03.02.21 · differential-geometry / manifolds

Rearrangement and self-indexing Morse functions

shipped3 tiersLean: none

Anchor (Master): Milnor Lectures on the h-Cobordism Theorem §§4-6; Smale On the structure of manifolds (1962); Kosinski Ch. VI-VII

Intuition Beginner

Building a shape by gluing on handles, one at a time, leaves you with a pile of handles attached in whatever order the height function happened to visit them. A low pass might come before a high valley; the bookkeeping is a jumble. The natural wish is to tidy it: glue all the simplest handles first, then the next-simplest, and so on up to the most complicated. Sorting the handles by how they attach is the goal of this unit.

The key freedom is that you may slide the heights of the level spots up and down, so long as you do not let two of them collide while one is trying to pass the other. Think of two climbers on the same rope at different heights. As long as their paths down the mountain never cross, you can raise one and lower the other freely, swapping which is higher. Two handles can be reordered exactly when their attaching paths miss each other.

The tidiest arrangement of all is the one where the height of each level spot is its own complexity number. A valley sits at height zero, the simplest passes at height one, the next at height two, on up to the peaks at the top. A height function arranged this way reads off the complexity of each spot directly from how high it sits. This is called self-indexing, because the height equals the index — the count of downhill directions.

The payoff is a clean ladder. The shape is built floor by floor, each floor adding only handles of one fixed complexity. That orderly ladder is the scaffolding on which the deepest theorems about smooth shapes are later raised.

Visual Beginner

On the left, a tall cobordism slab with four level spots scattered at random heights: a valley low down, a complicated peak just above it, then a simple pass higher up, then another. Arrows show two of the spots being slid past each other — one raised, one lowered — because the downhill paths streaming out of them never touch.

On the right, the finished, self-indexing picture: the same four spots now sit on evenly spaced floors, each floor labelled with a complexity number, and the height of every spot equals its number. The jumble has become a ladder. The takeaway: when downhill paths miss, heights can be sorted, and the tidiest sort puts each spot at the floor named by its index.

Worked example Beginner

Sorting a two-handle stack on the doughnut cobordism. Recall the doughnut surface, built from a disk and two bent-bar handles. Place it inside a slab and give the slab a height function whose level spots are: one valley (a $0$ -handle), then — by accident of how it was drawn — one of the two bars sitting below the other, even though both are the same kind of handle.

The two bars are both index- $1$ handles. Their attaching feet land on different parts of the rim, and the downhill streams pouring off them run to separate places. Because those streams never meet, we may raise the lower bar and lower the higher one until they sit at the same floor. Now both index- $1$ handles live together on floor one.

Next picture the cap that closes the doughnut off — an index- $2$ handle. It attaches above everything else, and nothing of lower complexity sits above it. So it rises to floor two.

The final stack reads, bottom to top: floor $0$ holds the valley, floor $1$ holds both bars, floor $2$ holds the cap. Heights $0, 1, 2$ match the complexity numbers $0, 1, 2$ . What this tells us: the messy original order has been combed into a clean ladder where each floor carries exactly one complexity of handle, and the floor number is the complexity itself.

Check your understanding Beginner

Formal definition Intermediate+

Throughout, $(W; V_{0}, V_{1})$ is a compact cobordism of dimension $n$ with a Morse function $f : W \to [0, 1]$ adapted to the boundary, $f^{- 1} (0) = V_{0}$ , $f^{- 1} (1) = V_{1}$ , all interior critical points nondegenerate, as in 03.02.20. Fix also a gradient-like vector field $ξ$ for $f$ : $ξ (f) > 0$ away from the critical points, and near each critical point $p$ of index $λ$ there are Morse coordinates $(u, v) \in R^{λ} \times R^{n - λ}$ with $f = f (p) - ∣ u ∣^{2} + ∣ v ∣^{2}$ and $ξ = (- u, v)$ .

