Differential Topology

Introduction

Differential topology is the study of smooth manifolds up to diffeomorphism, and of the invariants of a smooth map that survive a smooth deformation. Its characteristic tools are the regular value and the transversality that generalises it, the degree that counts preimages with signs, and the cobordism relation that asks when two closed manifolds bound a manifold of one dimension more. Each tool turns a geometric question into a finite combinatorial count, and each is robust: a regular value can be moved, a transverse intersection can be perturbed, and a degree is unchanged by a homotopy. This robustness is the reason the subject is a branch of topology and not merely of analysis, and it is why the differential-topological invariants computed here are topological invariants of the underlying space.

This article develops the subject from the definitions established in Smooth Manifolds and Differential Geometry. It treats Sard's theorem and its corollary that regular values are dense, the transversality of a smooth map to a submanifold and Thom's transversality theorem, the oriented and mod-two intersection numbers, the degree of a map between closed oriented manifolds of the same dimension with its regular-value formula, its homotopy invariance and its boundary and product formulas, the Brouwer fixed point theorem and the fundamental theorem of algebra as applications, the Whitney embedding and approximation theorems and the tubular neighbourhood theorem, the elements of Morse theory including the Morse lemma and the Morse inequalities, and the cobordism relation in outline.

The article assumes the manifold theory, the tangent space and the differential, the regular value theorem and the partition of unity of Smooth Manifolds and Differential Geometry, together with the topological background of Topological Spaces and Metric, Uniform and Complete Spaces. The Morse inequalities are stated in the de Rham form as standard. Sard's theorem and the approximation theorems are stated as standard; the measure-theoretic proof of Sard belongs to Part III, where the measure is available. The cobordism groups, the classifying spaces and the surgery obstruction are algebraic-topological constructions and are deferred. No physics is invoked.

Regular Values and Sard's Theorem

Critical Points and Regular Values

Definition. Let $F : M \to N$ be a smooth map between smooth manifolds of dimensions $m$ and $n$. A point $p \in M$ is a critical point of $F$ if the differential $dF_p : T_pM \to T_{F(p)}N$ is not surjective, and a regular point otherwise. The image $F(p)$ of a critical point is a critical value, and a point of $N$ that is not a critical value is a regular value; every point of $N \setminus F(M)$ is a regular value vacuously.

Theorem (regular value theorem). If $c \in N$ is a regular value of $F$, then $F^{-1}(c)$ is a regular submanifold of $M$ of dimension $m - n$, with $T_pF^{-1}(c) = \ker dF_p$ at each $p \in F^{-1}(c)$. If $c$ is a critical value the preimage may fail to be a manifold.

Proof. This is the implicit function theorem of Smooth Manifolds and Differential Geometry, applied at each point of the preimage.

Theorem (Sard). The set of critical values of a smooth map $F : M \to N$ has empty interior in $N$, equivalently the regular values are dense in $N$. The sharper statement, that the critical values form a null set, is proved in Part III, where the measure is available.

Proof sketch. The statement is local, and one reduces to a map between open sets of Euclidean spaces. For a map of class $C^k$ with $k$ large enough one writes the domain as a countable union of cubes on which the map is nearly affine and estimates the image of the critical set by a sum over the cubes; the total size of the images, measured as in Part III, is then seen to be arbitrarily small. The estimate uses the measure and the integration of Part III, where they are developed, and the theorem is quoted here as the standard analytic input of the subject.

Corollary. Regular values are dense in $N$: the set of critical values has empty interior in $N$, by Sard. In particular, if $\dim N \geq 1$ then $F$ has a regular value, and if $\dim M < \dim N$ then every value is critical, consistently with the fact that a smaller manifold cannot fill a larger one.

Corollary. If $F : M \to N$ is a smooth map and $\dim M = \dim N$, then the regular values are dense; at a regular value $c$, the preimage $F^{-1}(c)$ is finite when $M$ is compact.

Proof. At a regular value with $m = n$, the differential is an isomorphism at every preimage point, so $F$ is a local diffeomorphism there by the inverse function theorem; the preimage is discrete, and it is finite when $M$ is compact. Density is the corollary above.

