03.02.20 · differential-geometry / manifolds

Handles, surgery, and the cobordism category

shipped3 tiersLean: none

Anchor (Master): Milnor Lectures on the h-Cobordism Theorem §§2-3; Wallace Modifications and cobounding manifolds (1960); Kosinski Ch. VI-VII

Intuition Beginner

Think of building a clay model not by carving a single block but by gluing standard pieces onto a base. You start with a flat slab. Onto its rim you glue a curved bridge that arcs over and lands back on the slab, like the handle of a coffee mug. Each glued piece is a handle. A surprising fact of geometry is that every smooth shape, however intricate, can be assembled this way from a short menu of standard handles.

The menu is graded by how the handle attaches. A handle can be glued along two separate spots, dragging a bridge between them. It can be glued along a full ring, capping a tube. It can be a solid blob set down on its own. Each kind of attachment changes the shape in one controlled way: it can join two pieces, or open a hole, or close a hole, or fill a cavity.

This is the same bookkeeping as the rising-water picture from the height-function story, now made solid. There, each special height added a flat cell. Here, each special height adds a fat, three-dimensional handle. The flat cell is the skeleton; the handle is the cell with meat on its bones.

The payoff is a dictionary. A smooth shape becomes a list of handles. Surgery is the single move that turns one boundary shape into the next as you add a handle. And cobordism is the record of the whole journey: the solid region swept out between the shape you started with and the shape you ended with.

Visual Beginner

A solid slab sits on the left, a flat plate of clay. A bridge-shaped handle is glued onto its top, attached along two small disks where its feet land. The result, on the right, is a plate with a raised arch — the start of a doughnut. Below, the same handle is shown alone as a fat rectangle bent into an arch, with its two feet marked.

The takeaway from the picture: the handle is glued only along its feet, the two shaded disks. Everything else of the handle becomes new free surface. The solid lump filling the gap between the old plate and the new arch is the cobordism — the solid history of the change.

Worked example Beginner

Building a doughnut surface from handles. Start with a single solid disk, a round coin of clay. This is the base. Watch the rim of the model as we glue on two handles in turn.

First handle: glue a bent bar onto the rim, attached at two small arcs on the boundary, so the bar arches up and over and lands back down. The two feet are joined by the bar. The boundary of the model, which was one circle, is now reshaped — the bar has bridged across it. We have added a one-dimensional handle.

Second handle: glue another bent bar across the model the other way, so the two arches cross over each other and the surface closes up into a ring. The boundary disappears entirely; the model is now a closed doughnut surface.

So the doughnut surface is assembled from one disk plus two bent-bar handles. Count: one base, two handles. The two handles are exactly the two loops you can draw on a doughnut — one around the ring, one through the hole.

What this tells us. The number and kind of handles are not an accident of how we built it. Any handle assembly of the doughnut surface needs at least these handles: a base, and two bent bars for the two independent loops. Counting handles measures the shape, just as counting peaks and passes did for the height function.

Check your understanding Beginner

Formal definition Intermediate+

Fix a dimension $n$ and write $D^{k}$ for the closed unit $k$ -disk, $S^{k - 1} = \partial D^{k}$ . A point of the product $D^{λ} \times D^{n - λ}$ has a core coordinate in $D^{λ}$ and a cocore coordinate in $D^{n - λ}$ .

Definition (handle). A handle of index $λ$ in dimension $n$ is the product $H^{λ} = D^{λ} \times D^{n - λ}$ , regarded as a smooth manifold-with-corners. Its distinguished subsets are the core $D^{λ} \times {0}$ , the cocore ${0} \times D^{n - λ}$ , the attaching region $S^{λ - 1} \times D^{n - λ} \subset \partial H^{λ}$ , the attaching sphere $S^{λ - 1} \times {0}$ , and the belt sphere ${0} \times S^{n - λ - 1}$ .