Definition (ascending and descending manifolds). For a critical point $p$ , the descending (left-hand) manifold is the set of points whose forward $ξ$ -trajectory limits onto $p$ as $t \to - \infty$ flows them up out of $p$ ; concretely $$ W^u(p) = {x \in W : \lim_{t \to -\infty} \phi_t(x) = p}, $$ where $ϕ_{t}$ is the flow of $ξ$ . It is an embedded open disc of dimension $λ$ . The ascending (right-hand) manifold $W^{s} (p)$ , with $t \to + \infty$ , is an embedded open disc of dimension $n - λ$ . Their intersections with a regular level set $f^{- 1} (c)$ , for $c$ just below resp. just above $f (p)$ , are the descending sphere $S^{λ - 1}$ and ascending sphere $S^{n - λ - 1}$ of 03.02.20.

Definition (rearrangeable pair). Two critical points $p, q$ at distinct critical levels, with $f (p) < f (q)$ say, are independent for $ξ$ if no $ξ$ -trajectory runs from $q$ down to $p$ : equivalently, the descending manifold $W^{u} (q)$ is disjoint from the ascending manifold $W^{s} (p)$ . A finite set of critical points lying in a common interval of regular values is rearrangeable if it is pairwise independent.

Definition (self-indexing Morse function). A Morse function $f : W \to [0, n]$ adapted to the boundary is self-indexing (Smale's term: a nice function) if $$ f(p) = \operatorname{ind}(p) $$ for every critical point $p$ . Equivalently, every index- $λ$ critical point lies on the level $λ$ , and the half-integer levels $λ + \frac{1}{2}$ are regular and separate the index- $λ$ handles below from the index- $(λ + 1)$ handles above.

Definition (handlebody filtration). Given a self-indexing $f$ , set $W_{λ} = f^{- 1} [0, λ + \frac{1}{2}]$ . Then $V_{0} \subset W_{0} \subset W_{1} \subset \dots \subset W_{n} = W$ , and $W_{λ}$ is obtained from $W_{λ - 1}$ by attaching all the $λ$ -handles simultaneously along disjoint attaching tubes in $\partial_{+} W_{λ - 1}$ . This is the handle decomposition sorted by index.

Counterexamples to common slips

Independence is a property of the pair together with $ξ$ , not of the points alone. The same two critical points may be joined by a trajectory for one gradient-like field and independent for another; rearrangement often proceeds by first perturbing $ξ$ to remove an unwanted connecting trajectory.
Self-indexing does not require $f (p) < f (q)$ whenever $ind (p) < ind (q)$ strictly handle by handle within a level — all critical points of one index share a single level, so equal indices give equal values. The ordering is by index class, not a total order on points.
A trajectory from $q$ to $p$ forces $ind (q) > ind (p)$ generically (the dimension count $dim W^{u} (q) + dim W^{s} (p) > n$ is what permits an intersection), but in the non-generic case a trajectory can connect critical points of the same index; such a connection is an obstruction to independence that a small perturbation of $ξ$ removes.

Key theorem with proof Intermediate+

The structural result is Milnor's Rearrangement Theorem ^{[Milnor §4]}, LHC Theorem 4.1, with the self-indexing corollary as its repeated application.

Theorem (Rearrangement). Let $f : W \to [0, 1]$ be a Morse function with gradient-like field $ξ$ , and let $p_{1}, \dots, p_{k}$ be the critical points on a single critical level $c$ , together with possibly some critical points above and below. Suppose the critical points can be partitioned into two sets $A, B$ such that no $ξ$ -trajectory runs from a point of $B$ to a point of $A$ . Then for any $a < b$ in $(0, 1)$ avoiding the boundary critical values there is a new Morse function $f^{'}$ , with the same critical points and the same gradient-like field $ξ$ , such that $f^{'} = f$ outside a compact set, $f^{'} (p) < a$ for all $p \in A$ , and $f^{'} (p) > b$ for all $p \in B$ .

