03.02.28 · differential-geometry / manifolds

Pointer: surgery theory and the surgery exact sequence

shipped3 tiersLean: none

Anchor (Master): Wall Surgery on Compact Manifolds Ch. 9-17; Kervaire-Milnor Groups of homotopy spheres I (1963); Lück A Basic Introduction to Surgery Theory (2002); Ranicki Algebraic and Geometric Surgery (2002)

Intuition Beginner

The h-cobordism theorem answered one sharp question: when is a slab between two faces just a thickened face? Surgery theory asks the far broader question behind it. Fix a shape you understand well, say a sphere, and ask: how many genuinely different smooth shapes are bendy-equivalent to it — stretchable onto it without tearing, yet not the same shape? Counting these is the surgery programme. The h-cobordism theorem is the last step of every such count: once surgery has matched two shapes up to a slab, the theorem decides whether that slab collapses.

The tool is an operation on a shape, also called surgery. You cut out a tube, like coring an apple, and glue back a differently shaped plug into the hole. Each cut-and-plug changes the shape in a controlled way, the way swapping one handle of a coffee mug for another would. The programme is: start from the shape you want to reach, the model, and from a candidate that is bendy-equivalent to it; then perform a sequence of these cut-and-plug moves on the candidate, trying to massage it into the model.

Sometimes the massaging finishes cleanly and the candidate becomes the model. Sometimes it stalls, and what blocks it is a single bookkeeping number, an obstruction. The deep discovery is that this obstruction lives in a fixed list of groups depending only on the dimension and on the loops in the shape — the same group governs every shape of that kind. When the obstruction vanishes, the candidate can be massaged onto the model; when it does not, the candidate is genuinely a different shape.

This unit is a map of that country, a survey, not a full tour. It names the three milestones — the candidates, the obstruction, and the exact sequence that links them — and points to where each is built in full.

Visual Beginner

A surface is drawn with a small circular band marked on it, a collar around a loop. To the right, the same surface is redrawn with that band cut out, leaving two boundary circles, and a new piece — a flat disc pair, a differently shaped cap — glued in to fill the hole. An arrow labelled "one surgery" links the two pictures. Below, three boxes sit in a row connected by arrows: a box of candidate shapes, a box of cut-and-plug recipes, and a box holding a single number, the obstruction.

The takeaway from the picture: each surgery is a local cut-and-plug, the recipes are organised into a box, and whether the recipes can be carried through to reach the model is decided by one number sitting in the last box. The whole programme is the study of that row of three boxes and the arrows between them.

Worked example Beginner

Surgery that simplifies a surface. Take a torus, a doughnut surface, and run one surgery along the loop that goes the short way around the tube. Coring out a band around that loop leaves a tube with two boundary circles; capping each circle with a disc turns the doughnut into a sphere. One well-chosen cut-and-plug has simplified the torus all the way down to the sphere. This is the programme in miniature: the candidate was the torus, the model was the sphere, and a single surgery joined them.

When a cut-and-plug is forced to stall. Now imagine a candidate shape that is bendy-equivalent to a model, yet every cut-and-plug you try leaves behind a stubborn mismatch — a leftover that the moves can shuffle around but never erase. The leftover is the obstruction. In the surface case the leftover is measured by a single even number related to twisting, and only when it reads zero can the surgeries finish.

What this tells us. The torus-to-sphere move shows surgery working as advertised; the stalled case shows why an obstruction theory is needed at all. The bridge between them is the obstruction group: a fixed place where the leftover is recorded, the same place for every shape of a given dimension, so that "can we finish the surgeries?" becomes "does the recorded number vanish?".

Check your understanding Beginner

Formal definition Intermediate+

Fix a closed smooth manifold $X^{n}$ that serves as the model; the goal is to classify closed smooth manifolds homotopy equivalent to $X$ . The programme of Browder, Novikov, Sullivan, and Wall organises this into three sets and two maps.

Definition (structure set). The smooth structure set $S^{Diff} (X)$ is the set of equivalence classes of pairs $(M, h)$ where $M$ is a closed smooth manifold and $h : M \to X$ is a homotopy equivalence, with $(M_{0}, h_{0}) \sim (M_{1}, h_{1})$ when there is a diffeomorphism $g : M_{0} \to M_{1}$ such that $h_{1} \circ g$ is homotopic to $h_{0}$ . The base point is $(X, id)$ . The structure set measures how many smooth manifolds carry the homotopy type of $X$ .

