Introduction to Differential Topology

Foundations of Differential Topology Introduction Differential topology is the study of smooth geometric structures and their properties. Rather than focusing on rigid measurements like distance or angles (the domain of differential geometry), differential topology examines what properties of manifolds are preserved under smooth deformations. This course builds from the definition of smooth manifolds through to powerful invariants that classify low-dimensional spaces and capture essential topological information through calculus-based techniques. The key insight unifying differential topology is that smooth maps between manifolds can be understood locally—near any point, a smooth map behaves like a linear map between Euclidean spaces. This principle, formalized through the differential of a map, allows us to translate geometric questions into linear algebra, then back to global topology. Part 1: Foundations of Differential Topology What is a Smooth Manifold? A smooth manifold is a space that looks locally like Euclidean space, with a notion of smoothness built in. More precisely, an $n$-dimensional smooth manifold is a topological space where every point has a neighborhood that resembles an open set in $\mathbb{R}^n$, with all transitions between these local descriptions being smooth. The motivation for this definition comes from familiar examples: the circle $S^1$, the sphere $S^2$, and the torus are all smooth manifolds. Unlike these embedded examples, the abstract definition doesn't require the manifold to live in any particular ambient space—smoothness is intrinsic to the manifold itself. Charts and Atlases To make the notion of "looks locally like $\mathbb{R}^n$" precise, we use charts. A chart around a point $p$ is a smooth map $\phi\colon U \to \mathbb{R}^n$ where $U$ is a neighborhood of $p$, and $\phi$ is a homeomorphism onto an open set in $\mathbb{R}^n$ with smooth inverse. Think of a chart as a coordinate system that locally flattens the manifold into Euclidean space. An atlas is a collection of charts that together cover the entire manifold. The circle $S^1$, for example, can be covered by two overlapping charts: one parameterizing the circle without the north pole, and another without the south pole. Compatibility Through Transition Maps Where two charts overlap, their coordinate systems must be compatible. If $\phi1\colon U1 \to \mathbb{R}^n$ and $\phi2\colon U2 \to \mathbb{R}^n$ are two overlapping charts, the transition map $\phi2 \circ \phi1^{-1}$ takes points in $\mathbb{R}^n$ (the coordinates from chart 1) to points in $\mathbb{R}^n$ (the coordinates from chart 2). For a smooth manifold, all transition maps must be smooth—they must have continuous derivatives of all orders. This smoothness requirement ensures that the notion of smoothness on the manifold is consistent and independent of which chart we use. Without it, we could have conflicting notions of which functions are "smooth." Tangent Spaces and Directional Derivatives Once we have a smooth manifold, we can make sense of notions like "direction" and "rate of change" at each point. The tangent space $Tp M$ at a point $p$ of a manifold $M$ is the vector space of all tangent vectors at $p$. A tangent vector represents a direction at $p$, and geometrically corresponds to the velocity vector of a curve passing through $p$. More concretely, if $\gamma(t)$ is a smooth curve with $\gamma(0) = p$, then $\gamma'(0)$ is a tangent vector in $Tp M$. The key insight is that tangent vectors can be understood as directional derivatives. Given a tangent vector $v \in Tp M$ and a smooth function $f\colon M \to \mathbb{R}$, the tangent vector $v$ acts on $f$ by computing the directional derivative of $f$ in the direction of $v$: $$v(f) = \frac{d}{dt}\bigg|{t=0} f(\gamma(t))$$ where $\gamma$ is any curve with $\gamma'(0) = v$. This perspective shows why the definition of a smooth manifold is exactly what we need: because transitions between charts are smooth, the set of tangent vectors at $p$ forms a well-defined $n$-dimensional vector space, and directional derivatives are coordinate-independent. Differential Forms and Integration A differential form on a manifold is a smooth assignment of an alternating multilinear function to each tangent space. This definition may sound abstract, but differential forms are the natural objects for integration on manifolds. The simplest examples are smooth functions $f\colon M \to \mathbb{R}$, which are 0-forms. A 1-form $\omega$ assigns to each point $p$ a linear functional on $Tp M$—think of it as an "infinitesimal covector" at each point. Higher degree forms capture information about volumes and orientations. The power of differential forms lies in their calculus: there is an exterior derivative $d$ that takes $k$-forms to $(k+1)$-forms, and Stokes' theorem states that $$\intM d\omega = \int{\partial M} \omega$$ for any $(n-1)$-form $\omega$ on an $n$-dimensional manifold $M$ with boundary $\partial M$. This is the natural generalization