Handle attachment, CW homotopy type, and the Morse inequalities

Intuition Beginner

Imagine slowly flooding a mountainous island with water. As the water level rises, the dry land left above the surface changes shape. Most of the time, raising the level a little just makes the dry land a little smaller in a smooth, boring way — the shape does not really change. But at special heights, something new happens: two separate islands merge, or a lake first appears, or the last dry peak vanishes.

Morse theory is the study of exactly these special heights. The dry land above water level $a$ is the "sublevel set." The special heights are the heights of the peaks, the passes, and the pits of the landscape. Between two special heights nothing interesting happens to the shape; at a special height the shape gains one new feature.

This is the central bargain. A smooth height function turns a curved space into a recipe. The recipe says: build the space one piece at a time, adding one simple piece each time you pass a peak, a pass, or a pit.

The simple pieces are cells: a point, a line segment, a disk, a solid ball, and so on. The kind of pass you cross decides which cell you glue on. A pit adds a point. A pass adds a segment. A peak (in a surface) caps things off with a disk. Counting the passes of each kind already tells you a great deal about the holes in the space.

Visual Beginner

A doughnut (torus) standing upright, with a horizontal plane sweeping from bottom to top. Four special heights are marked: the bottom pit, the lower pass where the hole opens, the upper pass where the hole closes, and the top peak. To the right, the same four moments shown as cells being glued on: a point, then a segment, then a second segment, then a capping disk.

The takeaway from the picture: the torus is assembled from one point, two segments, and one disk. Those counts $1,2,1$ are exactly the number of independent holes of each dimension.

Worked example Beginner

The height of a standing doughnut. Stand a torus upright and measure each point by its height $f$. There are four special points where the surface is momentarily flat (level): the lowest point, two points on the inner ring (one where the central hole begins, one where it ends), and the highest point.

Walk the water level up from the floor.

• Just above the lowest point, the dry land is a tiny cap, like a single dot grown into a small disk. We have added a point: a $0$-cell.
• Crossing the first inner pass, the dry land grows a bridge that connects around. We have added a segment: a $1$-cell.
• Crossing the second inner pass, another bridge closes the loop the other way. We have added a second segment: another $1$-cell.
• Crossing the top, the last hole is capped. We have added a disk: a $2$-cell.

So the torus is built from one $0$-cell, two $1$-cells, and one $2$-cell. The number of connected pieces is $1$, the number of independent loops is $2$, and the number of enclosed voids is $1$.

What this tells us. The four special heights are not an accident of how we stood the doughnut up. Any reasonable height function on the torus must have at least one pit, at least two passes, and at least one peak. The shape of the doughnut forces those critical points to exist. Counting critical points is a way of measuring holes.

Check your understanding Beginner

Formal definition Intermediate+

Let $M$ be a smooth manifold and $f:M\to \mathbb{R}$ a smooth function. For $a\in \mathbb{R}$ write the sublevel set $$M^a=f^{-1}(-\infty ,a]=\{x\in M:f(x)\le a\}.$$ A point $p\in M$ is a critical point of $f$ if $d{f}_p=0$, and $c=f(p)$ is then a critical value. A critical point is nondegenerate if the Hessian $H_p(f)$ — the symmetric bilinear form on $T_{p}M$ given in any chart by the matrix $({\partial }^{2}f/\partial x^i\partial x^j)(p)$ — is nonsingular. The index $\lambda (p)$ is the dimension of a maximal subspace of $T_{p}M$ on which $H_p(f)$ is negative definite. A function all of whose critical points are nondegenerate is a Morse function; the unit 03.02.30 establishes that nondegenerate critical points are isolated and that the index is chart-independent.

Throughout, $M$ is taken so that each $M^a$ is compact (for instance $M$ closed, or $f$ proper and bounded below). This guarantees the flows below are complete and the critical values in any bounded interval are finite in number.