Definition (normal invariants). A degree-one normal map to $X$ is a degree-one map $f : M \to X$ from a closed manifold together with a bundle map $b : ν_{M} \to ξ$ covering $f$ , where $ξ \to X$ is a stable bundle. Modulo normal bordism, these form the set of normal invariants $N (X)$ . By the Pontryagin-Thom construction and the theory of the Spivak normal fibration, $N (X) ≅ [X, G / O]$ , homotopy classes of maps to the homogeneous space $G / O$ — the fibre of $B O \to B G$ measuring the difference between stable vector bundles and stable spherical fibrations.

Definition (surgery obstruction groups). The $L$ -groups $L_{n} (Z π)$ , for $π = π_{1} (X)$ , are the Witt groups of nonsingular $(- 1)^{n /2}$ -quadratic forms (for $n$ even) and quadratic formations (for $n$ odd) over the group ring $Z [π]$ with the involution $\overline{g} = g^{- 1}$ . They are $4$ -periodic: $L_{n} (Z π) ≅ L_{n + 4} (Z π)$ . For the simple case $π = 1$ , $$ L_n(\mathbb{Z}) = \begin{cases} \mathbb{Z} & n \equiv 0 \ (4) \ \text{(signature}/8) \ 0 & n \equiv 1 \ (4) \ \mathbb{Z}/2 & n \equiv 2 \ (4) \ \text{(Arf-Kervaire)} \ 0 & n \equiv 3 \ (4). \end{cases} $$

The surgery exact sequence. For a closed manifold $X^{n}$ with $n \geq 5$ there is an exact sequence of pointed sets ^{[Wall 1970]} $$ \cdots \to L_{n+1}(\mathbb{Z}\pi) \to \mathcal{S}^{\mathrm{Diff}}(X) \xrightarrow{\ \eta\ } \mathcal{N}(X) \xrightarrow{\ \theta\ } L_n(\mathbb{Z}\pi), $$ where $η$ forgets the homotopy equivalence down to its underlying normal map and $θ$ assigns to a normal map its surgery obstruction.

Counterexamples to common slips

The structure set is a set with a base point, not a group in general; only after Sullivan-Wall's identification with a generalised cohomology in the topological category, or via the algebraic surgery exact sequence of Ranicki, does it acquire a transitive group action. Treating $S (X)$ as an abelian group from the outset is an error.
$N (X) = [X, G / O]$ , not $[X, B O]$ : the relevant space is the fibre $G / O$ , because a normal map records a stable bundle reduction of a fixed spherical fibration, not a bundle outright.
The surgery obstruction depends on the whole normal map $(f, b)$ , not on $M$ alone. The same manifold can underlie normal maps with different obstructions.

Key theorem with proof Intermediate+

The structural heart of the subject is the fundamental theorem of surgery: a normal map can be improved to a homotopy equivalence by surgeries exactly when its surgery obstruction vanishes. We prove the half that contains the geometry — below the middle dimension surgery always proceeds — which is what makes the obstruction live in a single dimension.

Theorem (surgery below the middle dimension). Let $(f, b) : M^{n} \to X^{n}$ be a degree-one normal map with $X$ connected. Then by a finite sequence of surgeries on $M$ , compatible with the bundle data, $f$ can be made $⌊ n /2 ⌋$ -connected: $f_ : \pi_i(M) \to \pi_i(X) $i s ani so m or p hi s m f or$ i < n/2 $an d a s u r j ec t i o n f or$ i = \lfloor n/2 \rfloor$.* ^{[Browder 1972]}