Transversality

Transverse Maps and Submanifolds

Definition. Let $F : M \to N$ be a smooth map and let $Z \subseteq N$ be a regular submanifold. The map $F$ is transverse to $Z$, written $F \pitchfork Z$, if for every $p \in F^{-1}(Z)$ the subspaces

$$ dF_p(T_pM) \quad \text{and} \quad T_{F(p)}Z $$

together span $T_{F(p)}N$:

$$ dF_p(T_pM) + T_{F(p)}Z = T_{F(p)}N. $$

If $Z$ is a single point $c$, transversality says exactly that $c$ is a regular value. If $X \subseteq M$ and $Z \subseteq N$ are submanifolds and $F$ is the inclusion of $X$, the condition is written $X \pitchfork Z$ and reads $T_xX + T_xZ = T_xN$ at each $x \in X \cap Z$.

Definition. Two regular submanifolds $X, Z$ of $N$ are transverse at a point $x \in X \cap Z$ if $T_xX + T_xZ = T_xN$, and transverse if they are transverse at every point of $X \cap Z$.

Theorem (preimage theorem). If $F : M \to N$ is transverse to the regular submanifold $Z \subseteq N$, then $F^{-1}(Z)$ is a regular submanifold of $M$, of dimension

$$ \dim F^{-1}(Z) = \dim M - \dim N + \dim Z, $$

and at each $p \in F^{-1}(Z)$,

$$ T_p F^{-1}(Z) = (dF_p)^{-1}\bigl(T_{F(p)}Z\bigr). $$

Proof. The statement is local, so one may suppose $N = \mathbb{R}^n$ and $Z = \mathbb{R}^k \times \{0\}$, the local model of a submanifold. Write $F = (F_1, F_2)$ with $F_1 : M \to \mathbb{R}^k$ and $F_2 : M \to \mathbb{R}^{n-k}$. Transversality to $Z$ means that $d(F_2)_p$ is surjective whenever $F_2(p) = 0$. Hence $0$ is a regular value of $F_2$, and $F^{-1}(Z) = F_2^{-1}(0)$ is a submanifold of dimension $\dim M - (n - k)$ whose tangent space is $\ker d(F_2)_p = (dF_p)^{-1}(\mathbb{R}^k \times \{0\})$.

Corollary (transverse intersection). If $X$ and $Z$ are transverse regular submanifolds of $N$, then $X \cap Z$ is a regular submanifold of dimension $\dim X + \dim Z - \dim N$.

Proof. Apply the preimage theorem to the inclusion $X \hookrightarrow N$, which is transverse to $Z$.

Example. Two curves in a surface intersect transversely when their tangent lines are distinct at each intersection point, and the intersection is a finite set; this is the dimension count $1 + 1 - 2 = 0$. A curve and a surface in three-space intersect transversely when the curve crosses the surface and is not tangent to it, giving points; a curve tangent to the surface at a point may still be transverse there if it crosses, but a curve lying in the surface is not transverse.

The Transversality Theorem

Definition. Let $F : M \to N$ be smooth and let $Z \subseteq N$ be a submanifold. A perturbation of $F$ transverse to $Z$ is a smooth map $G$ homotopic to $F$ with $G \pitchfork Z$; the homotopy may be taken to be arbitrarily small in the fine topology on maps.

Theorem (transversality theorem, Thom). Let $F : M \to N$ be a smooth map of manifolds and $Z \subseteq N$ a regular submanifold. Then there are arbitrarily small smooth perturbations $G$ of $F$ with $G \pitchfork Z$. If in addition $F$ is transverse to $Z$ on a closed subset $A \subseteq M$ on which the transversality condition already holds, the perturbation may be taken to agree with $F$ on a neighbourhood of $A$.

Proof sketch. The space of smooth maps $M \to N$ is parametrised locally by finite-dimensional families, and the evaluation map for such a family is transverse to $Z$ by Sard's theorem; a regular parameter value then gives a map transverse to $Z$. Agreement on a closed set is achieved by choosing the perturbations supported away from it. The finite-dimensional approximation uses the approximation theorem below.