Definition (handle attachment). Let $W$ be a smooth $n$ -manifold whose boundary contains a piece $\partial_{+} W$ , and let $φ : S^{λ - 1} \times D^{n - λ} ↪ \partial_{+} W$ be a smooth embedding. Attaching a $λ$ -handle along $φ$ produces $W \cup_{φ} H^{λ} = (W ⊔ (D^{λ} \times D^{n - λ})) / (φ (u) \sim u for u \in S^{λ - 1} \times D^{n - λ}) .$ The quotient is naturally a topological manifold-with-corners; the straightening-the-angle construction (smoothing the right-angle ridge formed along the attaching region) endows it with a smooth structure, unique up to diffeomorphism rel $W$ . This is the smooth refinement of the CW cell-attachment of 03.02.31: the core $D^{λ} \times {0}$ is the $λ$ -cell, the attaching sphere is its boundary, and the cocore directions fatten the cell into a manifold.

Definition (cobordism). A cobordism $(W; V_{0}, V_{1})$ consists of a compact smooth $n$ -manifold $W$ together with a decomposition of its boundary $\partial W = V_{0} ⊔ V_{1}$ into two disjoint closed $(n - 1)$ -manifolds. The $V_{i}$ are the ends. Two closed $(n - 1)$ -manifolds are cobordant if some such $W$ exists; this is an equivalence relation ^{[Thom 1954]}. The product $V \times [0, 1]$ is the product cobordism, also written $V \times I$ .

Definition (Morse function on a cobordism). A Morse function on $(W; V_{0}, V_{1})$ is a smooth $f : W \to [0, 1]$ with $f^{- 1} (0) = V_{0}$ , $f^{- 1} (1) = V_{1}$ , no critical points on the boundary, and all interior critical points nondegenerate in the sense of 03.02.30. A gradient-like vector field for $f$ is a vector field $ξ$ with $ξ (f) > 0$ off the critical points and equal to the Euclidean gradient of the Morse normal form near each. Such a pair $(f, ξ)$ always exists: triangulate the existence proof of 03.02.31 relative to the boundary collar.

Definition (surgery). Let $V$ be a closed $(n - 1)$ -manifold and $φ : S^{λ - 1} \times D^{n - λ} ↪ V$ a smooth embedding (a framed embedded sphere). The surgery (or spherical modification) of $V$ along $φ$ is $χ (V, φ) = (V ∖ φ (S^{λ - 1} \times \overset{˚}{D}^{n - λ})) \cup_{φ ∣_{S^{λ - 1} \times S^{n - λ - 1}}} (D^{λ} \times S^{n - λ - 1}),$ removing a tubular neighbourhood $S^{λ - 1} \times D^{n - λ}$ of the embedded sphere and gluing in $D^{λ} \times S^{n - λ - 1}$ along the shared boundary $S^{λ - 1} \times S^{n - λ - 1}$ ^{[Wallace 1960]}. The two halves share the boundary $S^{λ - 1} \times S^{n - λ - 1}$ , so $χ (V, φ)$ is again a closed smooth $(n - 1)$ -manifold.

Counterexamples to common slips

A handle is not glued along its whole boundary. It is glued only along the attaching region $S^{λ - 1} \times D^{n - λ}$ , which is the part of $\partial H^{λ}$ where the core sphere lives. The complementary part $D^{λ} \times S^{n - λ - 1}$ becomes new boundary — that complementary part is exactly the surgered piece.
The index $λ$ and the codimension $n - λ$ are different data. A $1$ -handle in dimension $3$ ( $λ = 1$ , $n - λ = 2$ ) attaches along $S^{0} \times D^{2}$ — two solid disks — whereas a $2$ -handle in dimension $3$ attaches along $S^{1} \times D^{1}$ — an annulus. Same ambient dimension, opposite attaching pictures.
Surgery requires a framing: an embedding of the whole tube $S^{λ - 1} \times D^{n - λ}$ , not merely of the core sphere $S^{λ - 1}$ . Different framings of the same embedded sphere can give non-diffeomorphic results.

Key theorem with proof Intermediate+

The structural theorem joining the three definitions is Milnor's elementary-cobordism theorem ^{[Milnor §3]}, the smooth upgrade of Theorem 3.2 of 03.02.31.

