2.4 Manifolds and Surfaces · Homology & Cohomology

…the specialisations passed over form a simply extended manifoldness, whose true character is that in it a continuous progress from a point is possible only on two sides, forwards or backwards.

— Bernhard Riemann, On the Hypotheses which lie at the Bases of Geometry (1854), translated by W. K. Clifford (1873)

Stand in a field and the Earth looks flat: as far as you can tell, the ground around you is a piece of a plane. Many early cultures pictured the world that way. Seeing past it took careful reasoning: Aristotle noticed that the shadow the Earth casts on the Moon during an eclipse is always round, and only a ball casts a round shadow from every angle. Each small patch of ground looks flat; the whole Earth curves round and closes up. Spaces like this, flat in every small piece while the whole may be curved, twisted or closed up in surprising ways, are called manifolds. Spheres, tori, Klein bottles, the space we live in and the spacetime of relativity are all manifolds, and they are the natural home of homology and cohomology.

Cartographers already know how to handle such a space: no single flat map shows the whole Earth, so they bind many into an atlas. Mathematicians do the same, and the atlas will carry us to edges, to tangent planes, and to the question of whether “clockwise” makes sense on a surface at all. The reward at the end is a complete list of every closed surface there can be. Every one of them, however tangled, is a sphere with handles added, or with Möbius bands sewn in.

Where this is going

Much of the rest of this book happens on manifolds. The boundary operator $\partial$ of homology (§3.2 Chains and the Boundary Operator) is a combinatorial shadow of the boundary of a manifold, and its basic law $\partial\partial = 0$ echoes this chapter’s slogan “a boundary has no boundary”. Orientation decides whether the top homology group of a closed surface is $Z$ or $0$ (§3.4 Computing Homology). Differential forms and de Rham cohomology (§4.3 Differential Forms, §4.4 de Rham Cohomology) live on the smooth manifolds and tangent spaces built here. Poincaré duality (§4.6 Poincaré Duality) is a symmetry that only closed manifolds possess. And the classification of surfaces gives us a complete catalogue of two-dimensional test cases, on which homology will rediscover the classification all by itself.

Spaces that look flat up close

Start with the flat spaces themselves. The line $R$ is the set of real numbers. The plane $R^{2}$ is the set of pairs $(x, y)$ of real numbers, and ordinary space $R^{3}$ is the set of triples $(x, y, z) .$ In general, $R^{n}$ (read “R n”) is the set of lists $(x_{1}, x_{2}, \dots, x_{n})$ of $n$ real numbers, with the usual distance between two lists, $(x_{1} - y_{1})^{2} + \dots + (x_{n} - y_{n})^{2} .$ We call it $n$ -dimensional Euclidean space, after Euclid, whose geometry it is. Nobody can picture $R^{4},$ but every question about it is a question about lists of four numbers, and that is enough to do mathematics there.

A manifold of dimension $n$ is a space in which every point has a small neighbourhood that looks like $R^{n}$ — not in size or shape, but up to homeomorphism (§2.1 Spaces and Continuity): a rubber-sheet copy. The surface of the Earth is a 2-dimensional manifold, because around each point there is a patch that can be flattened onto a piece of the plane without tearing. A circle is a 1-dimensional manifold: magnify any point and you see a short arc, which straightens out into an interval of the line.

“Small neighbourhood” sounds like a weakening, but in topology small pieces are as good as big ones. The open interval $(- 1, 1)$ and the whole line are homeomorphic: the map $t \mapsto t / (1 - ∣ t ∣)$ stretches the interval over the entire line, pushing the two ends off to infinity, and $s \mapsto s / (1 + ∣ s ∣)$ undoes it. In the same way $x \mapsto x / (1 - ∣ x ∣)$ is a homeomorphism from the open unit ball of $R^{n}$ onto all of $R^{n},$ where $∣ x ∣$ is the distance from $x$ to the origin. So “every point has a neighbourhood homeomorphic to $R^{n} ”,$ “… to an open ball in $R^{n} ”$ and “… to an open subset of $R^{n} ”$ all say the same thing.

