Cobordism and Surgery Theory

Introduction

Cobordism is the equivalence relation on closed manifolds generated by bounding: two closed $n$-manifolds are cobordant if their disjoint union bounds a compact $(n+1)$-manifold. The equivalence classes form the cobordism groups $\Omega_n$, graded by dimension and by the structure imposed on the manifolds — orientation, complex structure, spin structure, framing — and the fundamental theorem of the subject, due to Thom, computes the unoriented cobordism ring as a polynomial algebra over $\mathbb{F}_2$ and the oriented cobordism ring rationally as a polynomial algebra on the complex projective spaces. Surgery is the complementary technique: a closed manifold is cut along an embedded $S^k\times D^{n-k}$ and glued back along $D^{k+1}\times S^{n-k-1}$, which changes the manifold in a controlled way, and the classification of the manifolds in a fixed homotopy type is expressed by the surgery exact sequence, whose terms are the normal invariants classified by the space $G/O$ and the surgery obstruction groups $L_n(\mathbb{Z}[\pi_1])$ of Wall.

The two ideas meet in the classification theorems of high-dimensional topology. The $h$-cobordism theorem of Smale states that in dimension at least five a simply connected cobordism that is a homotopy equivalence at both ends is a product; the $s$-cobordism theorem of Barden, Mazur and Stallings corrects the statement by the Whitehead torsion, and the correction is precisely the phenomenon realised by the lens spaces of Lens Spaces. The high-dimensional Poincaré conjecture, the finiteness of the number of smooth structures on a sphere up to diffeomorphism, the existence of exotic spheres and the classification of the manifolds homotopy equivalent to a given manifold are all corollaries of the surgery machinery.

The boundaries of the article. The three- and four-dimensional classification — the geometrisation theorem, the failure of the $h$-cobordism theorem in dimension four, the theorems of Freedman and Donaldson and the exotic structures — is that of Low-Dimensional Topology, and the surgery of three-manifolds along knots and links is that of Low-Dimensional Topology and Knot Theory; the present article states the general theory and cites the low-dimensional exceptions. The homotopy theory of the Thom spectra, the stable homotopy groups, the Steenrod operations, the characteristic classes and the bordism homology theories are those of Algebraic Topology and of Characteristic Classes, both written by other agents; the article states the computations and cites the machinery. The linear algebra of the quadratic forms and of their Witt classes over the group rings, which is the algebraic content of the surgery obstruction, is that of Topology on Linear Algebras with a degree-2 form, and the algebraic surgery of Ranicki is the general form of the theory. The Lie groups whose homogeneous spaces appear as the generators of the cobordism groups are those of Part I. The Morse theory and the gradient flows on which the handle decomposition of a cobordism rests are analytic, and the theorem that a cobordism admits a handle decomposition from a Morse function belongs to Part III, where the derivative and the limit are available; the results are used here as statements. The exotic spheres are homotopy-theoretic and differential-topological objects, and no physics is invoked.

Cobordism: Definition and Thom's Theorem

Definition. Let $\mathcal{S}$ denote one of the structure types: unoriented, oriented, complex (with a stable complex structure on the tangent bundle), spin, or framed. Two closed $n$-manifolds $M_0, M_1$ with structure $\mathcal{S}$ are $\mathcal{S}$-cobordant if there is a compact $(n+1)$-manifold $W$ with structure whose boundary is the disjoint union $M_0\sqcup M_1$ and whose structure restricts to the given structures, with the orientations or the framings matched on the two components. Cobordism is an equivalence relation, and the classes form an abelian group $\Omega_n^{\mathcal{S}}$ under disjoint union, with the class of the empty set as the identity and $[M]+[M]=0$ in the unoriented case. The graded sum

$$ \Omega_*^{\mathcal{S}} = \bigoplus_{n\geq0}\Omega_n^{\mathcal{S}} $$

is a graded ring under the product induced by the Cartesian product of manifolds.