Proof. Surgery on an embedded sphere $S^{k} ↪ M$ with framed (identity-action) normal bundle replaces a neighbourhood $S^{k} \times D^{n - k}$ by $D^{k + 1} \times S^{n - k - 1}$ ; on homotopy it kills the class $[S^{k}] \in π_{k} (M)$ while possibly creating a class in degree $n - k - 1$ . Suppose $f$ is already $(k - 1)$ -connected with $k < n /2$ , so by relative Hurewicz the first non-vanishing relative group is $π_{k} (f) = π_{k} (X, M)$ , finitely generated as a $Z [π]$ -module since $X$ is a finite complex. Represent a generator by a map of a pair; the boundary gives a class $x \in π_{k - 1} (M)$ in the kernel of $f_{*}$ , hence — because $f$ is $(k - 1)$ -connected and $k - 1 < n /2$ , so $2 (k - 1) < n$ — Whitney's embedding theorem represents $x$ by an *embedded* sphere $S^{k - 1} ↪ M$ with an embedded null-bordism present after one more step. The bundle map $b$ frames the normal bundle of this sphere, providing the framing surgery requires. Performing the surgery kills $x$ without disturbing lower homotopy, since the surgery is in the right degree range. The dual class created sits in degree $n - k > n /2$ , above where we are working, so it does not reopen a lower group. Inducting on $k$ up to $⌊ n /2 ⌋$ makes $f$ as connected as claimed; the dimension bound $2 k < n$ is what keeps every embedding and framing available. $□$

Bridge. Putting these together, the foundational reason the surgery obstruction lives in a single group is exactly this theorem: everything below the middle dimension is cleared away for free, so the entire residue of the problem is concentrated in the middle, where it becomes a quadratic form (for $n$ even) or a formation (for $n$ odd) over $Z [π]$ . This is exactly the passage that identifies geometry with algebra — the kernel $K_{n /2} (M) = ker (f_{*} : H_{n /2} (M) \to H_{n /2} (X))$ carries the intersection pairing, and the bridge is that completing the surgery means making this pairing vanish, which is possible iff its Witt class in $L_{n} (Z π)$ is zero. The h-cobordism theorem of 03.02.23 is the central insight that closes the loop: once a normal map has been improved to a homotopy equivalence, the trace of the surgeries is an h-cobordism, and its product structure is what delivers an actual manifold in the structure set. This pattern — clear the easy range, concentrate the obstruction in the middle — appears again in the Kervaire-Milnor computation below and builds toward the whole $L$ -theoretic classification; it generalises the single $\pm 1$ cancellation of 03.02.22 to a Witt-class vanishing.

Exercises Intermediate+

Exercise (medium, short-answer). Explain why the surgery obstruction of a normal map $(f, b) : M^n \to X^n$ can be made to live in a group depending only on $n \bmod 4$ and $\pi_1(X)$.

Hint

Use the below-the-middle theorem to clear all homotopy below $n /2$ , then ask what data remains.

Answer

Argument. By the surgery-below-the-middle theorem, $f$ is made $⌊ n /2 ⌋$ -connected. The only surviving kernel is $K_{n /2} (M)$ , a finitely generated $Z [π]$ -module carrying the equivariant intersection form, which is $(- 1)^{n /2}$ -symmetric and quadratically refined by the bundle data $b$ . Its Witt class — the form modulo hyperbolic (cancellable) summands — is the obstruction, and the Witt group of such forms over $Z [π]$ is by definition $L_{n} (Z π)$ , which depends only on $n mod 4$ (the sign and form-versus-formation alternation) and on $π$ .

Exercise (hard, short-answer). Given exactness of $\mathcal{S}(X) \xrightarrow{\eta} \mathcal{N}(X) \xrightarrow{\theta} L_n(\mathbb{Z}\pi)$, explain what it means for a normal invariant to lie in the image of $\eta$, and how the h-cobordism theorem is used to produce the corresponding element of the structure set.

Hint

Image of $η$ equals kernel of $θ$ ; a vanishing obstruction lets surgery finish.

Answer

Argument. By exactness, a normal invariant is in $im (η)$ iff its surgery obstruction $θ$ vanishes. When $θ = 0$ , the middle-dimensional form is Witt-null (hyperbolic up to stabilisation), so further surgeries kill $K_{n /2}$ and improve the normal map $f : M \to X$ to a homotopy equivalence. The trace of all these surgeries is a normal bordism $W$ from $M$ to a manifold $M^{'}$ with $M^{'} ≃ X$ ; rel the homotopy equivalence, $W$ is an h-cobordism, and by 03.02.23 (for $dim \geq 6$ , simply connected, or the $s$ -cobordism refinement otherwise) it is a product. The product structure identifies $M^{'}$ as a genuine smooth manifold homotopy equivalent to $X$ , i.e. a well-defined class $(M^{'}, h) \in S (X)$ mapping to the given normal invariant.