The small-sphere test

Which spaces are not manifolds? Here is a test you can do by eye. Put a small sphere around a point $p$ and look at where it meets the space, like a radar picture of the neighbourhood of $p .$ If $p$ lies on a curve, the small sphere meets the curve in two points, one on each side — Riemann’s “forwards or backwards”. If $p$ lies in the middle of a surface, the sphere meets it in a closed loop: a circle’s worth of directions in which you can walk away from $p .$ Any other answer is a warning sign.

The small-sphere test Pick a space · click it to move the probe · drag the sphere’s rim to resize it

looks flat .

Sphere radius 0.42

Figure 2.4.1 A glass probe sphere around a point, and in teal the set where it meets the shape. Ordinary points of a curve give two points; ordinary points of a surface give one circle. The crossing of the figure eight, the tip of the double cone and the line where two planes cross give something else however small the sphere, so these spaces are not manifolds. The tip of a single cone passes: topologically it is an ordinary point. Points on the rim of the disk give an arc, the mark of a boundary point.

The small-sphere picture depends on how the space sits in $R^{3},$ so it is a guide rather than a proof. To prove that a point is special we use something homeomorphisms must preserve. The simplest such thing is the number of pieces a space falls into when you remove a point, which you met as cut points in §2.1 Spaces and Continuity.

Proof

Let $x$ be the crossing point of the figure eight, and suppose some neighbourhood $U$ of $x$ were homeomorphic to an open interval, by a homeomorphism $h .$ Near $x$ the figure eight looks like the letter X, so inside $U$ we can find a smaller open neighbourhood $V$ of $x$ shaped like an X: four short arms meeting at $x .$ Its image $h (V)$ is an open, connected piece of the interval containing $h (x),$ so it is itself an open interval. Now remove the middle points. $V$ minus $x$ falls into four pieces, the four arms; an open interval minus one of its points falls into two. But $h$ restricts to a homeomorphism between these two spaces, and homeomorphic spaces have the same number of pieces. Four is not two, so no such $h$ exists.

The double cone works the same way. A small neighbourhood of the tip, minus the tip, falls into two pieces, the upper and the lower cone. A connected open piece of the plane minus one point stays connected, because a path can always walk around the missing point. Two is not one.

For two planes crossing along a line the cut-point trick is too weak: remove a point of the crossing line and what is left is still in one piece. The small-sphere picture — two circles that cross, instead of one circle — is the right instinct, and a tool called local homology (§3.6 Exact Sequences and Mayer–Vietoris) turns that instinct into a proof.

The definition of a manifold

Here is the official definition. The first condition is the one we have been discussing; the other two are housekeeping, which rule out pathological spaces that nobody wants to call manifolds.

Condition 1 is the heart of the matter. Condition 2 rules out a strange creature that you met while gluing spaces in §2.2 Gluing Spaces Together.

The line with two origins

Figure 2.4.2 Take two copies of

R

and glue each point

x \neq = 0

of the first to the same point of the second, leaving the two zeros unglued. Each origin has neighbourhoods that are open intervals (

U_{1}

in gold,

U_{2}

in teal), so the space is locally Euclidean. But any neighbourhood of

0_{1}

meets any neighbourhood of

0_{2} :

they cannot be separated.

A little history Riemann’s manifoldness

Bernhard Riemann 1826 – 1866

Hermann Weyl 1885 – 1955

The word “manifold” comes from Bernhard Riemann’s Mannigfaltigkeit, in the lecture he gave at Göttingen on 10 June 1854 to qualify as a university teacher (Riemann 1854). Riemann, a Lutheran pastor’s son, had entered the university to study theology. For the lecture he offered three topics, and Carl Friedrich Gauss, who was in the audience, chose the one on the foundations of geometry, against Riemann’s expectations. The lecture was published only in 1868, two years after Riemann died of tuberculosis in Italy, aged thirty-nine. Hermann Weyl, a student of Hilbert, gave the first modern, intrinsic definition — for surfaces, by neighbourhoods and coordinate charts — in his 1913 book on Riemann surfaces (Weyl 1913), the year he moved to a chair in Zürich and spent a year as Einstein’s colleague.