Theorem (elementary cobordism $\leftrightarrow$ one handle $\leftrightarrow$ one surgery). Let $(W; V_{0}, V_{1})$ carry a Morse function $f$ with exactly one critical point $p$ , of index $λ$ , and a gradient-like field $ξ$ . Then:

(i) $W$ is diffeomorphic to $V_{0} \times I$ with a single $λ$ -handle attached along an embedding $φ : S^{λ - 1} \times D^{n - λ} ↪ V_{0} \times {1}$ determined by the descending sphere of $ξ$ at $p$ ;

(ii) the upper end is the surgery $V_{1} ≅ χ (V_{0}, φ)$ ;

(iii) conversely, the trace of any surgery $χ (V_{0}, φ)$ — the manifold built by attaching the corresponding handle to $V_{0} \times I$ — is an elementary cobordism with one critical point of index $λ$ .

A cobordism with a single critical point is called an elementary cobordism.

Proof. Normalise $f (p) = 1/2$ and choose $ε > 0$ so that $f^{- 1} [\frac{1}{2} - ε, \frac{1}{2} + ε]$ contains only $p$ . By Theorem 3.1 of 03.02.31 applied to $f$ and to $1 - f$ , the regions $f^{- 1} [0, \frac{1}{2} - ε]$ and $f^{- 1} [\frac{1}{2} + ε, 1]$ are products $V_{0} \times I$ and $V_{1} \times I$ , so it suffices to analyse the slab $f^{- 1} [\frac{1}{2} - ε, \frac{1}{2} + ε]$ around the single critical level.

By the Morse lemma 03.02.30 take coordinates $(u, v) \in R^{λ} \times R^{n - λ}$ centred at $p$ with $f = \frac{1}{2} - ∣ u ∣^{2} + ∣ v ∣^{2}$ and $ξ$ the Euclidean gradient $(- u, v)$ . The descending sphere is ${∣ u ∣^{2} = ε, v = 0}$ , a sphere $S^{λ - 1}$ embedded in the lower level $f^{- 1} (\frac{1}{2} - ε) ≅ V_{0}$ ; the ascending sphere is ${u = 0, ∣ v ∣^{2} = ε}$ , a sphere $S^{n - λ - 1}$ in the upper level. Flowing $ξ$ backward from a tubular neighbourhood of the descending sphere produces an embedding $φ : S^{λ - 1} \times D^{n - λ} ↪ V_{0}$ . The slab is the union of the product region outside the chart with the chart block ${∣ u ∣^{2} \leq ε, ∣ v ∣^{2} \leq ε}$ , and that block is precisely $D^{λ} \times D^{n - λ}$ glued to $V_{0} \times I$ along $φ$ . This proves (i).

For (ii), the upper end $V_{1} = f^{- 1} (\frac{1}{2} + ε)$ agrees with $V_{0}$ away from the chart. Inside the chart, the level ${- ∣ u ∣^{2} + ∣ v ∣^{2} = ε}$ removes the locus where $∣ u ∣^{2} < ε$ at $v = 0$ — a copy of $S^{λ - 1} \times \overset{˚}{D}^{n - λ}$ , the open attaching tube — and replaces it by the set ${∣ u ∣^{2} \leq ε} \times {∣ v ∣^{2} = ε}$ , a copy of $D^{λ} \times S^{n - λ - 1}$ , glued along their common boundary ${∣ u ∣^{2} = ε, ∣ v ∣^{2} = ε} ≅ S^{λ - 1} \times S^{n - λ - 1}$ . That is exactly $χ (V_{0}, φ)$ .

For (iii), reverse the construction: given $φ$ , attach $H^{λ} = D^{λ} \times D^{n - λ}$ to $V_{0} \times I$ along $φ (\cdot) \times {1}$ and straighten the corner. The function $f = \frac{1}{2} - ∣ u ∣^{2} + ∣ v ∣^{2}$ in the handle coordinates, extended by the height coordinate on the product part, is Morse with one critical point of index $λ$ , and the gradient of this expression is gradient-like. The upper end is $χ (V_{0}, φ)$ by the computation in (ii). $□$

Bridge. This theorem is the foundational reason the topology of a manifold can be read off a single real-valued function: passing one nondegenerate critical point is exactly one handle attachment, which is exactly one surgery on the level set. It builds toward the rearrangement and self-indexing of 03.02.21, where the handles attached by a general Morse function are sorted by index into a clean filtration, and the central insight — that an elementary cobordism is a handle is a surgery — generalises the cell-attachment picture of 03.02.31 from homotopy type to smooth structure. The bridge is that the descending sphere of the gradient-like field carries the entire attaching datum, so the analytic object (the Morse function) and the combinatorial object (the handle, with its attaching framing) determine each other. This identification of cobordism with handle attachment appears again in 03.02.21 as the spine of the $h$ -cobordism proof, where putting these together lets one cancel handles algebraically.