Convention. $f$ decreases in the direction in which cells are attached: passing upward through a critical value of index $\lambda$ attaches a $\lambda$-cell. This is Milnor's convention ^{[Milnor §3]}. The opposite convention ($-f$) attaches a $(n-\lambda )$-cell and is used in some sources; every statement below is stated for the increasing-$f$ direction.

A gradient-like vector field for $f$ is a vector field $X$ with $X(f)>0$ away from the critical points, and which in Morse coordinates near each critical point equals the Euclidean gradient of the normal form. Such an $X$ exists on any manifold admitting a Morse function: choose a Riemannian metric and a chart-compatible patching by a partition of unity. Its normalized flow is the engine of both theorems below.

For the cell language let $D^{\lambda }$ be the closed $\lambda$-disk, $e^{\lambda }$ its interior (an open $\lambda$-cell), and $S^{\lambda -1}=\partial D^{\lambda }$. Attaching a $\lambda$-cell to a space $Y$ along a map $\phi :S^{\lambda -1}\to Y$ produces $Y{\cup }_{\phi }{e}^{\lambda }=(Y\bigsqcup D^{\lambda })/(x\sim \phi (x))$. This is the CW operation from 03.12.10.

Counterexamples to common slips

• The condition is on the Hessian, not the gradient: $f(x)=x^3$ on $\mathbb{R}$ has $d{f}_0=0$ but $f^{\prime \prime }(0)=0$, so $0$ is a degenerate critical point and the theorems below do not apply at it.
• A critical value can host several critical points. Theorem 3.2 below is stated for a single critical point on the level; with $k$ critical points of indices ${\lambda }_1,\dots ,{\lambda }_k$ on one level, one attaches $k$ cells of those dimensions simultaneously.
• $M^a$ deformation-retracting onto $M^b$ requires $[b,a]$ to be free of critical values, not merely to avoid a particular critical point. A critical point with value inside $(b,a)$ blocks the retraction even if $f$ is regular at the endpoints.

Key theorem with proof Intermediate+

The two structural theorems are Milnor's Theorem 3.1 and Theorem 3.2 ^{[Milnor §3]}.

Theorem 3.1 (Passing a regular interval). Let $f:M\to \mathbb{R}$ be smooth with $M^a$ compact and $f^{-1}[b,a]$ containing no critical point, $b<a$. Then $M^a$ deformation-retracts onto $M^b$; in particular $f^{-1}[b,a]$ is diffeomorphic to $M^b\times [b,a]$ and $M^a\simeq M^b$.

Proof. Choose a Riemannian metric $⟨,⟩$ and let $X=\nabla f/⟨\nabla f,\nabla f⟩$ on the region where $\nabla f\ne 0$; cut $X$ off to be compactly supported using a smooth bump $\rho$ that equals $1$ on $f^{-1}[b,a]$ and vanishes outside a slightly larger compact set. Since $f^{-1}[b,a]$ has no critical point and is compact, $\nabla f\ne 0$ there and $X$ is well defined and smooth. Let ${\varphi }_t$ be the flow of $X$. Wherever $\rho =1$, $$\frac{d}{dt}f({\varphi }_t(x))=⟨\nabla f,X⟩=\frac{⟨\nabla f,\nabla f⟩}{⟨\nabla f,\nabla f⟩}=1,$$ so along a trajectory $f$ increases at unit rate as long as the trajectory stays in $f^{-1}[b,a]$. Compactness makes the flow complete. The time-$(a-b)$ map ${\varphi }_{a-b}$ carries $M^b$ diffeomorphically onto $M^a$ and is the identity outside the cutoff region, and $f^{-1}[b,a]\cong M^b\times [b,a]$ via $(x,s)\mapsto {\varphi }_{s-b}(x)$ for $x\in f^{-1}(b)$.