Theorem (Thom; the unoriented cobordism ring). The unoriented cobordism ring is a polynomial algebra over $\mathbb{F}_2$ on one generator in each dimension not of the form $2^k-1$:

$$ \Omega_*^{O} = \mathbb{F}_2[x_i : i\neq 2^k-1] . $$

Consequently $\Omega_n^{O}$ is an $\mathbb{F}_2$-vector space whose dimension is the number of partitions of $n$ into parts not of the form $2^k-1$, and the low-dimensional values are

$n$	0	1	2	3	4	5	6	7	8
$\dim_{\mathbb{F}_2}\Omega_n^{O}$	1	0	1	0	2	1	3	1	5

with $\Omega_2^O = \mathbb{Z}/2$ generated by $\mathbb{RP}^2$ and $\Omega_4^O = (\mathbb{Z}/2)^2$ generated by $\mathbb{RP}^4$ and $\mathbb{RP}^2\times\mathbb{RP}^2$.

Proof sketch. The Pontryagin–Thom construction identifies $\Omega_n^{O}$ with the homotopy group $\pi_n(MO)$ of the Thom spectrum of the universal bundle over $BO$, and the Thom isomorphism together with the Steenrod operations computes the homotopy ring: the mod $2$ cohomology of $MO$ is a free module over the Steenrod algebra generated by the Thom class, and the indecomposables of the homotopy ring are detected by the Stiefel–Whitney numbers of the manifolds. The dimension count is the statement that a partition of $n$ into parts $i$ with $i\neq2^k-1$ contributes one monomial. The details of the Thom spectra and the Steenrod algebra belong to Algebraic Topology.

Theorem (Thom; Milnor; the oriented cobordism ring). The oriented cobordism ring is rationally a polynomial algebra on the classes of the complex projective spaces,

$$ \Omega_*^{SO}\otimes\mathbb{Q} = \mathbb{Q}[\mathbb{CP}^2,\mathbb{CP}^4,\mathbb{CP}^6,\ldots] , $$

and all of its torsion is of order $2$ (Wall). In low dimensions $\Omega_1^{SO} = \Omega_2^{SO} = \Omega_3^{SO} = 0$, $\Omega_4^{SO} = \mathbb{Z}$ generated by $\mathbb{CP}^2$ and detected by the signature, and $\Omega_5^{SO} = \mathbb{Z}/2$ generated by the Wu manifold $SU(3)/SO(3)$. A closed oriented manifold is a boundary if and only if all of its Stiefel–Whitney numbers and all of its Pontryagin numbers vanish.

Proof sketch. The rational computation uses the Pontryagin numbers, which are the characteristic numbers of the tangent bundle: a manifold with all Pontryagin numbers zero is rationally a boundary, and the products of the complex projective spaces realise the classes. The statement about the torsion is the theorem of Wall. The detection of the boundary condition by the Stiefel–Whitney and Pontryagin numbers combines the unoriented result with the signature theorem of Characteristic Classes.

Theorem (Pontryagin–Thom; the identification with homotopy). For each structure $\mathcal{S}$ with a classifying space $B$ and a Thom spectrum $MB$, there is a natural isomorphism

$$ \Omega_n^{\mathcal{S}} \cong \pi_n(MB), $$

so that the cobordism groups are the homotopy groups of the Thom spectra: $\Omega_*^{O} = \pi_*(MO)$, $\Omega_*^{SO} = \pi_*(MSO)$, $\Omega_*^{U} = \pi_*(MU)$, $\Omega_*^{Spin} = \pi_*(MSpin)$. The complex cobordism ring is the polynomial algebra $\Omega_*^{U} = \mathbb{Z}[x_1,x_2,\ldots]$ with $\deg x_i = 2i$ (Milnor, Novikov), and the ring $\pi_*(MU)$ carries the formal group law of Quillen; the spin cobordism groups are computed from $\Omega_*^{Spin}\otimes\mathbb{Z}[\tfrac12]$ by the forgetful maps and are detected by the Stiefel–Whitney numbers together with the $\alpha$-invariant in dimensions $\equiv1,2\pmod 8$.

Proof sketch. The Thom space of a bundle with structure $\mathcal{S}$ is the one-point compactification of the disc bundle, and a manifold with structure maps into the Thom spectrum by collapsing the complement of a tubular neighbourhood of the embedded manifold; the correspondence with the homotopy groups is formal. The computation of $\pi_*(MU)$ is by the Adams spectral sequence and the theory of the formal group.