Could a space be a 2-manifold and a 3-manifold at the same time? Intuition says no — a sheet of paper is not a block of wood — but remember that continuous maps can do wild things: Giuseppe Peano found in 1890 a continuous map from an interval onto a whole square (Peano 1890). The reassuring answer is a famous theorem.

L. E. J. Brouwer proved this in 1911 (Brouwer 1911). It is surprisingly hard to prove from scratch, and becomes easy with homology: §3.5 Maps, Invariance, and First Triumphs proves that $R^{m}$ and $R^{n}$ are not homeomorphic when $m \neq = n,$ and §3.6 Exact Sequences and Mayer–Vietoris finishes the job for open subsets.

Example Manifolds in every dimension

Dimension 0. Any set of isolated points: each point is its own neighbourhood, and $R^{0}$ is a single point.
Dimension 1. The line, any open interval, the circle $S^{1} .$ In fact every connected 1-manifold is homeomorphic either to $R$ or, if it is compact, to $S^{1} .$
Dimension 2. The plane, the open disk, the sphere $S^{2},$ the torus $T^{2},$ and the Klein bottle $K$ and the projective plane $RP^{2},$ which §2.2 Gluing Spaces Together built abstractly by gluing. The definition never asks a manifold to sit inside $R^{3},$ so the Klein bottle’s difficulties with ordinary space are no obstacle.
Dimension 3. Ordinary space $R^{3};$ the 3-sphere $S^{3},$ the points of $R^{4}$ at distance 1 from the origin; the 3-torus, a cube whose opposite faces are glued, so that leaving through one wall brings you back through the opposite one — the world of old video games, upgraded to three dimensions. And the space around a knot: remove a knotted string from $R^{3}$ and what is left is a 3-manifold, whose shape records the knot (§3.7 Knots and Links).
Dimension 4. The spacetime of general relativity: three dimensions of space and one of time.

Two constructions make new manifolds from old. An open subset of an $n$ -manifold is again an $n$ -manifold: around each of its points, a small enough chart neighbourhood fits inside it. And the product $M \times N$ of an $m$ -manifold and an $n$ -manifold is an $(m + n)$ -manifold, because a product of an open ball in $R^{m}$ with one in $R^{n}$ is an open subset of $R^{m + n} .$ The torus is the product $S^{1} \times S^{1} .$ Manifolds also arise as spaces of states, just as Riemann foresaw: the two angles of a double pendulum can each be anything on a circle, so the positions of the pendulum form a torus.

Charts, atlases and transition maps

A manifold is defined by its small neighbourhoods. To compute anything on it we need coordinates on those neighbourhoods, and cartography supplies the vocabulary. An atlas of the Earth uses many maps, each covering part of the globe, overlapping at their edges. On an overlap the same town appears on two pages with different coordinates, and a good atlas tells you how to convert from one to the other.

Read $φ$ (“phi”) as the map that prints one page of the atlas: it turns points of $M$ into coordinates. Its inverse $φ^{- 1}$ reads the page back, turning coordinates into points. Condition 1 of the definition of a manifold says precisely that $M$ has an atlas.

An atlas of the sphere Hover or tap the globe · click a map

point

p = (0.50, - 0.45, 0.74)

lies in 3 charts.

transition from

z > 0

x > 0

(u, v) \mapsto (v, 1 - u^{2} - v^{2})

, so

(0.50, - 0.45) \mapsto (- 0.45, 0.74)

Hover or tap the globe to move the point; click a map to make it the active chart.

Figure 2.4.3 The six hemisphere charts of the sphere. The gold point lies in two or three of them (the coloured caps), and each of those flat maps shows its coordinates. A point on the rim of a hemisphere is missing from that chart — every chart is an open set — but another chart always picks it up. The readout shows the transition map between two charts that both contain the point.

For example, take the northern chart ( $z > 0)$ and the eastern one ( $x > 0),$ which keeps $(y, z) .$ A point with northern coordinates $(u, v)$ is the point $(u, v, 1 - u^{2} - v^{2})$ of the sphere, and the eastern chart keeps its last two coordinates. So the transition map is

(u, v) ⟼ (v, 1 - u^{2} - v^{2}),

defined on the half-disk where $u > 0$ (the points that are in both hemispheres). Check it against Figure 2.4.3: the typical point has northern coordinates $(0.50, - 0.45),$ and $1 - 0.25 - 0.2025 \approx 0.74,$ so its eastern coordinates are $(- 0.45, 0.74),$ just as the eastern map shows. Being a composite of homeomorphisms, every transition map is a homeomorphism between open subsets of $R^{n} .$

The same argument shows that no compact manifold of positive dimension fits on a single page. Every flat map of the whole Earth must cheat somewhere; the Mercator projection, for instance, leaves out the poles.