In words: independent critical points can be pushed to different ranges of values, $A$ below $B$ , without changing the manifold, the critical points, or the gradient-like field.

Proof. The hypothesis says $W^{u} (B) \cap W^{s} (A) = \emptyset$ , where $W^{u} (B) = ⋃_{q \in B} W^{u} (q)$ and likewise for $A$ . Consider the set $K_{A} = W^{s} (A) \cup W^{u} (A)$ of points whose trajectory limits onto $A$ in forward or backward time, together with $A$ itself, and similarly $K_{B}$ . The closure of the union of all trajectories that pass through a neighbourhood of $A$ — that is, the set of $x$ whose forward limit lies in $A$ or whose backward limit lies in $A$ — is a compact set saturated by the flow. The disjointness $W^{u} (B) \cap W^{s} (A) = \emptyset$ , plus the fact that every non-critical trajectory limits onto critical points or the boundary at both ends, lets one build a smooth function $π : W \to [0, 1]$ that is constant along trajectories ( $ξ (π) = 0$ ), equal to $0$ on the trajectories through $A$ and to $1$ on the trajectories through $B$ . (Construct $π$ from the flow-out of disjoint closed neighbourhoods of $K_{A}$ and $K_{B}$ and extend by a $ξ$ -invariant partition of unity; invariance is what makes $π$ depend only on which trajectory a point lies on.)

Now choose a smooth $G : [0, 1] \times [0, 1] \to [0, 1]$ , $G = G (s, t)$ , with $\partial G / \partial t > 0$ for all $s$ (so $t \mapsto G (s, t)$ is an increasing reparametrisation of the value), such that $G (0, c) < a$ and $G (1, c) > b$ , and $G (s, \cdot) = id$ near $t = 0, 1$ . Define $$ f'(x) = G\big(\pi(x),, f(x)\big). $$ Then $f^{'}$ agrees with $f$ near the boundary, and $ξ (f^{'}) = G_{s} \cdot ξ (π) + G_{t} \cdot ξ (f) = G_{t} ξ (f)$ , which has the same sign as $ξ (f)$ because $G_{t} > 0$ . Hence $f^{'}$ has exactly the same critical points as $f$ , with the same indices (the Hessian is scaled by the positive factor $G_{t}$ ), and $ξ$ remains gradient-like for $f^{'}$ . At a critical point $p \in A$ the trajectory through $p$ has $π = 0$ , so $f^{'} (p) = G (0, f (p)) = G (0, c) < a$ ; at $p \in B$ , $π = 1$ and $f^{'} (p) = G (1, c) > b$ . $□$

Bridge. This is exactly the freedom to slide independent level spots past one another, made rigorous: the trajectory-constant function $π$ separates the two families, and reparametrising the value along $π$ is the foundational reason the heights can be shifted without disturbing the gradient flow. Iterating it across all indices builds toward the self-indexing function, where every index- $λ$ handle is sorted onto the level $λ$ ; this is exactly the smooth filtration $W_{0} \subset \dots \subset W_{n}$ , and the sorting generalises the cell-by-dimension filtration of a CW complex from 03.12.13 to the smooth handle category of 03.02.20. The central insight is that the descending sphere of a lower-index handle and the ascending sphere of a higher-index handle live in complementary dimensions, so they can be made disjoint, and disjointness is precisely what licenses the swap. Putting these together, rearrangement appears again in 03.02.20's programme as the combinatorial spine on which handle cancellation and the $h$ -cobordism theorem are built. The bridge is that the analytic move (reparametrise $f$ along $π$ ) and the geometric move (slide the handles) are one and the same operation.

Exercises Intermediate+

Exercise (hard, short-answer). Prove that the Rearrangement Theorem implies the existence of a Morse function on $(W; V_0, V_1)$ whose critical values are strictly increasing in the index: all index-$0$ points below all index-$1$ points, etc. (the "ordered" case, weaker than self-indexing).