Advanced results Master

The Kervaire-Milnor computation of exotic spheres. The first triumph of the surgery method, predating its full axiomatisation, is the computation of the group $Θ_{n}$ of smooth structures on the topological $n$ -sphere under connected sum ^{[Kervaire-Milnor 1963]}. A homotopy $n$ -sphere is stably parallelisable, so it admits a degree-one normal map to $S^{n}$ whose surgery up to the middle dimension is unobstructed; the residual obstruction is exactly an element of $L_{n} (Z) = (Z, 0, Z /2, 0)$ . Kervaire and Milnor assemble this into an exact sequence $$ 0 \to bP_{n+1} \to \Theta_n \to \pi_n^s / J \to (\text{coker into } L_{n+1}), $$ where $b P_{n + 1} \subseteq Θ_{n}$ is the subgroup of homotopy spheres bounding parallelisable manifolds, $π_{n}^{s}$ is the stable homotopy group of spheres, and $J$ is the image of the $J$ -homomorphism. The subgroup $b P_{n + 1}$ is finite cyclic and computed directly: for $n + 1 = 4 k$ it has order growing with the Bernoulli numbers via the signature, giving for instance $∣ b P_{4 k} ∣$ on the order of $2^{2 k - 2} (2^{2 k - 1} - 1) num (B_{k} /4 k)$ . The classical output $∣ b P_{8} ∣ = 28$ recovers Milnor's twenty-eight smooth structures on $S^{7}$ , the exotic spheres of 03.06.17, now seen not as a curiosity but as the $L_{8} (Z) = Z$ obstruction made geometric. The remaining $Z /2$ Kervaire pieces, $b P_{4 k + 2}$ , are where the deep Kervaire invariant problem lives, resolved (except $n = 126$ ) by Hill-Hopkins-Ravenel.

Wall's non-simply-connected theory. Wall's Surgery on Compact Manifolds ^{[Wall 1970]} extends the obstruction to arbitrary $π = π_{1} (X)$ by replacing $Z$ -coefficient forms with $Z [π]$ -coefficient quadratic forms and formations under the involution $\overline{g} = g^{- 1}$ , defining $L_{n} (Z π)$ in full generality and establishing the exact sequence with these coefficients. This is where the Whitehead-torsion refinement of 03.08.20 re-enters: in the simple-homotopy version one uses $L^{s}$ -groups built from based forms, and the structure set is the simple structure set, so that the h-cobordisms produced are products by the $s$ -cobordism theorem rather than merely by the simply connected h-cobordism theorem.

Topological invariance and the assembly map. Novikov's theorem on the topological invariance of rational Pontryagin classes ^{[Novikov 1964]} is the result that makes a topological surgery theory possible, with $G / Top$ replacing $G / O$ and a cleaner periodicity. Ranicki's algebraic theory ^{[Ranicki 2002]} reinterprets the whole exact sequence as the long exact sequence of an assembly map $A : H_{n} (X; L_{∙}) \to L_{n} (Z π)$ from the homology with coefficients in the $L$ -theory spectrum to the $L$ -group of the group ring; the structure set becomes the homotopy fibre, and the Novikov and Borel conjectures become statements about $A$ .

Synthesis. Surgery theory is the central insight that turns the classification of manifolds within a homotopy type into a computation in algebraic $L$ -theory, and the foundational reason it works is the concentration of all obstruction into the middle dimension proved above. Putting these together: the below-the-middle theorem clears the easy range, the middle pairing is exactly the quadratic form whose Witt class is the surgery obstruction, and the surgery exact sequence $S (X) \to N (X) \to L_{n} (Z π)$ identifies the gap between bendy-equivalence and diffeomorphism with the kernel and cokernel of the obstruction map. This is exactly the programme that generalises the h-cobordism theorem of 03.02.23: that theorem is the special case $X = D^{n}$ , where there is no middle-dimensional residue and the structure set is a single point, and it reappears at the end of every surgery as the step that converts a finished normal bordism into a manifold. The theory is dual, through Poincaré duality 03.12.16, to the structure of the intersection form, and the same alternation of symmetric and skew forms that gives the $4$ -periodicity is the load-bearing pattern of the whole subject. From the single $\pm 1$ cancellation of the handle calculus to the Witt group of $Z [π]$ , the bridge is unbroken.