Corollary. If $\dim M + \dim Z < \dim N$, then a map $F : M \to N$ can be perturbed to one with $F(M) \cap Z = \emptyset$, and if $\dim M + \dim Z = \dim N$ it can be perturbed to one meeting $Z$ in a finite set of transverse points. Consequently two submanifolds of complementary dimension can always be made to meet transversely in finitely many points.

Remark. Transversality is the differential-topological form of the statement that a generic configuration is as simple as the dimension count allows. It is used below to make intersections finite and to define the intersection number, and it underlies the degree theory and the Morse theory of the rest of the article.

Degree Theory

The Oriented Intersection Number

Let $M$ and $N$ be smooth manifolds with $M$ compact, and let $F : M \to N$ be a smooth map transverse to a closed submanifold $Z \subseteq N$ with

$$ \dim M + \dim Z = \dim N. $$

Transversality makes $F^{-1}(Z)$ a finite set. Suppose $M$, $N$ and $Z$ are oriented and $N$ is connected. At each $p \in F^{-1}(Z)$ the differential $dF_p$ maps $T_pM$ isomorphically onto a complement of $T_{F(p)}Z$; the sign of $p$ is $+1$ if the direct sum orientation of $dF_p(T_pM) \oplus T_{F(p)}Z$ agrees with the orientation of $T_{F(p)}N$ and $-1$ otherwise.

Definition. The oriented intersection number of $F$ with $Z$ is

$$ I(F, Z) = \sum_{p \in F^{-1}(Z)} \operatorname{sign}(p) \in \mathbb{Z}. $$

Proposition. The integer $I(F, Z)$ depends only on the homotopy class of $F$, not on the perturbation making it transverse.

Proof sketch. A homotopy between two transverse maps can itself be made transverse to $Z$ by the transversality theorem, and the preimage of $Z$ under the homotopy is then a compact oriented one-manifold with boundary the union of the two finite preimage sets, counted with signs; a compact oriented one-manifold has boundary of total signed count zero, which gives the invariance.

The Degree of a Map

Now let $M$ and $N$ be closed oriented manifolds of the same dimension $n$, with $N$ connected, and let $F : M \to N$ be smooth. At a regular value $c \in N$ the preimage $F^{-1}(c)$ is finite, and the sign of a preimage point $p$ is $+1$ if $dF_p$ preserves orientation and $-1$ if it reverses it.

Definition. The degree of $F$ is

$$ \deg(F) = \sum_{p \in F^{-1}(c)} \operatorname{sign}(p) $$

for any regular value $c$ of $F$; the right-hand side is independent of the regular value chosen, by the previous proposition applied to $Z = \{c\}$ and the connectedness of the complement of the critical values.

Theorem (properties of degree). Let $M$ and $N$ be closed oriented manifolds of dimension $n$ and let $F, G : M \to N$ be smooth.

(a) $\deg(F)$ is an integer, and $\deg(F) = 0$ if $F$ is not surjective.

(b) If $F$ and $G$ are homotopic then $\deg(F) = \deg(G)$.

(c) If $N$ is a closed oriented manifold and $F$ is a diffeomorphism then $\deg(F) = +1$ if $F$ preserves orientation and $-1$ if it reverses it.

(d) $\deg(F \circ G) = \deg(F)\deg(G)$ for composable maps, and $\deg(\mathrm{id}) = 1$.

(e) If $M = N = S^n$ then two maps are homotopic if and only if they have the same degree.

(f) If $M$ and $N$ are compact with boundary and $F$ maps $\partial M$ to $\partial N$, then $\deg(F|_{\partial M}) = \deg(F)$.

Proof sketch. Part (a) is the finiteness of the preimage and the definition; (b) is the homotopy invariance of the intersection number; (c) follows by taking a regular value and using the inverse function theorem; (d) follows from the chain rule and the multiplicativity of orientation signs; (e) is the Hurewicz and cellular approximation argument, standard from algebraic topology; (f) follows by applying the one-manifold boundary argument to a regular value of $F$.

Definition. The degree mod $2$, written $\deg_2(F)$, is the parity of $\# F^{-1}(c)$ for a regular value $c$; it is defined without any orientation and depends only on the homotopy class. It is the reduction of $\deg(F)$ modulo $2$ when an orientation is present.