Hint

Apply the theorem to $A = {all critical points of index \leq λ}$ , $B = {index > λ}$ , and use the dimension count to verify independence.

Answer

Argument. First make $ξ$ Morse-Smale by a small perturbation, so that $W^{u} (q) ⋔ W^{s} (p)$ for all pairs. Then a trajectory from $q$ to $p$ forces $dim W^{u} (q) + dim W^{s} (p) \geq n + 1$ , i.e. $ind (q) + (n - ind (p)) \geq n + 1$ , i.e. $ind (q) > ind (p)$ . Hence no trajectory runs from a lower-index to a higher-index point: taking $A = {ind \leq λ}$ and $B = {ind > λ}$ , there is no trajectory from $B$ to $A$ , so the theorem pushes $A$ below $B$ . Doing this for $λ = 0, 1, \dots, n - 1$ in turn (each step preserving the earlier separations, as those involve already-separated value ranges) orders all indices. $□$

Exercise (hard, short-answer). Starting from an ordered Morse function (index-$\lambda$ points all on a level $a_\lambda$, with $a_0 < a_1 < \cdots < a_n$), explain how to rescale to a self-indexing one with $f(p) = \operatorname{ind}(p)$, and why the rescaling preserves the gradient-like field up to positive reparametrisation.

Hint

Choose an increasing smooth $h : [0, 1] \to [0, n]$ sending each $a_{λ}$ to $λ$ and apply it to $f$ .

Answer

Argument. Pick a strictly increasing smooth $h : [f (V_{0}), f (V_{1})] \to [0, n]$ with $h (a_{λ}) = λ$ for each occupied index $λ$ and $h^{'} > 0$ throughout (interpolate smoothly between the prescribed values, possible since the $a_{λ}$ are strictly increasing). Set $\tilde{f} = h \circ f$ . Then $\tilde{f} (p) = h (a_{ind (p)}) = ind (p)$ , so $\tilde{f}$ is self-indexing. Moreover $ξ (\tilde{f}) = h^{'} (f) ξ (f)$ with $h^{'} > 0$ , so $ξ$ stays gradient-like for $\tilde{f}$ , the critical points and indices are unchanged, and the half-levels $λ + \frac{1}{2}$ are regular. $□$

Advanced results Master

The Morse-Smale chain complex. Fix a self-indexing Morse function $f$ on $(W; V_{0}, V_{1})$ with a gradient-like field $ξ$ chosen Morse-Smale: $W^{u} (p) ⋔ W^{s} (q)$ for every pair of critical points (a generic condition on $ξ$ , by the transversality theory of 03.15.01). Let $C_{λ}$ be the free abelian group on the index- $λ$ critical points, oriented by a choice of orientation of each descending disc $W^{u} (p)$ . For adjacent indices, $dim W^{u} (p) + dim W^{s} (q) = λ + (n - (λ - 1)) = n + 1$ , so on a regular level between the levels $λ$ and $λ - 1$ the descending sphere $S_{p}^{λ - 1}$ of $p$ and the ascending sphere $S_{q}^{n - λ}$ of $q$ meet in a finite set of points, each carrying a sign from the orientations. Their signed count is the intersection number $[p : q] \in Z$ , and $$ \partial p = \sum_{\operatorname{ind}(q) = \lambda - 1} [p : q], q $$ defines $\partial : C_{λ} \to C_{λ - 1}$ . The geometric heart is that $\partial^{2} = 0$ : a broken trajectory from an index- $λ$ point to an index- $(λ - 2)$ point through an intermediate point is one end of a $1$ -parameter family of trajectories whose other end is a second broken trajectory, so the boundary terms cancel in pairs. The homology of $(C_{*}, \partial)$ is $H_{*} (W, V_{0}; Z)$ .