of integration by parts to manifolds. Differential forms provide the language for stating many fundamental results in differential topology, especially those involving integration over submanifolds. Smooth Maps and Their Differentials Just as calculus on $\mathbb{R}^n$ studies smooth functions, differential topology studies smooth maps between manifolds. A smooth map $f\colon M \to N$ between two smooth manifolds is one that, when expressed in local charts, becomes an ordinary smooth function $\mathbb{R}^m \to \mathbb{R}^n$. That is, if $\phi$ and $\psi$ are charts around points in $M$ and $N$ respectively, then $\psi \circ f \circ \phi^{-1}$ must be smooth in the usual sense. The differential (or pushforward) of $f$ at a point $p$ is a linear map $$dfp \colon Tp M \to T{f(p)} N$$ defined by: for each tangent vector $v \in Tp M$ (viewed as a directional derivative), $$dfp(v)(g) = v(g \circ f)$$ for any smooth function $g\colon N \to \mathbb{R}$. Intuitively, $dfp$ tells us how $f$ stretches or contracts directions at $p$. If a curve through $p$ has velocity $v$, then its image curve under $f$ has velocity $dfp(v)$. The rank of $dfp$—the dimension of its image—measures how many directions are "enlarged" by $f$. If $dfp$ has rank less than $\dim N$, then $f$ is squashing the manifold down; if it has rank less than $\dim M$, then multiple directions are being mapped to the same direction. Local Classification: Embeddings, Immersions, and Submersions The rank of the differential $dfp$ determines the local geometry of $f$ near $p$. There are three main cases: Immersions: If $dfp$ is injective (rank equals $\dim M$), then $f$ is an immersion near $p$. Geometrically, immersions are locally homeomorphisms—they preserve local dimension. However, immersions need not be globally injective: different points can map to the same image. The figure-eight curve is an immersion of the circle into the plane that is not injective. Embeddings: An embedding is an immersion that is also a homeomorphism onto its image (in the subspace topology). Think of an embedding as an immersion with the additional property that it doesn't self-intersect. Every embedding is an immersion, but not every immersion is an embedding. Submersions: If $dfp$ is surjective (rank equals $\dim N$), then $f$ is a submersion near $p$. Submersions map manifolds onto manifolds of lower dimension in a controlled way. The projection map from a cylinder onto its circular base is a submersion. Local Diffeomorphisms: If $dfp$ is both injective and surjective—necessarily requiring $\dim M = \dim N$—then $dfp$ is an isomorphism, and $f$ is a local diffeomorphism near $p$. Local diffeomorphisms are locally invertible and preserve all smooth structure. A common confusion: the distinction between immersions and embeddings hinges on whether $f$ is a homeomorphism onto its image, not just locally injective. You can have an injective immersion that is not an embedding—for instance, a spiral curve spiraling infinitely in the plane has an injective differential everywhere but is not a homeomorphism onto its image. Part 2: Classification of Low-Dimensional Manifolds Compact One-Dimensional Manifolds The simplest manifolds are one-dimensional. A compact one-dimensional manifold without boundary is a closed curve. The classification is complete and beautiful: Every compact one-dimensional manifold without boundary is homeomorphic to the circle $S^1$. This may seem obvious—how many ways can you topologically deform a closed loop?—but it is a genuine theorem: any closed 1D manifold, no matter how it is presented, is the same as a circle. If we allow boundary (endpoints), then: Every compact one-dimensional manifold with boundary is homeomorphic to a closed interval $[0,1]$. These classifications use the fact that compact connected 1-manifolds without boundary have no "ends" to escape to, and their compactness prevents infinite spiraling. Closed Surfaces: The Two-Dimensional Classification The classification of two-dimensional closed manifolds (surfaces without boundary) is one of the crown jewels of topology, and it is remarkably complete. Classification Theorem for Closed Surfaces: Every closed, connected, orientable or non-orientable surface is homeomorphic to exactly one of the following: The sphere $S^2$ A connected sum of $g$ tori $T^2 \# \cdots \# T^2$ (denoted $Tg$), for $g \geq 1$ A connected sum of $k$ projective planes $\mathbb{RP}^2 \# \cdots \# \mathbb{RP}^2$, for $k \geq 1$ The sphere is the only simply-connected closed surface. Every other surface either has genus (handles) or is non-orientable. A torus $T^2$ (the doughnut surface pictured above) is the simplest non-simply-connected surface. The connected sum of two surfaces is formed by cutting a small disk from each and gluing them together along the boundary. Adding more tori increases the genus. The image above shows connected sums of tori—surfaces with multiple handles attached to a sphere. This diagram illustrates how a surface $W$ can be decomposed into simpler pieces: a