The Winding Number and the Fundamental Theorem of Algebra

Example (maps of the circle). For a smooth map $F : S^1 \to S^1$ the degree is the winding number: it is the integer $k$ such that $F$ lifts to the map $t \mapsto kt + \text{constant}$ on the universal cover $\mathbb{R}$, and it counts the signed number of times the image winds around the circle. A map of degree $k \neq 0$ is surjective, and the identity has degree $1$ while the antipodal map $z \mapsto -z$ has degree $-1$.

Example (the fundamental theorem of algebra). Let $p(z) = z^n + a_{n-1}z^{n-1} + \cdots + a_0$ be a complex polynomial of degree $n \geq 1$. Regard it as a map $\mathbb{C} \to \mathbb{C}$ and extend it to the one-point compactifications, giving a smooth map $\hat p : S^2 \to S^2$ with $\hat p(\infty) = \infty$. On the circle of large radius $R$ the term $z^n$ dominates, so $p$ is homotopic on the region $|z| \geq R$ to $z \mapsto z^n$; hence $\hat p$ is homotopic to the map $z \mapsto z^n$ on the whole sphere, of degree $n$. A map of nonzero degree is surjective, so $p$ attains $0$, and $p$ has a complex root.

The Brouwer Fixed Point Theorem

Theorem (Brouwer fixed point, smooth case). Every smooth map $f : \overline{B^n} \to \overline{B^n}$ of the closed unit ball into itself has a fixed point.

Proof. Suppose $f$ has no fixed point, so that $f(x) \neq x$ for all $x$ in the ball. Define $r : \overline{B^n} \to S^{n-1}$ by letting $r(x)$ be the point where the ray from $f(x)$ through $x$ meets the sphere; the map is smooth because $f$ is smooth and the division by $|x - f(x)|$ with $|x - f(x)| \geq \epsilon > 0$ is smooth. On the sphere $r$ is the identity, so $r \circ \iota = \mathrm{id}_{S^{n-1}}$ for the inclusion $\iota : S^{n-1} \hookrightarrow \overline{B^n}$, and taking degrees gives

$$ \deg(r \circ \iota) = \deg(\mathrm{id}_{S^{n-1}}) = 1 . $$

But the ball is contractible, so the inclusion $\iota$ is nullhomotopic; hence $r \circ \iota$ is nullhomotopic, and by the homotopy invariance of the degree, part (b), it has degree $0$. The two values of the degree of the same map contradict each other. Hence $f$ has a fixed point.

Remark. The same argument with degree mod $2$ gives the theorem for a continuous map, since a continuous map can be uniformly approximated by a smooth one by the approximation theorem below, and a fixed point of the approximation need not be a fixed point of the original; the standard deduction uses a limiting argument, which is a statement about the limit and belongs to Part III. The smooth case is what differential topology supplies directly.

Embeddings, Approximation and Tubular Neighbourhoods

The Whitney Theorems

Definition. A smooth map $F : M \to \mathbb{R}^k$ is an embedding if it is an immersion and a homeomorphism onto its image, which is then a regular submanifold of $\mathbb{R}^k$. A manifold is embeddable in $\mathbb{R}^k$ if it admits an embedding.

Theorem (Whitney embedding). Every smooth manifold of dimension $n$ admits a proper embedding into $\mathbb{R}^{2n}$, and a (not necessarily proper) embedding into $\mathbb{R}^{2n-1}$ for $n \geq 2$. In particular every smooth manifold is diffeomorphic to a regular submanifold of a Euclidean space.

Proof sketch. One first embeds $M$ into some $\mathbb{R}^N$ by a proper map built from a partition of unity, then uses Sard's theorem to project the image to lower dimensions without creating self-intersections or destroying injectivity of the differential: at each projection step the set of directions that create a self-intersection or a tangency is the set of critical values of an associated map of a manifold of controlled dimension into the space of directions, and Sard's theorem guarantees that a direction outside it exists. The two steps reduce the ambient dimension to $2n$, and a refinement gives $2n - 1$ when $n \geq 2$.