Why self-indexing is the right frame. The intersection-number boundary map is well-posed precisely because, in a self-indexing function, the ascending sphere of an index- $(λ - 1)$ point and the descending sphere of an index- $λ$ point sit on a common regular level $λ - \frac{1}{2}$ , where their dimensions $n - λ$ and $λ - 1$ add to $n - 1 = dim (level)$ , the complementary-dimension condition for a transverse $0$ -dimensional intersection. Without sorting by index, descending and ascending spheres of the wrong index pairs would clutter the intermediate levels, and the count would not be a chain map. This is the sense in which rearrangement is the combinatorial spine of the handle calculus: it produces exactly the filtration $W_{λ} = f^{- 1} [0, λ + \frac{1}{2}]$ for which $C_{λ} = H_{λ} (W_{λ}, W_{λ - 1})$ is free on the $λ$ -handles, recovering the cellular chain complex of 03.12.13 in its smooth, handle-theoretic incarnation.

Existence and the polar refinement. Smale's existence theorem ^{[Smale 1962]} states that every cobordism admits a self-indexing Morse function with a Morse-Smale gradient-like field. On a closed connected manifold one can further demand exactly one minimum and one maximum — Marston Morse's polar (or " $1$ - $1$ ") functions ^{[Morse 1960]} — by cancelling surplus index- $0$ and index- $n$ critical points against index- $1$ and index- $(n - 1)$ ones; this trims $C_{0}$ and $C_{n}$ to rank one, the normalisation used when the chain complex is run against simple connectivity in the $h$ -cobordism argument.

Synthesis. Rearrangement is the central insight that converts the unordered handle decomposition of 03.02.20 into a computable algebraic object. It is the foundational reason a smooth cobordism carries a finite free chain complex: sorting the handles by index produces the filtration $W_{0} \subset \dots \subset W_{n}$ , and this is exactly the smooth realisation of the CW filtration of 03.12.13, so the Morse-Smale complex $C_{*}$ is dual to — in fact identified with — the cellular chain complex computing $H_{*} (W, V_{0})$ . Putting these together, the self-indexing function and its Morse-Smale field generalise the height-function-on-a-torus picture of 03.02.30 into the full machine that builds toward handle cancellation and the $h$ -cobordism theorem: the intersection numbers $[p : q]$ are the entries of the boundary matrix, an acyclic such matrix signals geometrically cancelling handles, and the bridge from algebra back to geometry — Whitney's trick turning an algebraic $\pm 1$ into a single geometric intersection point — is the content of the next unit in the programme.

Full proof set Master

Proposition (independence is achievable by perturbing $ξ$ , generic case). Let $p, q$ be critical points with $ind (p) \geq ind (q)$ lying in a common interval of regular values, and suppose $ξ$ is Morse-Smale. Then there is no $ξ$ -trajectory from $p$ to $q$ unless $ind (p) > ind (q)$ ; in particular two critical points of equal index in a common regular interval are independent.

Proof. A $ξ$ -trajectory from $p$ to $q$ is a point of $W^{u} (p) \cap W^{s} (q)$ other than the rest points. Under the Morse-Smale hypothesis this intersection is transverse, so where non-empty it is a manifold of dimension $dim W^{u} (p) + dim W^{s} (q) - n = ind (p) + (n - ind (q)) - n = ind (p) - ind (q)$ . The flow $ξ$ acts freely on this intersection (away from rest points) with $1$ -dimensional orbits, so a non-empty trajectory set has dimension $\geq 1$ , forcing $ind (p) - ind (q) \geq 1$ . Hence a connecting trajectory requires $ind (p) > ind (q)$ . When $ind (p) = ind (q)$ the intersection has formal dimension $0$ yet must be flow-invariant and so contain no non-rest points: it is empty, and $p, q$ are independent. $□$

Proposition (the half-level slices are handlebodies). For a self-indexing $f$ with gradient-like $ξ$ , the sublevel $W_{λ} = f^{- 1} [0, λ + \frac{1}{2}]$ is obtained from $W_{λ - 1}$ by attaching one $λ$ -handle per index- $λ$ critical point, along disjoint embeddings of the attaching tubes into $\partial_{+} W_{λ - 1}$ .