Surgery

Definition. Let $M$ be a closed $n$-manifold and let $\phi : S^k\times D^{n-k}\hookrightarrow M$ be an embedding with trivial normal bundle. Surgery on $M$ along $\phi$ removes the interior of the image and glues in $D^{k+1}\times S^{n-k-1}$ along the common boundary $S^k\times S^{n-k-1}$, producing a new closed $n$-manifold $M'$:

$$ M' = (M\setminus \mathrm{int}\,\phi(S^k\times D^{n-k})) \cup_{\partial}\ (D^{k+1}\times S^{n-k-1}) . $$

The manifold $W = M\times I$ with a $(k+1)$-handle attached along $\phi$ is a cobordism from $M$ to $M'$, the trace of the surgery; the inclusion $M\hookrightarrow W$ induces the quotient on $\pi_k$ that kills the class $[\phi]$, so the trace is a homotopy equivalence only when that class is trivial, and in the nontrivial case it is the record of a genuine change of the homotopy type of the manifold.

Proposition (effect on homology). Let $k

Proof sketch. The complement of the tubular neighbourhood retracts onto the complement of the sphere, and the cocore disc $D^{n-k}$ in the glued copy is a relative cycle whose boundary is the original sphere; the effects on the homotopy and homology groups follow from the Seifert–van Kampen theorem and the Mayer–Vietoris sequence.

Definition. A normal map of degree one from a closed $n$-manifold $M$ to a space $X$ is a map $f : M\to X$ together with a bundle map $\nu_M\to\xi$ of the stable normal bundle to a bundle over $X$, such that the induced map on the fundamental classes is an isomorphism in degree $n$. The normal invariant of $f$ is the class of the bundle map in the set $[X,G/O]$ of homotopy classes of maps to the classifying space $G/O$ of the stable normal invariants. Surgery on a normal map replaces $M$ by a surgered manifold $M'$ equipped with a normal map to $X$ of the same degree.

Theorem (Wall; the surgery obstruction). Let $X$ be a Poincaré space of dimension $n\geq5$ with fundamental group $\pi$. There is a homomorphism, the surgery obstruction,

$$ \sigma : \Omega_n(X) \longrightarrow L_n(\mathbb{Z}[\pi]) , $$

from the group of normal maps of degree one over $X$ to the Wall $L$-group, which is the Witt group of the nonsingular $(-1)^{n/2}$-quadratic forms over the group ring $\mathbb{Z}[\pi]$ with the involution $g\mapsto g^{-1}$, compatible with the surgery. The obstruction vanishes if and only if the normal map is normally bordant to a homotopy equivalence; the surgery obstruction of a simply connected manifold is detected by the signature in dimension $4k$ and by the Kervaire invariant in dimension $4k+2$: $L_{4k}(\mathbb{Z}) = \mathbb{Z}$ and $L_{4k+2}(\mathbb{Z}) = \mathbb{Z}/2$, the odd-dimensional $L$-groups being the targets of the odd-dimensional obstructions and computed by the algebraic surgery of Ranicki.

Proof sketch. The obstruction is defined by the middle-dimensional intersection form of the surgered manifold: performing surgery below the middle dimension reduces the normal map to one which is a homotopy equivalence on the lower half of the homotopy groups, and the remaining obstruction is the class of the (−1)^{n/2}-quadratic form carried by the middle homology, an element of the Witt group of $\mathbb{Z}[\pi]$. That the vanishing of this class is sufficient is the clever part: it permits the middle-dimensional classes to be represented by disjoint embedded spheres, which are then surgered away, and the Whitney trick for cancelling the intersections is available in dimension at least five. The algebra of the quadratic forms is that of Topology on Linear Algebras with a degree-2 form.

Theorem (the surgery exact sequence). Let $X$ be a closed $n$-manifold with fundamental group $\pi$ and $n\geq5$. The structure set $\mathcal{S}(X)$ of the homotopy equivalences $M\to X$ from closed $n$-manifolds to $X$ modulo $h$-cobordism fits into the exact sequence

$$ L_{n+1}(\mathbb{Z}[\pi]) \longrightarrow \mathcal{S}(X) \longrightarrow [X,G/O] \longrightarrow L_n(\mathbb{Z}[\pi]) , $$

in which the second map assigns to a homotopy equivalence its normal invariant and the last map is the surgery obstruction. Consequently the classification of the closed manifolds homotopy equivalent to $X$ is reduced to the computation of the $L$-groups and of the set of normal invariants, and the structure set is the measure of the failure of the Poincaré duality space $X$ to determine its manifold representatives.