Define $r:M^a\times [0,1]\to M^a$ by $$r_t(x)=\{\begin{array}{ll}x& f(x)\le b,\\ {\varphi }_{-t(f(x)-b)}(x)& b\le f(x)\le a.\end{array}$$ Then $r_0=\mathrm{id}$, each $r_t$ fixes $M^b$, and $r_1$ maps $M^a$ onto $M^b$ by pushing every point down its trajectory until its value reaches $b$. The two cases agree on $f^{-1}(b)$, so $r$ is continuous, and it is a deformation retraction of $M^a$ onto $M^b$. $\square$

Theorem 3.2 (Passing a critical value: handle attachment). Let $p$ be a nondegenerate critical point of index $\lambda$ with $f(p)=c$, and suppose $p$ is the only critical point on $f^{-1}(c)$, with $M^{c+\epsilon }$ compact and $f^{-1}[c-\epsilon ,c+\epsilon ]$ containing no other critical point for some $\epsilon >0$. Then there is an attaching map $\phi :S^{\lambda -1}\to M^{c-\epsilon }$ and a homotopy equivalence $$M^{c+\epsilon }\simeq M^{c-\epsilon }{\cup }_{\phi }{e}^{\lambda }.$$

Proof. By the Morse lemma 03.02.30 choose coordinates $(u_1,\dots ,u_{\lambda },v_1,\dots ,v_{n-\lambda })$ in a neighbourhood $U$ of $p$, centred at $p$, with $$f=c-|u{|}^2+|v{|}^2,|u{|}^2=\sum u_i^2,|v{|}^2=\sum v_j^2.$$ Shrink $\epsilon$ so the chart contains $\{|u{|}^2+|v{|}^2\le 2\epsilon \}$. Let $e^{\lambda }=\{|u{|}^2\le \epsilon ,v=0\}$ be the descending cell: the disk of dimension $\lambda$ along which $f\le c$ and which meets $M^{c-\epsilon }$ only in its boundary sphere $\{|u{|}^2=\epsilon ,v=0\}$. Set $\phi$ equal to the inclusion of that boundary sphere.

The argument has two halves. First, $M^{c+\epsilon }$ deformation-retracts onto $M^{c-\epsilon }\cup e^{\lambda }$. Define a new function $F$ that agrees with $f$ outside $U$ and equals $f-\mu (|u{|}^2+2|v{|}^2)$ inside, where $\mu$ is a smooth cutoff with $\mu (0)>\epsilon$, ${\mu }^{\prime }\in (-1,0]$, and $\mu$ supported in $\{|u{|}^2+2|v{|}^2<2\epsilon \}$. The modification lowers $f$ near $p$ enough that $F^{-1}(-\infty ,c-\epsilon ]=M^{c-\epsilon }\cup H$, where $H$ is a neighbourhood (a "handle") of the descending cell, while $F$ and $f$ have the same critical points and $F^{-1}(-\infty ,c+\epsilon ]=M^{c+\epsilon }$. The region $F^{-1}[c-\epsilon ,c+\epsilon ]$ now contains no critical value, so Theorem 3.1 gives $M^{c+\epsilon }\simeq F^{-1}(-\infty ,c-\epsilon ]=M^{c-\epsilon }\cup H$.

Second, $M^{c-\epsilon }\cup H$ deformation-retracts onto $M^{c-\epsilon }\cup e^{\lambda }$. The handle $H$ retracts onto $M^{c-\epsilon }\cup e^{\lambda }$ by pushing the $v$-directions to $0$ and flowing the $u$-directions outward along $-\nabla f$ to the boundary sphere $|u{|}^2=\epsilon$; on the part of $H$ already outside the cell this retraction lands in $M^{c-\epsilon }$, and the two prescriptions agree on the overlap. Composing the two retractions exhibits $$M^{c+\epsilon }\simeq M^{c-\epsilon }{\cup }_{\phi }{e}^{\lambda },$$ with $\phi :S^{\lambda -1}=\partial e^{\lambda }\hookrightarrow M^{c-\epsilon }$. $\square$

Corollary (CW homotopy type). If $f:M\to \mathbb{R}$ is a Morse function on a closed manifold with critical points $p_1,\dots ,p_k$ of indices ${\lambda }_1,\dots ,{\lambda }_k$, then $M$ has the homotopy type of a CW complex with exactly one cell of dimension ${\lambda }_i$ for each $i$.