Exercises Intermediate+

Exercise (medium, symbolic). Let $(W; V_0, V_1)$ be an elementary cobordism with one critical point of index $\lambda$ in dimension $n$. Express the effect of passing the handle on the Euler characteristics: write $\chi(V_1) - \chi(V_0)$ in terms of $\lambda$ and $n$.

Hint

Surgery removes $S^{λ - 1} \times D^{n - λ}$ and glues in $D^{λ} \times S^{n - λ - 1}$ ; compare their Euler characteristics relative to the shared boundary.

Answer

$χ (V_{1}) - χ (V_{0}) = (- 1)^{λ} (χ (S^{λ}) - χ (S^{λ - 1}) \dots)$ , equivalently $(- 1)^{λ} + (- 1)^{n - 1 - λ} - (- 1)^{n - 1} \cdot 0$ collapses to $(- 1)^{λ} (1 - (- 1)^{n - λ})$ . Concretely: removing $S^{λ - 1} \times D^{n - λ}$ and gluing $D^{λ} \times S^{n - λ - 1}$ over the same boundary changes $χ$ by $χ (D^{λ} \times S^{n - λ - 1}) - χ (S^{λ - 1} \times D^{n - λ}) = χ (S^{n - λ - 1}) - χ (S^{λ - 1})$ , since the disk factors contribute $1$ . For $n - 1$ even this vanishes (odd-dimensional surgery preserves $χ$ ); for $n - 1$ odd it can be $\pm 2$ .

Exercise (medium, short-answer). Explain why a surgery requires an embedding of the framed tube $S^{\lambda-1} \times D^{n-\lambda}$ rather than just the core sphere $S^{\lambda-1}$, and give the geometric role of the framing.

Hint

The glued-in piece $D^{λ} \times S^{n - λ - 1}$ must match the removed piece along its boundary.

Answer

Rubric. Full credit: the surgery removes a tubular neighbourhood, which is a copy of $S^{λ - 1} \times D^{n - λ}$ , and the replacement $D^{λ} \times S^{n - λ - 1}$ is glued along the common boundary $S^{λ - 1} \times S^{n - λ - 1}$ . To specify this gluing one needs a trivialisation of the normal bundle of the embedded sphere — a framing — i.e. an embedding of the whole tube, not just its core. Different framings give different gluing maps and can yield non-diffeomorphic results, so the framing is genuine data.

Exercise (hard, short-answer). Show that the product cobordism $V \times I$ admits a Morse function with no critical points, and conversely that a cobordism whose Morse function has no critical points is diffeomorphic to a product.

Hint

Use the projection $V \times I \to I$ one way; use Theorem 3.1 of 03.02.31 (integrate a gradient-like field) the other way.

Answer

Argument. The projection $f (x, t) = t$ on $V \times I$ has $df = d t \neq = 0$ everywhere, so it is Morse with no critical points and ends $V \times {0}, V \times {1}$ . Conversely, if $f : W \to [0, 1]$ is Morse with $f^{- 1} (0) = V_{0}$ , $f^{- 1} (1) = V_{1}$ and no critical points, pick a gradient-like field $ξ$ with $ξ (f) = 1$ ; its flow is defined for all relevant times by compactness and carries $V_{0}$ diffeomorphically to each level $f^{- 1} (s)$ . The map $(x, s) \mapsto ϕ_{s} (x)$ for $x \in V_{0}$ is a diffeomorphism $V_{0} \times I \to W$ , so $W ≅ V_{0} \times I$ .

