zero set in the lower open disk.

By [L2] its signed zero count is $⟨e(E),[S^4]⟩$. On the positively oriented lower disk each local contribution is $\mathrm{sign}\det D{F}_t$, the index of the coordinate vector field $Y=F_t$ by [L4]. Hence [L3] gives $⟨e(E),[S^4]⟩=\mathrm{deg}(a\mapsto Y(a)\mathrm{/}\mathrm{\mid }Y(a)\mathrm{\mid })=\mathrm{deg}(a\mapsto g(a)v_0)$, since the boundary was fixed and $v_0$ is unit. Scaling a nonzero $v_0$ to unit length gives the displayed general formula.

For either basic clutching choose $v_0=1$. Both boundary maps are then $a\mapsto a$, of degree $+1$ by [L5]. For any other unit $v_0$, a path from $1$ to $v_0$ in $S^3$ gives a homotopy of the boundary maps, so their degree is unchanged. This proves both calibrations in the stated orientation convention.

Write $V=V_L$ or $V=V_R$ with its complex structure: for $V_L$ right multiplication by $i$ commutes with left multiplication by every quaternion and makes $V_L$ a complex rank-two bundle; for $V_R$ left multiplication by $i$ plays the same role for right multiplication.

The complexification $V_{\mathbb{C}}=V{\otimes }_{\mathbb{R}}\mathbb{C}$ carries the complexified complex structure $J_{\mathbb{C}}$; its $\pm i$ eigenspace projections are complex subbundles $E_{+}\cong V$ and $E_{-}\cong \bar{V}$ (locally over a trivializing chart they are the $\pm i$ eigenspaces of the constant matrix $J$), so $V_{\mathbb{C}}\cong V\oplus \bar{V}$ as complex bundles.

By [L5] and step 2.1, $c_2(V_{\mathbb{C}})=c_2(V)+c_2(\bar{V})=c_2(V)+c_2(V)=2c_2(V)$ because $c_2$ is even in the conjugate sign of [L4], and $c_1$ terms do not contribute in degree four on $S^4$ as $H^2(S^4)=0$.

By the Pontryagin convention $p_1(V_{\mathbb{R}})=-c_2(V_{\mathbb{C}})=-2c_2(V)$ (the sign is the one fixed in [L4]).

The complex orientation of $V_L$ is opposite to the fibre orientation: the commuting complex structure is right multiplication by $i$, whose complex basis $(1,j)$ gives the real ordered basis $(1,i,j,-k)$, the negative of the fixed fibre basis $(1,i,j,k)$; hence by [L2] and [L3] $c_2(V_L)=e(V_{L,\mathbb{R}})$ in the complex orientation $=-e(V_{L,\mathbb{R}})$ in the fibre orientation $=-u$.

The complex orientation of $V_R$ agrees with the fibre orientation: the commuting complex structure is left multiplication by $i$, whose complex basis $(1,j)$ gives the real ordered basis $(1,i,j,k)$; hence $c_2(V_R)=+u$.

Substituting steps 5.1 and 6.1 into step 4.1 gives $p_1(V_L)=-2(-u)=2u$ and $p_1(V_R)=-2u$, while step 1.1's Euler computation gives $e(V_L)=e(V_R)=u$, as asserted.