Proof. Order the critical values $c_1<\cdots <c_m$ and pick regular values $a_0<c_1<a_1<\cdots <c_m<a_m$ with $M^{a_0}=\varnothing$ and $M^{a_m}=M$. Across each $[a_{j-1},a_j]$ Theorem 3.1 gives a homotopy equivalence onto $M^{a_{j-1}}$ except at the single critical value $c_j$, where Theorem 3.2 (applied once per critical point on that level) attaches one cell per critical point of the corresponding index. Whitehead's theorem and the cellular-approximation that an attaching map $S^{\lambda -1}\to M^{a_{j-1}}$ may be homotoped into the $(\lambda -1)$-skeleton turn the iterated mapping-cone into a genuine CW complex with the stated cells ^{[Milnor §3]}. $\square$

Bridge. The corollary converts an analytic object — the critical-point data of a smooth function — into a combinatorial one, a CW complex whose cell counts $c_{\lambda }$ are the numbers of index-$\lambda$ critical points. This builds toward the Morse inequalities, which appear immediately below as the homological shadow of these cells, and appears again in 03.12.13 where cellular homology computes Betti numbers directly from a cell structure of exactly this kind. The bridge is that handle attachment makes the chain complex of 03.12.13 available with one generator per critical point, so Betti numbers and critical-point counts are forced into the same inequalities.

Exercises Intermediate+

Advanced results Master

The Morse inequalities. Fix a field of coefficients, let $b_{\lambda }={\dim }_{}{H}_{\lambda }(M)$ and let $c_{\lambda }$ be the number of index-$\lambda$ critical points of a Morse function $f$ on a closed $M^n$. The CW model of the corollary has chain groups $C_{\lambda }$ free of rank $c_{\lambda }$, and cellular homology 03.12.13 computes $H_{\lambda }(M)=\ker {\partial }_{\lambda }/\mathrm{im}{\partial }_{\lambda +1}$. Write $z_{\lambda }=\mathrm{rank}\ker {\partial }_{\lambda }$ and ${\beta }_{\lambda }=\mathrm{rank}\mathrm{im}{\partial }_{\lambda }$. Rank-nullity in degree $\lambda$ reads $c_{\lambda }=z_{\lambda }+{\beta }_{\lambda }$, and by definition $b_{\lambda }=z_{\lambda }-{\beta }_{\lambda +1}$.

The weak Morse inequalities are immediate: $c_{\lambda }=z_{\lambda }+{\beta }_{\lambda }\ge z_{\lambda }\ge z_{\lambda }-{\beta }_{\lambda +1}=b_{\lambda }$, so $$c_{\lambda }\ge b_{\lambda }\text{for every }\lambda .$$

The strong Morse inequalities come from an alternating-sum telescoping. For each $\lambda$, $$\sum _{i=0}^{\lambda }(-1{)}^{\lambda -i}{c}_i\ge \sum _{i=0}^{\lambda }(-1{)}^{\lambda -i}{b}_i,$$ with equality at $\lambda =n$, where it becomes the Euler-characteristic identity $\sum (-1{)}^i{c}_i=\sum (-1{)}^i{b}_i=\chi (M)$. The cleanest packaging is the Morse polynomial $M_t={\sum }_{\lambda }{c}_{\lambda }{t}^{\lambda }$ and the Poincaré polynomial $P_t={\sum }_{\lambda }{b}_{\lambda }{t}^{\lambda }$: $$M_t-P_t=(1+t)Q(t),$$ where $Q(t)={\sum }_{\lambda }{q}_{\lambda }{t}^{\lambda }$ has nonnegative integer coefficients $q_{\lambda }={\beta }_{\lambda +1}\ge 0$. The strong inequalities are exactly the statement $q_{\lambda }\ge 0$ read coefficient by coefficient through the factor $1+t$, and the weak inequalities follow by dropping the $\beta$-terms.