Proof sketch. The composition of the consecutive maps is zero by the construction of the obstruction; the exactness in the middle is the statement that a normal map with vanishing surgery obstruction is normally bordant to a homotopy equivalence, which is the main theorem of the obstruction theory; the exactness at the ends is formal. The map from the $L$-group to the structure set is realised by the surgery on a manifold with boundary.

The $h$-Cobordism and $s$-Cobordism Theorems

Theorem (Smale; the $h$-cobordism theorem). Let $W$ be a compact oriented smooth $n$-manifold with boundary the disjoint union $M_0\sqcup M_1$ of closed simply connected manifolds, $n\geq6$, and suppose that the inclusions $M_i\hookrightarrow W$ are homotopy equivalences. Then $W$ is diffeomorphic to $M_0\times I$, so that $M_0$ and $M_1$ are diffeomorphic and the cobordism is a product.

Proof sketch. A cobordism admits a handle decomposition from a Morse function on $W$, by the Morse theory of Part III; the handles are cancelled in pairs using the Whitney trick, which is available in dimension at least five, until no handles remain; the resulting product decomposition is the diffeomorphism. The simple connectivity supplies the embedded discs needed for the cancellation and the Whitney trick removes the intersections. The proof is the model of the surgery method: the handle cancellation is the geometric content of the vanishing of the surgery obstruction.

Theorem (Barden, Mazur, Stallings; the $s$-cobordism theorem). Let $W$ be a compact oriented smooth $n$-manifold with boundary $M_0\sqcup M_1$, $n\geq6$, such that the inclusions are homotopy equivalences and the induced homotopy equivalences are simple, that is, of vanishing Whitehead torsion. Then $W$ is diffeomorphic to $M_0\times I$. In general the obstruction to the product structure of an $h$-cobordism is the Whitehead torsion of the homotopy equivalence, an element of the Whitehead group $\mathrm{Wh}(\pi_1(W))$; the $h$-cobordism is a product if and only if the torsion vanishes, and the torsion can be varied only by the action of the group.

Proof sketch. The handle cancellation of Smale's proof is obstructed by the algebraic $K_1$ of the group ring, whose quotient is the Whitehead group; the obstruction of a handle pair is the torsion of the corresponding based chain complex, and the vanishing of the torsion is exactly the condition under which the pair can be cancelled. The constructions of the obstruction and its realisability are the content of the theorem.

Corollary (the high-dimensional Poincaré conjecture; Smale, Stallings, Zeeman). A closed topological $n$-manifold homotopy equivalent to $S^n$ with $n\geq5$ is homeomorphic to $S^n$; the smooth version fails in general, since by the Kervaire–Milnor computation the smooth homotopy $n$-spheres form the group $\Theta_n$, which is nontrivial in general, so a homotopy sphere need not be diffeomorphic to $S^n$. For $n=4$ the smooth statement is the open smooth Poincaré conjecture, and for $n=3$ the topological statement is the Poincaré conjecture, proved by Perelman.

Proof sketch. Remove the interiors of two disjoint embedded discs from the homotopy sphere; the resulting manifold has two boundary components, each a homotopy $(n-1)$-sphere, and the complement of the two discs is an $h$-cobordism, so the $h$-cobordism theorem makes it a product and produces the homeomorphism to $S^n$. The failure of the argument in the smooth category is exactly the failure of the $h$-cobordism theorem to produce a diffeomorphism: the product structure is only topological, and the discrepancy is measured by $\Theta_n$.

Homotopy Spheres and Exotic Spheres

Definition. The group of homotopy spheres $\Theta_n$ is the abelian group of the oriented homotopy $n$-spheres — closed smooth $n$-manifolds homotopy equivalent to $S^n$ — under the connected sum, modulo orientation-preserving diffeomorphism. The group $\Theta_n$ is finite abelian for $n\geq5$, and was computed by Kervaire and Milnor from the surgery exact sequence:

$$ 0 \longrightarrow bP_{n+1} \longrightarrow \Theta_n \longrightarrow \pi_n(G/O) \longrightarrow \mathbb{Z}/2 , $$

where $bP_{n+1}$ is the subgroup of the homotopy spheres bounding a parallelizable manifold and the last map is the Kervaire invariant.