Full proof set Master

Proposition (the surgery obstruction of a closed manifold normal self-map vanishes; the signature obstruction). Let $X^{4 k}$ be a closed oriented manifold and $(f, b) : M^{4 k} \to X^{4 k}$ a degree-one normal map. Then the surgery obstruction $θ (f, b) \in L_{4 k} (Z) = Z$ equals $\frac{1}{8} (sign (M) - sign (X))$ , and in particular it vanishes when $M$ and $X$ have equal signatures.

Proof. After surgery below the middle dimension, $f$ is $2 k$ -connected, and the only surviving kernel is $K_{2 k} (M) = ker (f_{*} : H_{2 k} (M; Q) \to H_{2 k} (X; Q))$ (working rationally to read off the signature). Since $f$ has degree one, $f_{*}$ is a split surjection on homology with the splitting given by $f^{!}$ (the Umkehr map), so $H_{2 k} (M) ≅ H_{2 k} (X) \oplus K_{2 k} (M)$ as a direct sum orthogonal with respect to the intersection form, because $f$ preserves the cup-product pairing and the $X$ -summand pairs with the kernel to zero by the projection formula. Hence the intersection form on $M$ splits as the form on $X$ plus the form on the kernel: $⟨, ⟩_{M} = ⟨, ⟩_{X} \oplus ⟨, ⟩_{K}$ . Taking signatures, which are additive over orthogonal direct sums, $sign (M) = sign (X) + sign (K_{2 k} (M))$ , so $sign (K) = sign (M) - sign (X)$ . The surgery obstruction in $L_{4 k} (Z) = Z$ is by definition $\frac{1}{8} sign (K)$ — the form on the kernel is even (the quadratic refinement from $b$ forces even diagonal), and an even nonsingular symmetric form has signature divisible by $8$ with Witt class detected by signature $/8$ (the rank- $8$ generator being the $E_{8}$ form). Therefore $θ (f, b) = \frac{1}{8} (sign (M) - sign (X))$ , vanishing precisely when the signatures agree. $□$

Proposition (vanishing obstruction permits completion to a homotopy equivalence). If $θ (f, b) = 0 \in L_{n} (Z π)$ with $n \geq 5$ and $f$ already $⌊ n /2 ⌋$ -connected, then finitely many further surgeries in the middle dimension improve $(f, b)$ to a normal map that is a homotopy equivalence.

Proof. For $n = 2 m$ even, the obstruction is the Witt class of the nonsingular $(- 1)^{m}$ -quadratic form $(K_{m} (M), λ, μ)$ over $Z [π]$ . Vanishing of the Witt class means that after adding hyperbolic summands $H (Z [π])$ — each created by a single surgery on a framed embedded $S^{m}$ with a complementary dual sphere, which the connectivity and the bundle data $b$ supply for $2 m \geq 5$ — the form becomes the standard hyperbolic form on a free module. A hyperbolic basis consists of pairs $(e_{i}, f_{i})$ with $λ (e_{i}, f_{i}) = 1$ , $λ (e_{i}, e_{j}) = λ (f_{i}, f_{j}) = 0$ , and $μ (e_{i}) = 0$ . The vanishing of $μ$ on the $e_{i}$ is exactly the condition (the quadratic, not merely homological, vanishing) under which each $e_{i}$ is represented by an embedded sphere with framed normal bundle, so surgery on it is defined; performing surgery on a maximal hyperbolic-isotropic set of such spheres kills $K_{m} (M)$ entirely without creating new middle homology, because each dual $f_{i}$ caps off the sphere created. With $K_{m} = 0$ and all lower and (by Poincaré duality 03.12.16) all higher kernels already zero, $f_{*}$ is an isomorphism on all homology with $Z [π]$ -coefficients; since $f$ is a $π_{1}$ -isomorphism by $1$ -connectivity, the homology Whitehead theorem upgrades this to a homotopy equivalence. For $n$ odd the same conclusion follows from the vanishing of the formation obstruction, where the relevant move cancels a pair of complementary surgeries. $□$

This proposition is the geometric content of exactness at $N (X)$ : the surgery obstruction map $θ$ has kernel exactly the normal invariants that lift to the structure set, and the h-cobordism theorem of 03.02.23 supplies the final product structure that names the resulting manifold.