Reeb's theorem. A closed smooth manifold $M^n$ admitting a Morse function with exactly two critical points is homeomorphic to $S^n$ ^{[Reeb 1952]}. The two critical points are a minimum (index $0$) and a maximum (index $n$). Theorem 3.2 makes each sublevel half a single Morse cell, and the Morse lemma makes a neighbourhood of each critical point a standard disk; the two closed disks $M^{1/2}$ and the closure of its complement are glued along an $(n-1)$-sphere by a homeomorphism, and $D^n{\cup }_h{D}^n\cong S^n$ for any boundary homeomorphism $h$. The conclusion is homeomorphism, not diffeomorphism: Milnor's exotic $7$-spheres carry Morse functions with two critical points yet are not diffeomorphic to the standard $S^7$, which is why the theorem cannot be upgraded to the smooth category.

Lacunary functions and perfection. A Morse function is perfect (over a field) when $c_{\lambda }=b_{\lambda }$ for all $\lambda$, equivalently $M_t=P_t$, equivalently $Q\equiv 0$. A lacunary Morse function — one with no two critical points of consecutive indices — is automatically perfect, because all cellular boundary maps vanish for degree reasons, forcing ${\beta }_{\lambda }=0$. The height function on ${\mathbb{CP}}^n$, with one critical point in each even index $0,2,\dots ,2n$, is lacunary and hence perfect, recovering $b_{2k}=1$, $b_{2k+1}=0$.

Full proof set Master

Proof of the strong Morse inequalities. With $C_{\bullet }$ the cellular chain complex, ${\partial }_{\lambda }:C_{\lambda }\to C_{\lambda -1}$, set $z_{\lambda }=\mathrm{rank}\ker {\partial }_{\lambda }$ and ${\beta }_{\lambda }=\mathrm{rank}\mathrm{im}{\partial }_{\lambda }$. Rank-nullity gives $c_{\lambda }=z_{\lambda }+{\beta }_{\lambda }$ and homology gives $b_{\lambda }=z_{\lambda }-{\beta }_{\lambda +1}$. Form the partial alternating sums $$S_{\lambda }^c=\sum _{i=0}^{\lambda }(-1{)}^{\lambda -i}{c}_i,S_{\lambda }^b=\sum _{i=0}^{\lambda }(-1{)}^{\lambda -i}{b}_i.$$ Substitute and telescope. Each $c_i=z_i+{\beta }_i$ and $b_i=z_i-{\beta }_{i+1}$, so $$S_{\lambda }^c-S_{\lambda }^b=\sum _{i=0}^{\lambda }(-1{)}^{\lambda -i}({\beta }_i+{\beta }_{i+1}).$$ The inner sum telescopes: consecutive terms $(-1{)}^{\lambda -i}{\beta }_{i+1}$ and $(-1{)}^{\lambda -(i+1)}{\beta }_{i+1}$ cancel in pairs, leaving the single surviving term ${\beta }_{\lambda +1}$ (with ${\beta }_0=0$). Hence $$S_{\lambda }^c-S_{\lambda }^b={\beta }_{\lambda +1}\ge 0,$$ which is the strong inequality $S_{\lambda }^c\ge S_{\lambda }^b$. At $\lambda =n$ there are no cells in degree $n+1$, so ${\beta }_{n+1}=0$ and the inequality is the equality $\chi =\sum (-1{)}^i{c}_i=\sum (-1{)}^i{b}_i$. $\square$