Theorem (Whitney approximation). Every continuous map between smooth manifolds is homotopic to a smooth map, and if it is already smooth on a closed subset it may be approximated there by smooth maps agreeing with it on a neighbourhood of that subset.

Tubular Neighbourhoods

Definition. Let $S \subseteq M$ be a regular submanifold of a Riemannian manifold $(M, g)$ with metric $g$ from Smooth Manifolds and Differential Geometry. The normal bundle of $S$ in $M$ is the subbundle $NS \subseteq TM|_S$ whose fibre at $p$ is the orthogonal complement of $T_pS$ with respect to $g_p$. The exponential map of the metric, defined, sends a neighbourhood of the zero section of $NS$ diffeomorphically onto a neighbourhood of $S$ in $M$.

Theorem (tubular neighbourhood). Every regular submanifold $S$ of a smooth manifold $M$ has an open neighbourhood $U$, the tubular neighbourhood, that is the total space of a vector bundle over $S$ — the normal bundle when a metric is chosen — and the inclusion $S \hookrightarrow U$ is a homotopy equivalence.

Proof sketch. Choose a Riemannian metric, form the normal bundle, and apply the inverse function theorem to the restriction of the metric exponential map to a small disc bundle; the resulting diffeomorphism exhibits $U$ as a disc bundle over $S$, and a disc bundle deformation retracts onto its zero section.

Morse Theory

Morse Functions and the Morse Lemma

Definition. Let $f : M \to \mathbb{R}$ be a smooth function. A critical point $p$ of $f$ is nondegenerate if the Hessian bilinear form on $T_pM$,

$$ \operatorname{Hess}_p f(v, w) = V(W(f))(p), $$

where $V, W$ are vector fields extending $v, w$, is nondegenerate. The index of a nondegenerate critical point is the index of the Hessian quadratic form at that point — the number of negative squares in a diagonal form of $\operatorname{Hess}_p f$, well defined by Sylvester's law of inertia. The function $f$ is a Morse function if all its critical points are nondegenerate.

The Hessian is well defined: the value $V(W(f))(p)$ depends only on $v$ and $w$, because the difference between two extensions changes the expression by terms involving $[V, W](f)(p)$, which vanishes at a critical point.

Theorem (Morse lemma). Let $p$ be a nondegenerate critical point of $f$ of index $\lambda$. Then there are coordinates $x^1, \ldots, x^n$ on a neighbourhood of $p$ with $x^i(p) = 0$ and

$$ f = f(p) - (x^1)^2 - \cdots - (x^\lambda)^2 + (x^{\lambda+1})^2 + \cdots + (x^n)^2 . $$

Proof sketch. The statement is local and reduces to a quadratic form, which the classification of quadratic forms of Quadratic Forms and Polarisation puts into the displayed diagonal shape; the change of coordinates is produced by the inverse function theorem. Nondegeneracy is exactly the invertibility needed.

Corollary. A nondegenerate critical point is isolated; hence a Morse function on a closed manifold has finitely many critical points, and the critical values are isolated.

Theorem (existence of Morse functions). Morse functions exist on every smooth manifold; in fact the Morse functions form a dense open subset of $C^\infty(M)$ in the fine topology.

Proof sketch. By Sard's theorem applied to the map $p \mapsto \operatorname{Hess}_p f$ in local coordinates, a generic perturbation of $f$ avoids the critical values of the Hessian determinant; the openness is the openness of nondegeneracy.

The Morse Inequalities

Definition. For a Morse function $f$ on a closed manifold $M$ and a real number $a$ write $M_a = f^{-1}((-\infty, a])$ for the sublevel set. Let $c_k$ be the number of critical points of index $k$ and $b_k = \dim_{\mathbb{R}} H^k_{dR}(M)$ the $k$-th Betti number, the de Rham cohomology being taken as standard.

Theorem (weak Morse inequalities). $c_k \geq b_k$ for every $k$, and

$$ \sum_{k=0}^{n} (-1)^k c_k = \sum_{k=0}^{n} (-1)^k b_k = \chi(M), $$

the Euler characteristic.