Connections Master

This unit is the downstream completion of the handle and cobordism programme of 03.02.20: the elementary cobordism that 03.02.20 attaches to a single surgery is the atom of the whole theory, and the cobordism category there is the ambient setting in which a normal bordism between $M$ and a model lives; surgery theory is what happens when one runs that category with bundle data and a connectivity target rather than blindly.

The h-cobordism theorem of 03.02.23 is the closing step of every surgery argument: once a normal map is improved to a homotopy equivalence, the trace of the surgeries is an h-cobordism, and its product structure — by the simply connected theorem, or by the Whitehead-torsion-controlled $s$ -cobordism refinement when $π_{1} \neq = 1$ — is what converts the algebraic vanishing of the obstruction into an actual manifold in the structure set; without 03.02.23 the exact sequence would compute an obstruction but never deliver a shape.

The exotic spheres of 03.06.17 are the first and most famous surgery computation made concrete: the twenty-eight smooth structures on $S^{7}$ are the order of $b P_{8}$ inside $Θ_{7}$ , and the surgery viewpoint reinterprets Milnor's combinatorial Pontryagin-class invariant as the $L_{8} (Z) = Z$ signature obstruction, so the present unit is the structural home of that example.

The non-simply-connected obstruction routes through the Whitehead torsion and $s$ -cobordism theory of 03.08.20: the simple structure set uses $L^{s}$ -groups built from based quadratic forms over $Z [π]$ , and the torsion of the normal bordism is exactly the datum that 03.08.20 measures, so the two units describe the same $K$ - and $L$ -theoretic obstruction package from the algebraic and the geometric sides.

The middle-dimensional intersection form whose Witt class is the surgery obstruction is the equivariant refinement of the pairing of 03.12.16: Poincaré duality is what makes the kernel $K_{n /2} (M)$ self-dual and so a nonsingular form at all, and the $4$ -periodicity of the $L$ -groups is the algebraic shadow of the alternation between the symmetric and skew cases of that duality.

Historical & philosophical context Master

Surgery theory grew out of the same handle calculus that proved the h-cobordism theorem, but turned outward: where Smale asked when a cobordism is a product, Milnor, Kervaire, Browder, Novikov, Sullivan, and Wall asked how to manufacture and classify manifolds within a homotopy type. The decisive first computation was Kervaire and Milnor's Groups of homotopy spheres I (Annals of Mathematics 77, 1963, 504–537) ^{[Kervaire-Milnor 1963]}, which organised the smooth structures on spheres into the finite abelian groups $Θ_{n}$ and computed them by surgering framed manifolds down to spheres, the obstruction falling into the groups later named $L_{*} (Z)$ . William Browder's Surgery on Simply-Connected Manifolds (Springer, 1972) ^{[Browder 1972]} and, in the foundational papers of the mid-1960s, Sergei Novikov's work — including his theorem on the topological invariance of rational Pontryagin classes (Izv. Akad. Nauk SSSR 28, 1964) ^{[Novikov 1964]} — established the simply connected theory and the role of the Spivak normal fibration. C. T. C. Wall's Surgery on Compact Manifolds (Academic Press, 1970) ^{[Wall 1970]} is the field's monument: it built the $L$ -groups $L_{n} (Z π)$ over arbitrary group rings and stated the surgery exact sequence in the generality used ever since.

Philosophically, surgery theory marks the maturation of the algebraicisation that the h-cobordism theorem began. The h-cobordism theorem reads a diffeomorphism type off a single vanishing homology group; surgery theory reads the entire moduli of manifolds in a homotopy type off the kernel and cokernel of a map into a Witt group of forms over a group ring. The manifold becomes, in Ranicki's later reformulation ^{[Ranicki 2002]}, the homotopy fibre of an assembly map — a fully homotopy-theoretic object. The persistent four-dimensional exception, where the middle dimension coincides with the obstruction dimension and the Whitney trick fails, remains the standing reminder that the entire algebraic machine rests, as the h-cobordism theorem did, on a single geometric fact about embedding discs.