Stereographic projection: two charts are enough

Six charts are more than we need. A classical construction, used since antiquity to draw maps of the stars, covers the sphere with two. Start one dimension down, with the circle $S^{1} = {(x, y) x^{2} + y^{2} = 1} .$ Stand at the north pole $N = (0, 1)$ and shine a light through a point $P$ of the circle. The ray hits the horizontal line $y = 0$ at a single point, and the position of that point is a number $φ_{N} (P) :$ the coordinate of $P$ as seen from the north pole.

Stereographic projection Drag P around the circle

P = (0.85, 0.53)

φ_{N} (P) = \frac{x}{1 - y} = 1.80

φ_{S} (P) = \frac{x}{1 + y} = 0.55

product:

φ_{N} (P) \cdot φ_{S} (P) = 1.00

, so on the overlap

φ_{S} = 1/ φ_{N}

Figure 2.4.4 Projection from the north pole (gold) and from the south pole (teal). Each is a chart that covers all of the circle except its own pole. As

P

approaches the north pole,

φ_{N} (P)

runs off to infinity, while

φ_{S} (P)

approaches

0 .

Wherever both are defined, the two coordinates multiply to

1 .

Notice what this says: a circle with one point removed is homeomorphic to a line. Conversely, adding a single point “at infinity” to the line closes it up into a circle.

The same construction works one dimension up. Projecting the sphere from its north pole $N = (0, 0, 1)$ onto the plane $z = 0$ gives

φ_{N} (x, y, z) = (\frac{x}{1 - z}, \frac{y}{1 - z}),

a homeomorphism from $S^{2} ∖ {N}$ onto the whole plane. Projecting from the south pole gives $φ_{S} (x, y, z) = (x / (1 + z), y / (1 + z)),$ and a short calculation, like the one for the circle, shows that the transition map is

(u, v) ⟼ \frac{( u , v )}{u ^{2} + v ^{2}},

called inversion in the unit circle: it turns the punctured plane inside out, swapping the inside of the unit circle with the outside and fixing the circle itself. In particular, a sphere with one point removed is homeomorphic to the plane. We used this in §2.3 Homotopy to show that every loop on the sphere shrinks to a point, and we will use it again in §4.8 Curvature and Characteristic Classes.

Edges: manifolds with boundary

The closed disk $D^{2} = {(x, y) x^{2} + y^{2} \leq 1},$ the cylinder $S^{1} \times [0, 1]$ and the Möbius band are not manifolds in the sense above. At a point of the rim, no neighbourhood looks like the plane; the small-sphere test returns an arc instead of a circle. But they fail in a gentle and useful way: near a rim point they look like a half-plane. We enlarge the definition to allow this.

The $n$ -dimensional half-space is $H^{n} = {(x_{1}, \dots, x_{n}) \in R^{n} ∣ x_{n} \geq 0},$ the points on or above the hyperplane $x_{n} = 0 .$ That hyperplane is its edge, written $\partial H^{n} .$ For $n = 2,$ $H^{2}$ is the upper half-plane and its edge is the horizontal axis.

Interior and boundary points

Figure 2.4.5 In the closed disk, an interior point

p

has a neighbourhood that a chart

φ

flattens onto an open disk inside the half-plane, away from its edge. A boundary point

q

only has neighbourhoods shaped like half-disks, and a chart

ψ

sends

q

to the edge of the half-plane. The boundary of the disk is the circle:

\partial D^{2} = S^{1} .