Theorem (strong Morse inequalities). For every $k$,

$$ c_k - c_{k-1} + c_{k-2} - \cdots \pm c_0 \geq b_k - b_{k-1} + b_{k-2} - \cdots \pm b_0 . $$

Proof sketch. As the level passes a critical value of index $\lambda$, the sublevel set changes by the attachment of a $\lambda$-handle $D^\lambda \times D^{n-\lambda}$ along $S^{\lambda-1} \times D^{n-\lambda}$; the handle attachment is the topological content of the Morse lemma, and it produces a chain complex whose $k$-th chain group has rank $c_k$ and whose homology is $H^k_{dR}(M)$. The rank inequalities for a chain complex then give the stated bounds. The handle attachment in the smooth category requires the gradient flow of $f$, whose existence and regularity are the differential-equation theory of Part III; the topological chain-level statement is what is used here, and it is standard.

Example. A Morse function on the torus $T^2$ has at least one critical point of each index $0, 1, 2, 1$, namely $c_0 \geq 1$, $c_1 \geq 2$, $c_2 \geq 1$, with total signed count $1 - 2 + 1 = 0 = \chi(T^2)$, which matches the Betti numbers $1, 2, 1$ of the torus.

Remark. Morse theory is the bridge between the differential topology of a function and the algebraic topology of the manifold, and the handles it attaches are the same handles used in the surgery classification of manifolds. The handle decomposition is the differential-topological input to the algebraic-topological theory.

Cobordism in Outline

The Cobordism Relation

Definition. Two closed smooth $n$-manifolds $M_0$ and $M_1$ are cobordant (or bordant) if there is a compact smooth $(n+1)$-manifold $W$ with boundary

$$ \partial W = M_0 \sqcup M_1, $$

the boundary components carrying the orientations induced by that of $W$, the manifold $M_0$ being oriented as a boundary component and $M_1$ with the opposite orientation. The manifold $W$ is a cobordism from $M_0$ to $M_1$. A closed manifold bounds if it is cobordant to the empty manifold, equivalently if it is the boundary of a compact manifold.

Proposition. Cobordism is an equivalence relation on the closed oriented $n$-manifolds: reflexivity is given by the cylinder $M_0 \times [0, 1]$, symmetry by reversing the cobordism, and transitivity by gluing two cobordisms along the common boundary component.

Proof. The cylinder has boundary $M_0 \sqcup \overline{M_0}$, giving reflexivity up to the orientation convention; reversing the interval exchanges the two boundary components; and gluing along a common boundary component uses a collar neighbourhood, whose existence is the tubular neighbourhood theorem, and smooths the corner.

Definition. The set of cobordism classes of closed oriented $n$-manifolds forms an abelian group $\Omega_n^{SO}$ under disjoint union, with zero the class of the empty manifold and inverses given by orientation reversal. Similarly the unoriented closed $n$-manifolds form a group $\mathrm{N}_n$ under disjoint union and cobordism, with inverses not needed because every element is its own inverse, so that $\mathrm{N}_n$ is a vector space over $\mathbb{F}_2$.

Example. $\Omega_0^{SO} \cong \mathbb{Z}$, the class of a point, and a closed oriented zero-manifold is a finite signed set; every closed oriented one-manifold bounds, so $\Omega_1^{SO} = 0$; a closed oriented surface bounds a three-manifold, so $\Omega_2^{SO} = 0$; and $\Omega_3^{SO} = 0$, while $\Omega_4^{SO} \cong \mathbb{Z}$ generated by the complex projective plane with its complex orientation.

Remark. The computation of the cobordism groups, the classifying spaces, the Pontryagin–Thom construction and the relation to the characteristic classes constitute the cobordism theory and the surgery theory, and they are not developed here. What belongs to differential topology is the relation itself and the fact that it is transverse-friendly: a cobordism can be put into general position relative to a map into a fixed space, which is the mechanism that converts the classification of manifolds up to cobordism into a computation about the topology of the target space. In the unoriented case the mod-two degree of a map to a sphere, defined above, is the first of the numerical invariants of the theory: by Thom's theorem a closed $n$-manifold bounds if and only if all its Stiefel–Whitney numbers vanish, and the characteristic classes are standard.