Bibliography Master

@article{kervairemilnor1963,
  author  = {Kervaire, Michel A. and Milnor, John W.},
  title   = {Groups of homotopy spheres: {I}},
  journal = {Annals of Mathematics},
  volume  = {77},
  number  = {3},
  pages   = {504--537},
  year    = {1963}
}

@book{wall1970surgery,
  author    = {Wall, C. T. C.},
  title     = {Surgery on Compact Manifolds},
  series    = {London Mathematical Society Monographs},
  publisher = {Academic Press},
  year      = {1970},
  note      = {2nd ed., ed. A. A. Ranicki, AMS, 1999}
}

@book{browder1972surgery,
  author    = {Browder, William},
  title     = {Surgery on Simply-Connected Manifolds},
  series    = {Ergebnisse der Mathematik und ihrer Grenzgebiete},
  volume    = {65},
  publisher = {Springer-Verlag},
  year      = {1972}
}

@article{novikov1964,
  author  = {Novikov, Sergei P.},
  title   = {Homotopy equivalent smooth manifolds {I}},
  journal = {Izv. Akad. Nauk SSSR Ser. Mat.},
  volume  = {28},
  pages   = {365--474},
  year    = {1964},
  note    = {English transl.: AMS Translations 48 (1965), 271--396}
}

@book{milnor1965hcobordism,
  author    = {Milnor, John W.},
  title     = {Lectures on the h-Cobordism Theorem},
  series    = {Princeton Mathematical Notes},
  publisher = {Princeton University Press},
  year      = {1965},
  note      = {Notes by L. Siebenmann and J. Sondow}
}

@book{ranicki2002,
  author    = {Ranicki, Andrew},
  title     = {Algebraic and Geometric Surgery},
  series    = {Oxford Mathematical Monographs},
  publisher = {Oxford University Press},
  year      = {2002}
}

@book{luck2002surgery,
  author    = {L{\"u}ck, Wolfgang},
  title     = {A Basic Introduction to Surgery Theory},
  series    = {ICTP Lecture Notes},
  volume    = {9},
  publisher = {ICTP, Trieste},
  year      = {2002}
}

Prerequisites

03.02.20
03.02.23

Tier anchors

beginner: Milnor Lectures on the h-Cobordism Theorem §3 (surgery on a level set); Ranicki Algebraic and Geometric Surgery Ch. 1 (informal programme)
intermediate: Browder Surgery on Simply-Connected Manifolds Ch. I-III; Wall Surgery on Compact Manifolds Ch. 1-3, 9-10 (the surgery exact sequence)
master: Wall Surgery on Compact Manifolds Ch. 9-17; Kervaire-Milnor Groups of homotopy spheres I (1963); Lück A Basic Introduction to Surgery Theory (2002); Ranicki Algebraic and Geometric Surgery (2002)

References

Kervaire, M. and Milnor, J. — Groups of homotopy spheres I (1963) · Annals of Mathematics 77, pp. 504--537; the exact sequence relating Theta_n, bP_{n+1}, and stable homotopy; the surgery computation of exotic spheres
Wall, C. T. C. — Surgery on Compact Manifolds (1970) · Academic Press; Ch. 9-10: the surgery obstruction groups L_n(Z pi) and the surgery exact sequence S(X) -> N(X) -> L_n(Z pi)
Browder, W. — Surgery on Simply-Connected Manifolds (1972) · Springer Ergebnisse 65; the simply connected surgery obstruction as signature/8 or Arf-Kervaire invariant; normal maps and the Spivak fibration
Novikov, S. P. — Homotopy equivalent smooth manifolds I (1964) · Izv. Akad. Nauk SSSR 28, pp. 365--474 (AMS Transl. 48, 1965); topological invariance of rational Pontryagin classes and the foundations of surgery
Milnor, J. — Lectures on the h-Cobordism Theorem (1965) · Princeton Mathematical Notes; §3: surgery (spherical modification) on a level set, the base case the surgery programme generalises
Ranicki, A. — Algebraic and Geometric Surgery (2002) · Oxford Mathematical Monographs; the surgery exact sequence, L-theory as the obstruction theory, the assembly map

Estimated time

beginner: 16m
intermediate: 44m
master: 92m