Could one point be a boundary point for one chart and an interior point for another? No, but the proof needs the same tool as invariance of dimension, local homology (§3.6 Exact Sequences and Mayer–Vietoris): it detects a hole around an interior point that is missing around a point on the edge of $H^{n} .$ Granting this, a manifold in the earlier sense is exactly a manifold with boundary whose boundary is empty. Some examples:

the interval $[0, 1]$ is a 1-manifold with boundary $\partial [0, 1] = {0, 1},$ two points;
the closed disk has boundary $\partial D^{2} = S^{1},$ and the solid ball $D^{3}$ has boundary $\partial D^{3} = S^{2};$
the cylinder $S^{1} \times [0, 1]$ has two boundary circles, its top and bottom;
the Möbius band has one boundary circle: run your finger along its edge and you go twice around the band before coming back (§2.2 Gluing Spaces Together).

Smooth manifolds and tangent spaces

Topology cannot see corners: the surface of a cube and the round sphere are the same topological manifold. Calculus can. To speak of velocities, derivatives and integrals on a manifold we must know which functions on it are smooth, and the definition of a topological manifold does not tell us.

The natural idea is to declare a function $f : M \to R$ smooth if it is smooth in coordinates, that is, if $f \circ φ^{- 1}$ is smooth for every chart $φ .$ Here “smooth” means infinitely differentiable: all partial derivatives of all orders exist and are continuous. But there is a catch. If $f$ looks smooth on one page of the atlas, does it look smooth on an overlapping page $ψ ?$ On the second page,

f \circ ψ^{- 1} = (f \circ φ^{- 1}) \circ (φ \circ ψ^{- 1}),

and a composite of smooth maps is smooth (that is the chain rule). So the answer is yes, provided the transition map $φ \circ ψ^{- 1}$ is itself smooth. That is the condition we impose.

Most smooth manifolds you will meet arrive as solution sets of equations, and there is a convenient test for those.

This is a form of the implicit function theorem from calculus. Near each point of the level set, the equation $F = c$ can be solved for one of the three coordinates as a smooth function of the other two, which presents the surface locally as a graph — and a graph is a chart, as the northern hemisphere was. The sphere is the level set $x^{2} + y^{2} + z^{2} = 1,$ and $\nabla F = (2 x, 2 y, 2 z)$ is nonzero on it. The torus is the level set $(x^{2} + y^{2} - R)^{2} + z^{2} = r^{2},$ with $0 < r < R :$ the points at distance $r$ from a circle of radius $R .$ The pretzel surfaces in the gallery at the end of this chapter were drawn by the computer in exactly this way, as the level set $F = 0$ of a function that blends several tori together.

Tangent spaces: the microscope view

Zoom in on a point of a smooth surface with a powerful microscope, and the surface looks more and more like a flat plane: the tangent plane at that point. Unlike the neighbourhoods of a topological manifold, which are flat only up to rubber-sheet deformation, the tangent plane is flat in the honest, linear sense: it is a vector space, in which arrows can be added and scaled (§1.5 Linear Algebra).

The cleanest way to define it is through motion. Let $M$ be a smooth surface in $R^{3}$ and $p$ a point of $M .$ A smooth curve in $M$ through $p$ is a smooth map $γ$ from a small interval $(- ε, ε)$ to $R^{3}$ with $γ (t) \in M$ for every $t$ and $γ (0) = p :$ a particle moving on the surface, passing through $p$ at time $0 .$ Its velocity at that moment is the vector $γ^{'} (0)$ of the derivatives of its three coordinates.

Why is $T_{p} M$ a plane? Use a chart. Near $p,$ write the surface as $X (u, v),$ a smooth parametrization with $X (u_{0}, v_{0}) = p$ (the inverse of a chart, made smooth). A curve in the surface is $γ (t) = X (u (t), v (t)),$ and by the chain rule its velocity at time $0$ is

γ^{'} (0) = u^{'} (0) \frac{\partial X}{\partial u} + v^{'} (0) \frac{\partial X}{\partial v} .

So every velocity is a combination of the two coordinate velocities $\partial X / \partial u$ and $\partial X / \partial v,$ and every combination occurs (take $u (t) = u_{0} + a t$ and $v (t) = v_{0} + b t).$ Hence $T_{p} M$ is the plane spanned by these two vectors, which are linearly independent because $X$ is a genuine parametrization.