Exercise (hard, short-answer). Let $\varphi : S^{n-1} \times D^1 \hookrightarrow V$ be an embedding (so $\lambda = n$, a top-index surgery on an $(n-1)$-manifold... reinterpret as $\lambda$ with $n-\lambda = 0$: an $n$-handle). Describe the attaching region, the surgery, and what such a handle does to a connected end.

Hint

When $n - λ = 0$ the cocore disk is a point; the attaching region is $S^{λ - 1} \times D^{0} = S^{n - 1}$ .

Answer

Argument. For an $n$ -handle ( $λ = n$ , $n - λ = 0$ ) the cocore $D^{0}$ is a point, the attaching region is $S^{n - 1} \times D^{0} = S^{n - 1}$ , the belt sphere $S^{- 1}$ is empty. Attaching it caps off a boundary sphere $S^{n - 1}$ with a disk $D^{n}$ . The corresponding surgery removes $S^{n - 1} \times D^{0}$ (a sphere component of the boundary) and glues in $D^{n} \times S^{- 1} = \emptyset$ : it deletes a spherical boundary component entirely. A maximum of a Morse function is exactly such an $n$ -handle, closing off the top of the manifold.

Advanced results Master

The cobordism category. Fix $n$ . The objects of $Cob_{n}$ are closed oriented $(n - 1)$ -manifolds. A morphism $V_{0} \to V_{1}$ is a diffeomorphism class, rel boundary, of cobordisms $(W; V_{0}, V_{1})$ with $\partial W = \overline{V_{0}} ⊔ V_{1}$ (the bar denoting reversed orientation). Composition of $(W; V_{0}, V_{1})$ and $(W^{'}; V_{1}, V_{2})$ is the glued cobordism $W \cup_{V_{1}} W^{'}$ , smoothed along the common boundary by the collar theorem; the identity on $V$ is the class of $V \times I$ . Associativity holds because gluing three cobordisms in either order yields diffeomorphic results rel boundary. This category is symmetric monoidal under disjoint union, with unit the empty $(n - 1)$ -manifold, and it carries duals: $\overline{V}$ is dual to $V$ , with the bent cobordisms $V \times I$ read as evaluation and coevaluation. A topological quantum field theory is a symmetric monoidal functor out of $Cob_{n}$ , by the Atiyah-Segal axioms; the handle calculus below is what makes such a functor computable, since every morphism factors into elementary pieces.

Factorisation into elementary cobordisms. Every cobordism $(W; V_{0}, V_{1})$ admits a Morse function $f : W \to [0, 1]$ adapted to the boundary, by a relative version of the genericity argument of 03.02.30: perturb any boundary-respecting function to make its interior critical points nondegenerate, using Sard's theorem on the difference with a generic linear functional in a chart. Ordering the finitely many critical values $0 < c_{1} < \dots < c_{k} < 1$ and inserting regular levels $a_{0}, \dots, a_{k}$ between them, the slabs $f^{- 1} [a_{j - 1}, a_{j}]$ are elementary cobordisms (after a small perturbation separating critical points onto distinct levels). Hence $W = W_{1} \circ W_{2} \circ \dots \circ W_{k}$ is a composite of elementary cobordisms in $Cob_{n}$ , one per critical point. By the elementary-cobordism theorem this is the same as saying $W$ is obtained from $V_{0} \times I$ by attaching one handle per critical point, the handle decomposition of $W$ relative to $V_{0}$ .

Handles, framings, and the attaching data. The attaching map of a $λ$ -handle is an embedding $φ : S^{λ - 1} \times D^{n - λ} ↪ \partial_{+} W$ . Its isotopy class is the only datum affecting the diffeomorphism type of the result, and it splits into an isotopy class of the embedded attaching sphere $S^{λ - 1} ↪ \partial_{+} W$ together with a framing of its normal bundle, an element of $π_{λ - 1} (S O (n - λ))$ once a base framing is fixed. For surfaces ( $n = 2$ ) the framings are a $Z /2$ choice (orientable versus Möbius band of a $1$ -handle); in higher dimensions the framing group grows and the surgery output depends on it. Wallace and Milnor's recognition that a surgery and its inverse $χ (χ (V, φ), ψ) ≅ V$ are realised by an elementary cobordism and its turned-upside-down mirror is the geometric content of the relation " $W$ from $V_{0}$ by a $λ$ -handle" $=$ " $\overline{W}$ from $V_{1}$ by an $(n - λ)$ -handle": the same handle read from the top is a handle of complementary index, with attaching and belt spheres exchanged.