Theorem (Kervaire–Milnor; the order of $bP_{4k}$). For $k\geq2$ the group $bP_{4k}$ is cyclic of order

$$ |bP_{4k}| = 2^{2k-2}(2^{2k-1}-1)\,\text{numerator}\!\left(\frac{4B_k}{k}\right) , $$

where $B_k$ denotes the $k$-th Bernoulli number in the indexing $B_1=\tfrac16$, $B_2=\tfrac1{30}$, $B_3=\tfrac1{42},\ldots$ The first values are

$k$	2	3	4	5
$\|bP_{4k}\|$	28	992	8128	261632

so that $bP_8 = \mathbb{Z}/28$, $bP_{12} = \mathbb{Z}/992$, $bP_{16} = \mathbb{Z}/8128$; the obstruction detecting the order is the signature of the bounding parallelizable manifold.

Proof sketch. A homotopy sphere bounding a parallelizable manifold has a bounding manifold with a framing, and the surgery obstruction of the framed manifold is its signature; the signature of a closed almost parallelizable $(4k)$-manifold is divisible by $8$ and the possible values are governed by the Bernoulli number through the Hirzebruch signature theorem. The group is cyclic because the possible signatures form a cyclic subgroup, and the order is the index computed by the $L$-genus. The signature theorem belongs to Characteristic Classes.

Example (Milnor's exotic spheres). $\Theta_7 = bP_8 = \mathbb{Z}/28$, so there are exactly $28$ oriented diffeomorphism classes of smooth structures on the topological seven-sphere; the $28$ classes are the Milnor spheres and their connected sums, and the spheres bounding parallelizable manifolds are detected by the signature of an $8$-manifold through the Hirzebruch signature theorem. The first exotic sphere discovered by Milnor is the total space of an $S^3$-bundle over $S^4$ associated with a map $S^3\to SO(4)$: it has a Morse function with exactly two critical points, so by Reeb's theorem it is homeomorphic to $S^7$, and its smooth structure is not standard, which is detected by the signature of the bounding disc bundle. The computation of $\Theta_7$ is the first complete answer to the question of the number of smooth structures on a sphere.

Theorem (the table of $\Theta_n$). The groups of homotopy spheres in low dimensions are

$n$	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18
$\Theta_n$	0	0	0	?	0	0	$\mathbb{Z}/28$	$\mathbb{Z}/2$	$(\mathbb{Z}/2)^3$	$\mathbb{Z}/6$	$\mathbb{Z}/992$	0	$\mathbb{Z}/3$	$\mathbb{Z}/2$	$\mathbb{Z}/8128\oplus\mathbb{Z}/2$	$\mathbb{Z}/2$	$(\mathbb{Z}/2)^4$	$\mathbb{Z}/8\oplus\mathbb{Z}/2$

The entry $\Theta_4$ is unknown: it is the smooth four-dimensional Poincaré conjecture, which is open, and the failure of the $h$-cobordism theorem in dimension four is what makes the group ill-behaved there. For $n\geq5$ the group is finite abelian, and for $n=3$ the vanishing is the Poincaré conjecture.

Proof sketch. The table is computed from the exact sequence of Kervaire–Milnor: the $bP$ subgroups are computed from the signature formula above, the quotients $\Theta_n/bP_{n+1}$ from the stable homotopy groups $\pi_n(G/O)$, and the Kervaire invariant obstruction from the theory of the framed manifolds. The computation of $\pi_n(G/O)$ is homotopy-theoretic and belongs to Algebraic Topology; the $J$-homomorphism and the Adams conjecture enter the identification of the image.

Theorem (the Kervaire invariant one problem; Hill–Hopkins–Ravenel, Lin–Wang–Xu). The Kervaire invariant $K : \pi_n(G/O)\to\mathbb{Z}/2$ is nonzero only if $n = 2^j-2$ for some $j$, that is only in dimensions $2,6,14,30,62$ and $126$; the elements of Kervaire invariant one exist in exactly those six dimensions, the cases up to $62$ classically and dimension $126$ by the theorem of Lin–Wang–Xu. Consequently the surgery obstruction of the Kervaire invariant is concentrated in those dimensions, and the manifolds of Kervaire invariant one are the exceptional objects in the classification of the framed manifolds.