Summary

Sard's theorem says that the critical values of a smooth map have empty interior, equivalently that the regular values are dense, and the regular value theorem makes the preimage of a regular value a submanifold whose tangent space is the kernel of the differential. Transversality is the relative form of the same condition: a map $F$ is transverse to a submanifold $Z$ when the image of the differential together with the tangent space of $Z$ spans the ambient tangent space, and then $F^{-1}(Z)$ is a submanifold of the expected dimension. Thom's transversality theorem says that any map can be perturbed into a transverse one, so generic intersections have the dimension the count predicts and are finite when the dimensions are complementary.

The oriented intersection number of a map with a submanifold of complementary dimension is the signed count of the intersection points, and it is invariant under homotopy; at a point it is the degree of a map between closed oriented manifolds of the same dimension, and the degree is an integer, additive under composition, invariant under homotopy, equal to $\pm 1$ for a diffeomorphism, and a complete invariant of homotopy for maps of spheres. It gives the winding number of a circle map, the fundamental theorem of algebra as the surjectivity of a degree-$n$ map of the sphere, and the Brouwer fixed point theorem by the contradiction between a retraction of the ball onto its boundary and the trivial degree of a map factoring through a contractible space.

Every smooth manifold embeds in a Euclidean space, by Whitney's embedding theorem, continuous maps are homotopic to smooth ones by the approximation theorem, and every submanifold has a tubular neighbourhood that is a disc bundle and a homotopy equivalence. A Morse function has nondegenerate critical points, the Morse lemma puts each into the normal form $-(x^1)^2 - \cdots + (x^{\lambda})^2 + (x^{\lambda+1})^2 + \cdots + (x^n)^2$ up to a constant, the sublevel sets change by handle attachments, and the counts of critical points satisfy the Morse inequalities with respect to the Betti numbers of the de Rham cohomology. Two closed manifolds are cobordant when together they bound a compact manifold of one dimension more; cobordism is an equivalence relation and the cobordism classes form a group under disjoint union, whose computation is a theorem of algebraic topology and belongs to the article being written in parallel.

Summary of Notation

Symbol	Meaning
$F \pitchfork Z$	The map $F$ is transverse to the submanifold $Z$: $dF_p(T_pM) + T_{F(p)}Z = T_{F(p)}N$
$X \pitchfork Z$	Transverse intersection of submanifolds: $T_xX + T_xZ = T_xN$
Regular value $c$	Value at which every preimage point is a regular point; $F^{-1}(c)$ is a submanifold
$\operatorname{sign}(p)$	Orientation sign of a transverse preimage point $p$
$I(F, Z) = \sum_p \operatorname{sign}(p)$	Oriented intersection number; homotopy invariant
$\deg(F)$, $\deg_2(F)$	Degree and degree mod $2$ of a map of closed oriented $n$-manifolds
$\deg(F \circ G) = \deg F \deg G$	Multiplicativity of degree
Winding number	Degree of a map $S^1 \to S^1$; the integer $k$ in the lift $t \mapsto kt + \mathrm{const}$
$\overline{B^n}$, $S^{n-1}$	Closed unit ball and its boundary sphere
Whitney embedding	$M^n$ embeds in $\mathbb{R}^{2n}$, and in $\mathbb{R}^{2n-1}$ for $n \geq 2$
$NS$	Normal bundle of a submanifold, with respect to a Riemannian metric
Tubular neighbourhood	Disc bundle neighbourhood of a submanifold; a homotopy equivalence onto $S$
$\operatorname{Hess}_p f$	Hessian bilinear form at a critical point
Morse function, index $\lambda$	Function with nondegenerate critical points; number of negative squares of the Hessian
$c_k$, $b_k$	Number of critical points of index $k$; $k$-th Betti number $\dim H^k_{dR}(M)$
Morse inequalities	$c_k \geq b_k$ and the alternating stronger forms; $\sum(-1)^k c_k = \chi(M)$
$\partial W = M_0 \sqcup M_1$	Cobordism $W$ from $M_0$ to $M_1$
$\Omega_n^{SO}$, $\mathrm{N}_n$	Oriented and unoriented cobordism groups of closed $n$-manifolds