Synthesis. The elementary-cobordism theorem is the central insight that ties the three languages of this unit into one. It is the foundational reason a Morse function controls smooth topology: a single critical point of index $λ$ is exactly a $λ$ -handle and exactly a surgery, and this identification is dual to the upside-down reading that turns a $λ$ -handle into an $(n - λ)$ -handle. Putting these together, the handle decomposition generalises the CW structure of 03.02.31 from homotopy type to diffeomorphism type, and the cobordism category packages every smooth $n$ -manifold-with-boundary as a composite of elementary morphisms. This builds toward the rearrangement theorem of 03.02.21, where the handles are sorted by index, and it identifies cobordism with handle attachment in the precise sense that the descending sphere is the surgery datum — the bridge on which the entire $h$ -cobordism programme of Smale rests.

Full proof set Master

Proposition (surgery is its own inverse up to cobordism). Let $V$ be a closed $(n - 1)$ -manifold and $φ : S^{λ - 1} \times D^{n - λ} ↪ V$ a framed embedding, with trace the elementary cobordism $(W; V, V^{'})$ , $V^{'} = χ (V, φ)$ . Then there is a framed embedding $ψ : S^{n - λ - 1} \times D^{λ} ↪ V^{'}$ — the belt sphere with its induced framing — such that $χ (V^{'}, ψ) ≅ V$ , and the trace of $ψ$ is the cobordism $W$ read from $V^{'}$ to $V$ , an elementary cobordism with one critical point of index $n - λ$ .

Proof. In the Morse model $f = \frac{1}{2} - ∣ u ∣^{2} + ∣ v ∣^{2}$ on the chart block, the cobordism $W$ has the single critical point $p$ of index $λ$ for $f$ . Replace $f$ by $1 - f = \frac{1}{2} + ∣ u ∣^{2} - ∣ v ∣^{2}$ . This is again Morse on $W$ , with the same critical point $p$ , now of index $n - λ$ (the roles of the $u$ - and $v$ -directions, hence of positive and negative Hessian eigenvalues, are exchanged), and with ends interchanged: $(1 - f)^{- 1} (0) = V^{'}$ and $(1 - f)^{- 1} (1) = V$ . The descending sphere of $1 - f$ at $p$ is the locus ${u = 0, ∣ v ∣^{2} = ε}$ , which is the belt sphere $S^{n - λ - 1}$ of the original handle, embedded in $V^{'}$ ; flowing the reversed gradient-like field backward from a tube around it gives the framed embedding $ψ : S^{n - λ - 1} \times D^{λ} ↪ V^{'}$ . By the elementary-cobordism theorem applied to $1 - f$ , the upper end of this surgery is $χ (V^{'}, ψ) = (1 - f)^{- 1} (1) = V$ , and $W$ is its trace. The index is $n - λ$ because the Hessian of $1 - f$ at $p$ has $n - λ$ negative eigenvalues. $□$

Proposition (existence of an adapted Morse function). Every cobordism $(W; V_{0}, V_{1})$ admits a Morse function $f : W \to [0, 1]$ with $f^{- 1} (0) = V_{0}$ , $f^{- 1} (1) = V_{1}$ , no boundary critical points, and all interior critical points nondegenerate.

Proof. Choose collar neighbourhoods $V_{0} \times [0, \frac{1}{3})$ and $V_{1} \times (\frac{2}{3}, 1]$ of the two ends and a smooth $g_{0} : W \to [0, 1]$ equal to the collar coordinate near each end (so $g_{0} = 0$ on $V_{0}$ , increasing into the interior, $= 1$ on $V_{1}$ ), with $d g_{0} \neq = 0$ on the collars; such a $g_{0}$ exists by patching collar projections with a partition of unity. On the compact interior $K = g_{0}^{- 1} [\frac{1}{3}, \frac{2}{3}]$ , cover by finitely many charts $U_{1}, \dots, U_{m}$ . Perturb $g_{0}$ chart by chart: on $U_{i}$ replace $g$ by $g - ⟨ a_{i}, x ⟩ ρ_{i} (x)$ with $a_{i} \in R^{n}$ small and $ρ_{i}$ a bump supported in $U_{i}$ . By the Sard argument of 03.02.30, for almost every $a_{i}$ the function becomes Morse on $supp ρ_{i}$ without creating boundary critical points or leaving $[0, 1]$ , and successive perturbations preserve nondegeneracy already achieved (an open condition) on the earlier supports. After $m$ steps the result is Morse on $K$ and unchanged near the boundary, hence the desired $f$ . $□$