Tangent planes Drag in the tangent plane to turn the velocity · click the surface to move the point · try the microscope

unit radius

p^= (0.00, 0.00, 0.00)

velocity

v = (0.00, 0.00, 0.00)

check:

p^\cdot v = 0.00

— every velocity at

p

is perpendicular to the radius.

Direction 35°

Microscope

Figure 2.4.6 A curve (teal) through a point

p

of a sphere or a torus, and its velocity arrow, which always lies in the glassy tangent plane. Turn the direction of the curve and the arrow sweeps out the whole plane; on the sphere the readout checks that each velocity is perpendicular to the radius, and on the torus it writes the velocity as a combination of the two coordinate velocities. The microscope zooms in until the surface and its tangent plane can no longer be told apart.

Remark Tangent vectors without a surrounding space

Our definition used the surrounding $R^{3} .$ For an abstract manifold, tangent vectors are defined intrinsically — for instance as velocities of curves written in a chart, with the velocities in two charts identified by the derivative of the transition map. All the standard definitions give an $n$ -dimensional vector space $T_{p} M$ at each point of an $n$ -manifold. A smooth map $f : M \to N$ carries curves through $p$ to curves through $f (p),$ and so it carries velocities to velocities. The result is a linear map $d f_{p} : T_{p} M \to T_{f (p)} N,$ the differential of $f$ at $p :$ the best linear approximation to $f$ near $p .$ It pushes vectors forward, which is why it is also written $f_{*},$ with a lower star. Together, all the tangent spaces form the tangent bundle $T M,$ which returns in §4.8 Curvature and Characteristic Classes; and differential forms (§4.3 Differential Forms) are devices that measure tangent vectors.

Orientation: which way is anticlockwise?

Draw a small circle on a sheet of paper, with an arrow running anticlockwise. You can draw the same arrow at every point of the sheet, consistently. Can you do the same on every surface? To answer, we first need to say precisely what a “sense of rotation” is, and linear algebra does it with a single number: the determinant.

An ordered basis of $R^{2}$ is a pair $(u, v)$ of vectors that do not lie on a common line. Turning $u$ towards $v$ the short way is either anticlockwise or clockwise, and the determinant tells you which:

det (u, v) = u_{1} v_{2} - u_{2} v_{1}

is positive for anticlockwise pairs and negative for clockwise ones. Its absolute value is the area of the parallelogram spanned by $u$ and $v .$

Orienting the plane Drag the tips of u and v

det (u, v) = (2) (1.75) - (0.5) (0.75) = 3.13

Turning from u to v is anticlockwise: a positive basis.

Figure 2.4.7 The parallelogram is green when the pair

(u, v)

is positive (turning from

u

v

is anticlockwise) and rose when it is negative. Swapping the two vectors, reversing one of them, or reflecting the whole picture in a line flips the sign of the determinant; rotating the pair does not.

In $R^{2}$ the two classes are “anticlockwise” and “clockwise”; in $R^{3}$ they are “right-handed” and “left-handed”. Why is this an equivalence relation? A basis is related to itself by the identity matrix, of determinant $1 > 0 .$ If $A$ changes the first basis into the second, then $A^{- 1}$ changes the second into the first, and $det A^{- 1} = 1/ det A$ has the same sign. If $A$ and then $B$ are changes of basis, the composite $B A$ has $det (B A) = det B \cdot det A,$ positive when both are. Orientation is yet another quotient: all the bases of a space, with everything forgotten except a sign.

Orienting a surface

An orientation of a surface is a choice of orientation of each tangent plane — a choice of “anticlockwise” at every point — that varies continuously as the point moves. For a topological surface, which has no tangent planes, one instead chooses a sense of rotation on each small disk, consistently where the disks overlap. For a manifold given by an atlas, the same idea becomes an oriented atlas: one whose transition maps all have positive Jacobian determinant (the determinant of their matrix of partial derivatives), so that “anticlockwise” on one page of the atlas means “anticlockwise” on every other (Lee 2013, Ch. 15).

The plane is orientable: use the same “anticlockwise” everywhere. So is the sphere: at each point, call a rotation positive if it looks anticlockwise to someone standing outside the sphere. The surprise is that not every surface is orientable.