Proof of the polynomial identity $M_t-P_t=(1+t)Q(t)$. Using $c_{\lambda }=z_{\lambda }+{\beta }_{\lambda }$ and $b_{\lambda }=z_{\lambda }-{\beta }_{\lambda +1}$, $$M_t-P_t=\sum _{\lambda }(c_{\lambda }-b_{\lambda })t^{\lambda }=\sum _{\lambda }({\beta }_{\lambda }+{\beta }_{\lambda +1})t^{\lambda }.$$ Set $Q(t)={\sum }_{\lambda }{\beta }_{\lambda +1}{t}^{\lambda }$ with nonnegative integer coefficients. Then $$(1+t)Q(t)=\sum _{\lambda }{\beta }_{\lambda +1}{t}^{\lambda }+\sum _{\lambda }{\beta }_{\lambda +1}{t}^{\lambda +1}=\sum _{\lambda }{\beta }_{\lambda +1}{t}^{\lambda }+\sum _{\lambda }{\beta }_{\lambda }{t}^{\lambda }=\sum _{\lambda }({\beta }_{\lambda }+{\beta }_{\lambda +1})t^{\lambda },$$ using ${\beta }_0=0$ to reindex the second sum. The two displays coincide, giving $M_t-P_t=(1+t)Q(t)$. $\square$

Proof of Reeb's theorem. Let $p_0$ (index $0$, value $0$) and $p_1$ (index $n$, value $1$, after rescaling) be the only critical points. By the Morse lemma, near $p_0$ the function is $|x{|}^2$ and near $p_1$ it is $1-|y{|}^2$. For small $\epsilon$, $M^{\epsilon }$ is a closed disk $D^n$ (a single index-$0$ cell with its standard-disk neighbourhood), and likewise $\{f\ge 1-\epsilon \}$ is a closed disk $D^n$. By Theorem 3.1 the level $f^{-1}(\epsilon )$ is diffeomorphic to $f^{-1}(1-\epsilon )$, both being $(n-1)$-spheres, and the region $f^{-1}[\epsilon ,1-\epsilon ]$ is a product $S^{n-1}\times [\epsilon ,1-\epsilon ]$ with no critical points. Collapsing the product, $M=D_{-}^n{\cup }_h{D}_{+}^n$ glued along their boundary spheres by a homeomorphism $h$. Any two $n$-disks glued along a boundary homeomorphism yield a space homeomorphic to $S^n$ (extend $h$ radially over one disk to a homeomorphism $D^n\to D^n$, then the gluing is the standard $S^n=D^n{\cup }_{\mathrm{id}}{D}^n$). Hence $M\cong S^n$. The argument is topological; in the smooth category the radial extension of $h$ need not be a diffeomorphism, which is exactly the phenomenon of exotic spheres. $\square$

Existence of the descending sphere as attaching map. In the Morse normal form $f=c-|u{|}^2+|v{|}^2$, the descending disk $\{|u{|}^2\le \epsilon ,v=0\}$ has boundary $\{|u{|}^2=\epsilon ,v=0\}$ lying in $f^{-1}(c-\epsilon )\subset M^{c-\epsilon }$. That boundary is an embedded $(\lambda -1)$-sphere, and the inclusion is the attaching map $\phi$ of Theorem 3.2. Its homotopy class in $M^{c-\epsilon }$ is the only datum of the attachment that affects homotopy type, and Theorem 3.1 guarantees that the choice of $\epsilon$ within the regular range does not change it. $\square$

Connections Master

The Morse lemma, the index, and nondegeneracy that this unit consumes are established in 03.02.30; that unit owns the local normal form $f=f(p)-|u{|}^2+|v{|}^2$ and the genericity of Morse functions via Sard's theorem, and this unit is the first place that local data is converted into global topology through the sublevel-set filtration.

Cellular homology 03.12.13 is the computational engine behind the Morse inequalities: the cell structure produced by handle attachment feeds directly into the cellular chain complex, and the rank-nullity bookkeeping that yields $c_{\lambda }\ge b_{\lambda }$ is exactly the rank-nullity of the cellular boundary maps. The Betti numbers in every inequality here are the homology groups defined and computed there.

The CW machinery — attaching maps, skeleta, adjunction spaces, and Whitehead's theorem — comes from 03.12.10, and the corollary of Theorem 3.2 is a statement entirely in that language: a smooth manifold with a Morse function is, up to homotopy, a CW complex assembled by the attaching operation defined in that unit.