Proposition (composition is well-defined in $Cob_{n}$ ). If $W ≅ W_{1}$ rel $\partial$ and $W^{'} ≅ W_{1}^{'}$ rel $\partial$ with matching middle boundary $V_{1}$ , then $W \cup_{V_{1}} W^{'} ≅ W_{1} \cup_{V_{1}} W_{1}^{'}$ rel $\partial$ , so composition descends to diffeomorphism classes.

Proof. Let $F : W \to W_{1}$ and $G : W^{'} \to W_{1}^{'}$ be diffeomorphisms restricting to the identity on the boundaries, in particular to the same map on the shared $V_{1}$ . The collar-neighbourhood theorem gives collars of $V_{1}$ in $W$ and in $W^{'}$ on which $F$ and $G$ are, after an isotopy rel $V_{1}$ , of product form $(id_{V_{1}}, t)$ . Then $F$ and $G$ agree on a neighbourhood of $V_{1}$ and patch to a smooth bijection $F \cup G : W \cup_{V_{1}} W^{'} \to W_{1} \cup_{V_{1}} W_{1}^{'}$ , which is a diffeomorphism rel the outer boundary. The isotopy adjustment changes nothing rel $\partial$ , so the glued class is well-defined and associativity follows from the associativity of set-theoretic gluing together with the same collar-smoothing on each interface. $□$

Connections Master

The handle attachment of this unit is the smooth refinement of the CW cell-attachment in 03.02.31: there a critical point of index $λ$ attaches a $λ$ -cell and determines homotopy type, while here the same critical point attaches a fattened $λ$ -handle $D^{λ} \times D^{n - λ}$ and determines diffeomorphism type. The core of the handle is exactly the cell, and the cocore directions are the new smooth data the homotopy-theoretic version discards.

The Morse-lemma normal form and the index of 03.02.30 supply the local model $f = c - ∣ u ∣^{2} + ∣ v ∣^{2}$ from which the descending sphere, the attaching map, and the surgery are all read off; without the chart-independence of the index established there, the index of a handle would not be well-defined, and the factorisation of a cobordism into elementary pieces would carry no invariant.

The rearrangement and self-indexing theory of 03.02.21 takes the unordered handle decomposition produced here and sorts the handles by index into a filtration, building the Morse-Smale chain complex whose boundary maps count intersections of ascending and descending spheres; that complex computes $H_{*} (W, V_{0})$ and is the algebraic engine of the $h$ -cobordism theorem, for which the elementary-cobordism dictionary of this unit is the geometric input.

The cobordism relation defined here is the geometric side of the bordism groups of 03.06.12 and 03.06.13, where the same equivalence is studied homotopy-theoretically through the Pontryagin-Thom construction and Thom spectra; this unit supplies the handle-and-surgery description of a single bordism that the spectrum-level account suppresses, and surgery is the move by which one represents a bordism class by a connected or highly-connected manifold.

Historical & philosophical context Master

René Thom introduced the cobordism relation and computed the unoriented bordism ring in Quelques propriétés globales des variétés différentiables (Commentarii Mathematici Helvetici 28, 1954, 17–86) ^{[Thom 1954]}, for which he received the Fields Medal in 1958; the geometric equivalence "two closed manifolds cobound a compact manifold" is his. Andrew Wallace, in Modifications and cobounding manifolds (Canadian Journal of Mathematics 12, 1960, 503–528) ^{[Wallace 1960]}, introduced spherical modifications — the operation now called surgery — and identified their cobordism traces, independently of and concurrently with Milnor's handle formulation.