A walk round a band Choose a band · play the walk, or drag the F along the band · drag elsewhere to turn

At the start, just ahead of its faint ghost. Play the walk, or drag the F along the band.

start two laps

Figure 2.4.8 A flat creature — the letter F with a turning arrow — walks once around the middle of a band, while a faint ghost stays at the start. On the cylinder it comes home unchanged. On the Möbius band it comes home as its own mirror image: the F is reversed and its arrow turns the other way. A second lap undoes the reflection.

Proof

Suppose we had chosen an orientation, and let the creature walk once around the central circle, carrying its arrow. At the start, its arrow agrees with the chosen orientation. As it walks, its arrow and the orientation at its current position both vary continuously, and there are only two possible orientations at each point, so they cannot drift apart without a jump. (Precisely: the set of moments at which the two agree is both open and closed in the time interval, so by connectedness it is the whole interval.) Hence they still agree when the creature arrives home. But the creature arrives home mirror-reversed, with its arrow turning the opposite way from the arrow it set off with — and both arrows are supposed to agree with the single orientation chosen at the starting point. This is impossible, so no orientation exists.

A loop along which a traveller comes home mirror-reversed is called an orientation-reversing loop. A thin strip around such a loop is a Möbius band. Conversely, on a connected surface on which no loop reverses orientation, choose an orientation at one point and carry it along paths to every other point; two paths to the same point together form a loop, which does not reverse orientation, so the result does not depend on the path, and we have built an orientation. This gives a practical test.

Reading orientability from a gluing word

For surfaces made from a single polygon by an edge word (§2.2 Gluing Spaces Together), orientability can be read off the word. Orient the polygon anticlockwise. If a letter appears once as $a$ and once as $a^{- 1},$ the two edges are glued the way neighbouring floor tiles meet, and a creature crossing from one to the other keeps its orientation. If a letter appears twice the same way round, as $\dots a \dots a \dots,$ then the strip of the polygon joining the two edges becomes a Möbius band when they are glued.

So the torus $ab a^{- 1} b^{- 1},$ the sphere $a a^{- 1}$ and the surfaces $Σ_{g}$ with words $a_{1} b_{1} a_{1}^{- 1} b_{1}^{- 1} \dots a_{g} b_{g} a_{g}^{- 1} b_{g}^{- 1}$ are orientable, while the Klein bottle $aba b^{- 1}$ (the letter $a$ appears twice as $a)$ and the projective plane $aa$ are not.

Key idea The torus and the Klein bottle are different

Orientability is a topological property: a homeomorphism carries Möbius bands to Möbius bands. The Klein bottle contains a Möbius band and the torus does not, so they are not homeomorphic — even though both are made from one square, both have Euler characteristic $0$ (§2.6 The Euler Characteristic) and, as §3.3 Homology Groups will show, both have the same homology mod 2. That the torus contains no Möbius band anywhere, and not just none that we have spotted, is guaranteed by its orientation: the anticlockwise arrows drawn on the square survive the gluing $ab a^{- 1} b^{- 1} .$ A second proof comes from homology over the integers in §3.4 Computing Homology, where the top homology group of the torus is $Z$ and that of the Klein bottle is $0 .$

One-sided surfaces, and the space around them

In §2.2 Gluing Spaces Together the Möbius band was called one-sided: an ant walking along it reaches the back of its starting point without crossing the edge. Sidedness is a property of a surface sitting in a surrounding space, which supplies the “back”; orientability belongs to the surface alone — the creature in Figure 2.4.8 never leaves the band. For surfaces in ordinary space the two notions agree: a surface in $R^{3}$ is two-sided exactly when it is orientable. And closed surfaces in space have no choice.

Its one-dimension-lower cousin, the Jordan curve theorem, appears in §3.5 Maps, Invariance, and First Triumphs, and both can be proved with homology (Hatcher 2002, Cor. 3.45–3.46). The consequence is striking: the Klein bottle and the projective plane cannot be built in ordinary space without passing through themselves. Every glass Klein bottle you have seen is an immersion, with a circle along which the surface crosses itself, and Boy’s surface is an immersion of the projective plane. With a fourth dimension to move in, the crossings can be avoided: the Klein bottle embeds in $R^{4} .$ Hassler Whitney proved in 1944 that every smooth $n$ -manifold can be embedded in $R^{2 n}$ — so every surface fits in $R^{4}$ — and immersed in $R^{2 n - 1}$ when $n \geq 2$ (Whitney 1944a; Whitney 1944b). The definition of a manifold never needed a surrounding space; Whitney’s theorem says that one can always be found.