The downstream Morse complex sharpens these inequalities into a chain complex whose homology is the homology of $M$, by counting gradient trajectories between critical points of adjacent index; the cell counts $c_{\lambda }$ become the ranks of the Morse chain groups, and Smale's handle-cancellation refinement of Theorem 3.2 is the geometric input to the $h$-cobordism theorem, the route to the high-dimensional Poincaré conjecture.

Historical & philosophical context Master

Marston Morse introduced the relations now bearing his name in his 1925 paper in the Transactions of the American Mathematical Society ^{[Morse 1925]}, where the inequalities between critical-point counts and connectivity numbers (the period's term for Betti numbers) first appear for functions on manifolds. He consolidated and extended the theory in his 1934 American Mathematical Society Colloquium volume The Calculus of Variations in the Large ^{[Morse 1934]}, whose title names the ambition: to extract global topological conclusions from the critical points of variational functionals.

René Thom, in a 1949 Comptes Rendus note ^{[Thom 1949]}, gave the cell-decomposition viewpoint — a Morse function partitions the manifold into cells indexed by critical points — that underlies the modern CW formulation. Georges Reeb proved the two-critical-point sphere recognition theorem in 1952 ^{[Reeb 1952]}. John Milnor's 1963 Morse Theory ^{[Milnor §3]} gave the streamlined gradient-flow proofs of Theorems 3.1 and 3.2 reproduced above, and his 1956 discovery of exotic $7$-spheres is the reason Reeb's theorem stops at homeomorphism. The handle-cancellation refinement by Stephen Smale in the early 1960s turned this circle of ideas into the proof of the $h$-cobordism theorem and the high-dimensional Poincaré conjecture.

Bibliography Master

@book{milnor1963morse,
  author    = {Milnor, John W.},
  title     = {Morse Theory},
  series    = {Annals of Mathematics Studies},
  number    = {51},
  publisher = {Princeton University Press},
  year      = {1963},
  note      = {Based on lecture notes by M. Spivak and R. Wells}
}

@article{morse1925,
  author  = {Morse, Marston},
  title   = {Relations between the critical points of a real function of $n$ independent variables},
  journal = {Transactions of the American Mathematical Society},
  volume  = {27},
  number  = {3},
  pages   = {345--396},
  year    = {1925}
}

@book{morse1934,
  author    = {Morse, Marston},
  title     = {The Calculus of Variations in the Large},
  series    = {American Mathematical Society Colloquium Publications},
  volume    = {XVIII},
  publisher = {American Mathematical Society},
  year      = {1934}
}

@article{reeb1952,
  author  = {Reeb, Georges},
  title   = {Sur les points singuliers d'une forme de {P}faff compl{\`e}tement int{\'e}grable ou d'une fonction num{\'e}rique},
  journal = {Comptes Rendus de l'Acad{\'e}mie des Sciences de Paris},
  volume  = {222},
  pages   = {847--849},
  year    = {1952}
}

@article{thom1949,
  author  = {Thom, Ren{\'e}},
  title   = {Sur une partition en cellules associ{\'e}e {\`a} une fonction sur une vari{\'e}t{\'e}},
  journal = {Comptes Rendus de l'Acad{\'e}mie des Sciences de Paris},
  volume  = {228},
  pages   = {973--975},
  year    = {1949}
}

@article{smale1962,
  author  = {Smale, Stephen},
  title   = {On the structure of manifolds},
  journal = {American Journal of Mathematics},
  volume  = {84},
  number  = {3},
  pages   = {387--399},
  year    = {1962}
}

@article{milnor1956,
  author  = {Milnor, John W.},
  title   = {On manifolds homeomorphic to the 7-sphere},
  journal = {Annals of Mathematics},
  volume  = {64},
  number  = {2},
  pages   = {399--405},
  year    = {1956}
}