Stephen Smale recast the differential topology of high-dimensional manifolds in terms of handle decompositions in On the structure of manifolds (American Journal of Mathematics 84, 1962, 387–399) ^{[Smale 1962]} and the companion Generalized Poincaré conjecture in dimensions greater than four (Annals of Mathematics 74, 1961), proving the $h$ -cobordism theorem and the high-dimensional Poincaré conjecture by the handle calculus assembled from the elementary-cobordism dictionary of this unit. John Milnor's Lectures on the h-Cobordism Theorem (Princeton Mathematical Notes, 1965) ^{[Milnor §3]}, with notes by Larry Siebenmann and John Sondow, gave the proof its canonical short form, organising it around exactly the handle, surgery, and gradient-like-vector-field structures developed above. The categorical reading of cobordism as a symmetric monoidal category, latent in this geometry, was made explicit by Michael Atiyah's axioms for topological quantum field theory in 1988.

Bibliography Master

@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}
}

@article{thom1954,
  author  = {Thom, Ren{\'e}},
  title   = {Quelques propri{\'e}t{\'e}s globales des vari{\'e}t{\'e}s diff{\'e}rentiables},
  journal = {Commentarii Mathematici Helvetici},
  volume  = {28},
  pages   = {17--86},
  year    = {1954}
}

@article{wallace1960,
  author  = {Wallace, Andrew H.},
  title   = {Modifications and cobounding manifolds},
  journal = {Canadian Journal of Mathematics},
  volume  = {12},
  pages   = {503--528},
  year    = {1960}
}

@article{smale1962structure,
  author  = {Smale, Stephen},
  title   = {On the structure of manifolds},
  journal = {American Journal of Mathematics},
  volume  = {84},
  number  = {3},
  pages   = {387--399},
  year    = {1962}
}

@article{smale1961poincare,
  author  = {Smale, Stephen},
  title   = {Generalized {P}oincar{\'e}'s conjecture in dimensions greater than four},
  journal = {Annals of Mathematics},
  volume  = {74},
  number  = {2},
  pages   = {391--406},
  year    = {1961}
}

@incollection{atiyah1988tqft,
  author    = {Atiyah, Michael},
  title     = {Topological quantum field theories},
  booktitle = {Publications Math{\'e}matiques de l'IH{\'E}S},
  volume    = {68},
  pages     = {175--186},
  year      = {1988}
}

@book{kosinski1993,
  author    = {Kosinski, Antoni A.},
  title     = {Differential Manifolds},
  series    = {Pure and Applied Mathematics},
  volume    = {138},
  publisher = {Academic Press},
  year      = {1993}
}

Prerequisites

03.02.02
03.02.03
03.02.30
03.02.31

Tier anchors

beginner: Milnor Lectures on the h-Cobordism Theorem §1 informal; Matsumoto An Introduction to Morse Theory Ch. 3 (handle picture)
intermediate: Milnor Lectures on the h-Cobordism Theorem §§1-2 (Theorems 3.1-3.5); Kosinski Differential Manifolds Ch. VI
master: Milnor Lectures on the h-Cobordism Theorem §§2-3; Wallace Modifications and cobounding manifolds (1960); Kosinski Ch. VI-VII

References

03-modern-geometry/Bott-LecturesOnMorseTheory.pdf · Handle attachment from a Morse function on a manifold-with-boundary; gradient-like vector fields
Milnor — Lectures on the h-Cobordism Theorem (Princeton Mathematical Notes, 1965) · §§1-3: Morse functions on a cobordism, elementary cobordisms, handles, and rearrangement (notes by L. Siebenmann and J. Sondow)
Smale — On the structure of manifolds (1962) · American Journal of Mathematics 84, pp. 387--399; handle decompositions and the handlebody calculus
Wallace — Modifications and cobounding manifolds (1960) · Canadian Journal of Mathematics 12, pp. 503--528; spherical modifications (surgery) and their cobordism traces
Thom — Quelques propriétés globales des variétés différentiables (1954) · Commentarii Mathematici Helvetici 28, pp. 17--86; the cobordism relation and the bordism groups

Estimated time

beginner: 16m
intermediate: 42m
master: 78m