Proof sketch. The problem is translated into a statement about the stable homotopy groups of spheres and the equivariant homotopy of the sphere spectrum, where the non-existence of the elements in dimensions above $126$ is proved by the computation of the slice spectral sequence; the existence in dimensions $2,6,14,30,62$ is by explicit construction and the existence in dimension $126$ by the refinements of the same equivariant computation. The homotopy theory belongs to Algebraic Topology.

Examples and Computations

Example (the unoriented cobordism ring in low dimensions). The even-dimensional polynomial generators of $\Omega_*^{O}$ can be taken to be the real projective spaces $\mathbb{RP}^{2i}$, whose Stiefel–Whitney numbers are the appropriate ones, and the odd-dimensional generators are supplied by the Dold manifolds; the monomials in the generators form an $\mathbb{F}_2$-basis, one for each partition of the degree into parts not of the form $2^k-1$. The dimension count for $n$ is therefore the number of such partitions: the values for $n=0,\ldots,8$ are $1,0,1,0,2,1,3,1,5$, obtained by counting the partitions of $n$ with the forbidden parts $1,3,7$ excluded. The class of $\mathbb{RP}^2$ generates $\Omega_2^O$, and the classes of $\mathbb{RP}^4$ and $\mathbb{RP}^2\times\mathbb{RP}^2$ together generate $\Omega_4^O$.

Example (oriented generators). The class of $\mathbb{CP}^{2k}$ generates the rational summand in dimension $4k$; the class of the Wu manifold $SU(3)/SO(3)$, a closed simply connected $5$-manifold with $w_2\neq0$ and $w_3\neq0$, generates $\Omega_5^{SO} = \mathbb{Z}/2$; and $\mathbb{CP}^2$ has signature $1$, so the signature detects the torsion-free part in dimension four. The class of $\mathbb{HP}^2$, of real dimension $8$, is a generator of the rank-two group $\Omega_8^{SO}$ alongside the class of $\mathbb{CP}^4$ — the two are distinguished by their Pontryagin numbers $(p_1^2,p_2)(\mathbb{HP}^2) = (4,7)$ and $(p_1^2,p_2)(\mathbb{CP}^4) = (25,10)$ — while the class of $\mathbb{CP}^2\times\mathbb{CP}^2$ satisfies $[\mathbb{HP}^2] = 3[\mathbb{CP}^2\times\mathbb{CP}^2] - 2[\mathbb{CP}^4]$ in $\Omega_8^{SO}$.

Example (the structure set of the sphere and of the lens spaces). Applied to $X = S^n$ with $n\geq5$, the surgery exact sequence gives the computation of $\Theta_n$ again; applied to $X = S^1\times S^{n-1}$ it computes the smooth structures on the torus-like manifolds; and applied to the lens spaces $X = L(p,q)$ it relates the structure set to the Whitehead group of $\mathbb{Z}/p$, the arithmetic of Lens Spaces. The failure of the sequence to give the classification in dimension four is the failure of the $h$-cobordism theorem: the terms of the sequence are still defined, but the conclusion that the vanishing of the obstruction yields a product fails, and the exotic structures of Low-Dimensional Topology are the result.

Remark (the low-dimensional exceptions). In dimension one and two the cobordism and surgery theory is the classification of the surfaces, and every closed surface is a boundary or a product as the case may be; in dimension three the classification is the geometrisation theorem, and the surgery obstructions are trivially computable because the topology is determined by the toral structure; in dimension four the classification is the theorem of Freedman in the topological category and the theorem of Donaldson in the smooth category, the $h$-cobordism theorem fails, and the whole high-dimensional framework breaks. The surgery theory of this article is therefore the theory of dimension at least five, and it is the standard against which the low-dimensional exceptional cases are measured.