Proof. The slab $f^{- 1} [λ - \frac{1}{2}, λ + \frac{1}{2}]$ contains exactly the index- $λ$ critical points, all on the single level $λ$ . By the elementary-cobordism theorem of 03.02.20 each contributes a $λ$ -handle attached to $\partial_{+} W_{λ - 1} = f^{- 1} (λ - \frac{1}{2})$ along the descending sphere of $ξ$ . The attaching tubes are disjoint: the attaching tube of the handle at $p$ is a neighbourhood of the descending sphere $S_{p}^{λ - 1} \subset f^{- 1} (λ - \frac{1}{2})$ , and for two index- $λ$ points $p \neq = p^{'}$ the descending manifolds $W^{u} (p), W^{u} (p^{'})$ are disjoint embedded discs whose level-set slices are disjoint spheres (distinct trajectories never coincide). Shrinking the tubular neighbourhoods makes the tubes disjoint, so all the $λ$ -handles attach simultaneously. The result is $W_{λ}$ after straightening corners. $□$

Proposition ( $\partial \circ \partial = 0$ for the Morse-Smale complex). With the intersection-number boundary $\partial p = \sum_{q} [p : q] q$ on a Morse-Smale self-indexing complex, $\partial^{2} = 0$ .

Proof sketch (the standard compactness-of-the-moduli argument). Fix $p$ of index $λ$ and $r$ of index $λ - 2$ . The coefficient of $r$ in $\partial^{2} p$ is $\sum_{q} [p : q] [q : r]$ , summed over index- $(λ - 1)$ points $q$ . Consider the moduli space $M (p, r)$ of unparametrised $ξ$ -trajectories from $p$ to $r$ ; by transversality it is a $1$ -manifold (dimension $ind (p) - ind (r) - 1 = 1$ ). Its ends, by the broken-trajectory compactification, are in bijection with pairs of trajectories $p \to q \to r$ through intermediate index- $(λ - 1)$ points — exactly the terms $[p : q] [q : r]$ with sign. A compact $1$ -manifold has an even number of boundary points counted with sign, so $\sum_{q} [p : q] [q : r] = 0$ . The compactness and gluing that justify this are the content of 03.15.01; here they are imported. Hence $\partial^{2} = 0$ . $□$

Connections Master

The rearrangement and self-indexing theory of this unit is the sorting step that organises the unordered handle decomposition of 03.02.20 into the filtration $W_{0} \subset W_{1} \subset \dots \subset W_{n}$ ; the elementary-cobordism dictionary established there — one critical point is one handle is one surgery, with the descending sphere carrying the attaching datum — is exactly the input the Rearrangement Theorem permutes, and the gradient-like vector field defined there is the object whose ascending and descending manifolds make independence a checkable condition.

The Morse functions, the index, and the Morse lemma of 03.02.30 supply the local normal form $f = c - ∣ u ∣^{2} + ∣ v ∣^{2}$ that defines the gradient-like field near each critical point and hence the ascending and descending spheres whose disjointness drives the swap; the chart-independence of the index proved there is what makes "sort by index" a well-defined instruction, and the worked torus example there is the seed that this unit grows into a full handlebody ladder.

The Morse-Smale chain complex built here from a self-indexing function is the smooth, handle-theoretic realisation of the cellular chain complex of 03.12.13: the filtration by half-levels is a CW filtration, $C_{λ}$ is free on the index- $λ$ handles, and the intersection-number boundary map agrees with the cellular boundary, so both compute $H_{*} (W, V_{0})$ ; the analytic underpinning of the boundary map — transversality of ascending and descending manifolds and compactness of the trajectory moduli — is developed in 03.15.01, whose gradient-flow Morse homology is this complex read on a closed manifold.