manifold $M$ with genus $g$ (on the left) and a surface with a non-orientable cross-cap (on the right). For non-orientable surfaces, the projective plane $\mathbb{RP}^2$ plays the role that the torus plays for orientable surfaces. A non-orientable surface cannot be consistently oriented globally—a tangent vector flips sign when you go around certain loops. Genus and Classification The genus is the fundamental invariant classifying surfaces: For orientable surfaces, the genus $g$ counts the number of "handles" (tori) attached to the sphere. A sphere has genus 0, a torus has genus 1, a figure-eight surface (two tori joined) has genus 2. For non-orientable surfaces, we similarly use a genus counting the number of cross-caps (projective planes) attached to a sphere. The crucial fact is that two closed surfaces are homeomorphic if and only if they have the same orientability and the same genus. This provides a complete classification: once you determine whether a surface is orientable and compute its genus, you have completely determined its topological type. The genus is typically computed using the Euler characteristic, another topological invariant, via the formula: For an orientable surface of genus $g$: $\chi = 2 - 2g$ For a non-orientable surface of non-orientable genus $k$: $\chi = 2 - k$ where $\chi = V - E + F$ for any triangulation (Euler's formula). Part 3: Transversality and Intersection Theory Defining Transverse Intersection One of the most powerful concepts in differential topology is transversality, which captures the idea of submanifolds intersecting "generically" without special alignment. Two submanifolds $X$ and $Y$ of an ambient manifold $M$ intersect transversely at a point $p \in X \cap Y$ if their tangent spaces together span the entire tangent space of $M$ at $p$: $$Tp X + Tp Y = Tp M$$ (where $+$ denotes the sum of subspaces). Intuitively, transversality means the submanifolds are not tangent to each other—they cross cleanly, at an angle. For example, two smooth curves in the plane intersect transversely at a point if they cross (not if they are tangent). In 3D, a curve and a surface intersect transversely if the curve pokes through the surface rather than lying along it. The tangent space condition captures exactly this geometric notion. If two submanifolds are transverse at $p$, then the intersection near $p$ has dimension $\dim(X) + \dim(Y) - \dim(M)$. This is the expected dimension—and having the expected dimension is what we mean by "generic" intersection. Stability Under Perturbation A key reason transversality matters is its stability: transverse intersections are robust under small perturbations. If $X$ and $Y$ intersect transversely at a point $p$, then small smooth perturbations of $X$ and $Y$ will still have an intersection near $p$, and the intersection will remain transverse. The intersection points do not suddenly appear or disappear. Conversely, if $X$ and $Y$ intersect non-transversely (tangentially), then a generic small perturbation can break the intersection entirely, move the intersection points, or create new intersections. This stability is why transversality appears throughout topology: it describes intersections that persist under deformation. It also explains why "generic" maps in differential topology are transverse—most maps have this property, and the non-transverse ones form a measure-zero exception. Algebraic Intersection Number When two submanifolds intersect transversely, we can count their intersection points algebraically, assigning each a sign based on orientation. Suppose $X$ and $Y$ are oriented submanifolds that intersect transversely at isolated points. At each intersection point $p$, the orientations of $X$ and $Y$ induce an orientation on the direct sum $Tp X \oplus Tp Y$. This orientation either agrees with the orientation of $Tp M$ (giving sign $+1$) or opposes it (giving sign $-1$). The algebraic intersection number is the sum of these signs over all intersection points: $$X \cdot Y = \sum{p \in X \cap Y} \text{sign}(p)$$ This count is remarkably stable: if we continuously deform $X$ and $Y$ while maintaining transversality, the algebraic intersection number remains constant (though individual intersection points may move or pair up and cancel). This invariance is powerful because it means we can compute intersection numbers by deforming to simpler configurations. In many cases, intersection numbers can be computed purely combinatorially or algebraically without explicit knowledge of the submanifolds. Transversality in Action Transversality underlies many fundamental results. One striking application is the Brouwer fixed-point theorem: any continuous map from a closed ball to itself has a fixed point. The differential topology proof uses transversality: the graph of the map, a submanifold of the product space, is forced to intersect the diagonal (the graph of the identity) transversely by a dimension count argument, guaranteeing at least one intersection point—the fixed point. <extrainfo> More generally, transversality