Summary

Cobordism is the equivalence relation of bounding; the cobordism groups $\Omega_n^{\mathcal{S}}$ are the homotopy groups of the Thom spectra, the unoriented ring is the polynomial algebra $\mathbb{F}_2[x_i : i\neq2^k-1]$ by Thom, the oriented ring is rationally $\mathbb{Q}[\mathbb{CP}^2,\mathbb{CP}^4,\ldots]$ with $2$-torsion only, and the complex and spin cobordism rings are computed by Milnor, Novikov and the $\alpha$-invariant. Surgery cuts out an embedded $S^k\times D^{n-k}$ and glues back $D^{k+1}\times S^{n-k-1}$, killing the homotopy class of the embedded sphere and effecting the vanishing of the obstruction below the middle dimension.

The classification in dimension at least five is the surgery exact sequence $L_{n+1}(\mathbb{Z}[\pi])\to\mathcal{S}(X)\to[X,G/O]\to L_n(\mathbb{Z}[\pi])$, with the Wall $L$-groups of quadratic forms over the group ring as the obstruction groups: the obstruction is the signature in dimension $4k$ and the Kervaire invariant in dimension $4k+2$, and the structure set measures the difference between the Poincaré duality space and its manifold representatives. The $h$-cobordism theorem of Smale makes a simply connected $h$-cobordism of dimension at least five a product, the $s$-cobordism theorem of Barden, Mazur and Stallings corrects it by the Whitehead torsion, and the corollaries are the high-dimensional Poincaré conjecture and the group of homotopy spheres $\Theta_n$, finite abelian for $n\neq3,4$. The order of $bP_{4k}$ is $2^{2k-2}(2^{2k-1}-1)\,\mathrm{num}(4B_k/k)$, giving $28$, $992$, $8128$ in dimensions $8$, $12$, $16$, so that $\Theta_7 = \mathbb{Z}/28$ — Milnor's twenty-eight smooth structures on the seven-sphere — and the Kervaire invariant one problem, reduced by Hill–Hopkins–Ravenel to the single case of dimension $126$ and completed there by Lin–Wang–Xu, is the exceptional phenomenon of the framed theory. In dimensions three and four the enterprise is replaced by the geometrisation theorem and by the theorems of Freedman and Donaldson, and the exotic phenomena of dimension four are the failure of the general theory.

Summary of Notation

Symbol	Meaning
$\Omega_n^{\mathcal{S}}$, $\Omega_*^{\mathcal{S}}$	Cobordism groups and ring, $\mathcal{S}\in\{O,SO,U,Spin,\text{framed}\}$
$MO$, $MSO$, $MU$, $MSpin$	Thom spectra of the structure types; $\Omega_n^{\mathcal{S}}\cong\pi_n(M\mathcal{S})$
$[M]$	Cobordism class of a closed manifold
$x_i$	Polynomial generators; $i\neq2^k-1$ in the unoriented case
$S^k\times D^{n-k}\hookrightarrow M$	Surgery datum; the surgered manifold replaces it by $D^{k+1}\times S^{n-k-1}$
$W$, trace	Cobordism traced by the surgery, $W = M\times I\cup(k+1)$-handle
$G/O$	Classifying space of the stable normal invariants; $[X,G/O]$ the normal invariants
$L_n(\mathbb{Z}[\pi])$	Wall surgery obstruction group; $L_{4k}(\mathbb{Z})=\mathbb{Z}$, $L_{4k+2}(\mathbb{Z})=\mathbb{Z}/2$
$\sigma$	Surgery obstruction homomorphism $\Omega_n(X)\to L_n(\mathbb{Z}[\pi])$
$\mathcal{S}(X)$	Structure set of the homotopy equivalences to $X$ modulo $h$-cobordism
$\mathrm{Wh}(\pi)$	Whitehead group; the $s$-cobordism obstruction
$\Theta_n$	Group of oriented homotopy $n$-spheres under connected sum
$bP_{n+1}$	Homotopy spheres bounding parallelizable manifolds; $\vert bP_{4k}\vert = 2^{2k-2}(2^{2k-1}-1)\mathrm{num}(4B_k/k)$
$B_k$	Bernoulli numbers in the indexing $B_1=\frac16,B_2=\frac1{30},B_3=\frac1{42},\ldots$
Kervaire invariant	The obstruction $\pi_n(G/O)\to\mathbb{Z}/2$; nonzero only for $n=2^j-2$