Historical & philosophical context Master

The idea of ordering critical points by index has roots in Marston Morse's own work: his The existence of polar non-degenerate functions on differentiable manifolds (Annals of Mathematics 71, 1960, 352–383) ^{[Morse 1960]} constructed functions with a single maximum and single minimum, an early instance of normalising a Morse function's critical-value structure to a prescribed pattern. The decisive step — that handles can be freely reordered by index and that a self-indexing function exists on any cobordism — is due to Stephen Smale, who introduced "nice functions" with $f (p) = ind (p)$ in On the structure of manifolds (American Journal of Mathematics 84, 1962, 387–399) ^{[Smale 1962]} and used the resulting handlebody filtration as the organising device of his proof of the high-dimensional Poincaré conjecture in Generalized Poincaré's conjecture in dimensions greater than four (Annals of Mathematics 74, 1961, 391–406) ^{[Smale 1961]}.

John Milnor's Lectures on the h-Cobordism Theorem (Princeton Mathematical Notes, 1965), with notes by Larry Siebenmann and John Sondow, gave the rearrangement argument its canonical form as the First Cancellation/Rearrangement theory of §4, organised around the gradient-like vector field and its trajectory-constant separating function. Philosophically, the move is a turning point in twentieth-century topology: it converts a purely analytic object, a smooth function and its gradient flow, into a combinatorial one, an ordered list of handles with intersection numbers, and thereby makes the smooth topology of high-dimensional manifolds amenable to linear algebra over $Z [π_{1}]$ . The reduction of a geometric classification problem to the algebra of a chain complex — later abstracted into surgery theory and the algebraic $K$ - and $L$ -theory of Wall and others — begins with the humble observation that level spots whose downhill paths miss can be slid past one another.

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{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}
}

@article{morse1960polar,
  author  = {Morse, Marston},
  title   = {The existence of polar non-degenerate functions on differentiable manifolds},
  journal = {Annals of Mathematics},
  volume  = {71},
  number  = {2},
  pages   = {352--383},
  year    = {1960}
}

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

@book{matsumoto2002morse,
  author    = {Matsumoto, Yukio},
  title     = {An Introduction to Morse Theory},
  series    = {Translations of Mathematical Monographs},
  volume    = {208},
  publisher = {American Mathematical Society},
  year      = {2002}
}

Prerequisites

03.02.20
03.02.30

Tier anchors

beginner: Milnor Lectures on the h-Cobordism Theorem §4 informal; Matsumoto An Introduction to Morse Theory Ch. 3 (ordering the handles)
intermediate: Milnor Lectures on the h-Cobordism Theorem §4 (Theorems 4.1, 4.8); Kosinski Differential Manifolds Ch. VI §§4-5
master: Milnor Lectures on the h-Cobordism Theorem §§4-6; Smale On the structure of manifolds (1962); Kosinski Ch. VI-VII

References

03-modern-geometry/Bott-LecturesOnMorseTheory.pdf · Gradient-like vector fields; ordering critical values; the first deformation theorem
Milnor — Lectures on the h-Cobordism Theorem (Princeton Mathematical Notes, 1965) · §4: Rearrangement of cobordisms (Theorem 4.1), self-indexing Morse functions (Theorem 4.8), gradient-like vector fields (notes by L. Siebenmann and J. Sondow)
Smale — On the structure of manifolds (1962) · American Journal of Mathematics 84, pp. 387--399; self-indexing functions and the handlebody filtration
Smale — Generalized Poincaré's conjecture in dimensions greater than four (1961) · Annals of Mathematics 74, pp. 391--406; ordering handles by index in the proof
Morse — The existence of polar non-degenerate functions on differentiable manifolds (1960) · Annals of Mathematics 71, pp. 352--383; functions with one maximum and one minimum, an antecedent of self-indexing

Estimated time

beginner: 16m
intermediate: 42m
master: 80m