arguments show that maps from a manifold to itself generically have fixed points, and the number of fixed points (counted algebraically) is determined by the map's degree and the manifold's topology. </extrainfo> Part 4: Calculus-Based Invariants Degree of a Map Between Spheres The degree of a continuous map $f\colon S^n \to S^n$ is a fundamental invariant that captures how many times the domain sphere "wraps around" the target. For a smooth map, the degree can be defined using transversality. Pick any regular value $y \in S^n$ (a value that is not in the image of any critical point of $f$). Then $f^{-1}(y)$ is a finite set of points. At each point $p \in f^{-1}(y)$, the differential $dfp$ is an isomorphism (since $y$ is regular), and we assign $p$ a sign $\pm 1$ based on whether $dfp$ preserves or reverses orientation. The degree is the sum of these signs: $$\deg(f) = \sum{p \in f^{-1}(y)} \text{sign}(p)$$ This definition is independent of the choice of regular value $y$. Key properties of degree: The degree is an integer. $\deg(f) = 0$ means $f$ is not surjective (generically). $\deg(f) = 1$ means $f$ is a diffeomorphism (orientation-preserving). $\deg(f) = -1$ means $f$ is a diffeomorphism with orientation-reversing. The degree is invariant under homotopy: if $f$ and $g$ are homotopic, then $\deg(f) = \deg(g)$. The homotopy invariance is crucial: the degree measures something intrinsic about the map's topology, not its specific smooth structure. It is a powerful tool for distinguishing maps up to homotopy and for proving that certain maps cannot exist. For example, there is no continuous map $S^1 \to S^1$ of degree 0, because every map from the circle to itself winds around a nonzero number of times. Morse Theory: The Topology of Critical Points Morse theory is the study of smooth functions on manifolds through their critical points—places where the differential vanishes. For a smooth function $f\colon M \to \mathbb{R}$, a point $p$ is a critical point if $dfp = 0$ (as a linear functional on $Tp M$). The value $f(p)$ is a critical value. At a non-degenerate critical point, the Hessian (matrix of second derivatives) is invertible. The index of a non-degenerate critical point is the number of negative eigenvalues of the Hessian—it counts the number of directions in which $f$ is curving downward. For example, at a local minimum, all eigenvalues are positive (index 0). At a saddle point, some are positive and some negative. At a local maximum, all are negative (index $\dim M$). The Key Insight: Non-degenerate critical points reveal the topology of the manifold. As you increase the function value from $-\infty$ to $+\infty$, at each critical point of index $k$, the manifold gains a $k$-dimensional "handle." The number and indices of critical points determine how complicated the manifold's topology is. More precisely, there exist Morse functions (smooth functions with only non-degenerate critical points) on any manifold, and the Morse complex—a chain complex built from the critical points—has homology equal to the singular homology of the manifold. Morse Inequalities and Topological Information The Morse inequalities relate the critical points of a Morse function to the topological structure of the manifold: Let $ck$ denote the number of critical points of index $k$, and let $bk$ denote the $k$-th Betti number (the rank of the $k$-th homology group). Then: $$ck \geq bk$$ for each $k$, with equality when $cj = 0$ for $j \neq k$. This says: the number of critical points of index $k$ is at least the $k$-th Betti number. If a Morse function has the minimum possible number of critical points, then $ck = bk$ for all $k$, and the manifold's topology is completely determined by the critical point structure. Conversely, if a smooth function on a manifold has very few critical points, that manifold must have simple topology. A function with only one critical point (a non-degenerate minimum) exists only on contractible manifolds (spaces homotopy-equivalent to a point). This connection between calculus (critical points) and topology (homology) is one of the deepest ideas in differential topology. It shows that smooth functions contain topological information, and that topology can be "read off" from calculus. Summary Differential topology interweaves smooth geometry with algebraic topology through several powerful ideas: Manifolds and charts provide the setting where calculus makes sense on curved spaces. Smooth maps and their differentials allow us to translate geometric questions into linear algebra. Low-dimensional classification shows how completely smooth manifolds in dimensions 1 and 2 can be understood—a rarity in higher dimensions. Transversality captures the idea of generic intersection and provides stable, computable invariants. Degree, Morse theory, and critical points connect calculus to topology, showing that smooth functions encode topological information. These ideas together form a toolkit for understanding the shape of smooth manifolds and the maps between them.