Connected sums: building surfaces like Lego

How many different closed surfaces are there? To find out, we need a way of building new surfaces from old ones.

Connected sum Step through · drag to rotate

Two tori

Figure 2.4.9 The connected sum of two tori in four steps: cut a small disk out of each (the gold circles), join the two holes by a tube, and smooth the result. A tube is only a thickened circle, so joining by a tube is the same as gluing the circles directly. The connected sum

T^{2} # T^{2}

is the surface of genus two.

It takes some work to show that the result does not depend on where the disks are cut or how the circles are glued, so that $M # N$ is well defined up to homeomorphism. Granting that, connected sum behaves very much like addition.

It is commutative and associative: $M # N ≅ N # M$ and $(L # M) # N ≅ L # (M # N) .$
The sphere is its zero: $S^{2} # M ≅ M .$ Removing a disk from a sphere leaves a disk, and gluing a disk into the hole in $M$ just fills the hole back in.
Adding a torus adds a handle: $M # T^{2}$ is $M$ with a handle attached. In particular $Σ_{g} = T^{2} # \dots # T^{2},$ with $g$ tori, is the surface of genus $g$ .
Adding a projective plane adds a cross-cap. The projective plane with a disk removed is a Möbius band, so $M # RP^{2}$ is $M$ with a disk cut out and a Möbius band sewn into the hole along its single edge.

Dyck’s theorem means that mixing handles and cross-caps never produces anything new: any mixture with at least one cross-cap is a connected sum of projective planes alone.

The classification of closed surfaces

We can now state the theorem promised in §2.2 Gluing Spaces Together. It says that the closed surfaces we already know are all the closed surfaces there are.

The number $g$ of handles is the genus of an orientable surface; the sphere has genus $0 .$ For $N_{k},$ the number $k$ of cross-caps is sometimes called its non-orientable genus.

Closed surfaces Choose a surface · drag to rotate

Σ_{2} = T^{2} # T^{2}

orientable: yes
genus: 2
Euler characteristic: $χ = - 2$
gluing word: $a_{1} b_{1} a_{1}^{- 1} b_{1}^{- 1} a_{2} b_{2} a_{2}^{- 1} b_{2}^{- 1}$

Genus 2: two handles, the connected sum of two tori.

Figure 2.4.10 The classification as a gallery. The orientable surfaces are spheres with

g

handles; each handle carries a gold loop around its tube and a teal loop around its hole. The non-orientable surfaces cannot be built in

R^{3}

without crossing themselves, so they are drawn as immersions: the projective plane as Boy’s surface and the Klein bottle as the classic bottle.

Surface	Orientable	$χ$	Edge word	$H_{1}$
sphere $S^{2}$	yes	$2$	$a a^{- 1}$	$0$
torus $T^{2} = Σ_{1}$	yes	$0$	$ab a^{- 1} b^{- 1}$	$Z^{2}$
genus $g,$ $Σ_{g}$	yes	$2 - 2 g$	$a_{1} b_{1} a_{1}^{- 1} b_{1}^{- 1} \dots$	$Z^{2 g}$
projective plane $RP^{2} = N_{1}$	no	$1$	$aa$	$Z /2$
Klein bottle $K = N_{2}$	no	$0$	$aabb$	$Z \oplus Z /2$
$k$ cross-caps, $N_{k}$	no	$2 - k$	$a_{1} a_{1} \dots a_{k} a_{k}$	$Z^{k - 1} \oplus Z /2$

Why “exactly one”? Two different surfaces in the list are never homeomorphic: orientability separates the two families, and within each family the Euler characteristic, $2 - 2 g$ or $2 - k,$ pins down the number of handles or cross-caps (§2.6 The Euler Characteristic). The last column of the table, computed with homology in §3.4 Computing Homology, separates them too: homology alone recovers the whole classification.

Remark How the proof goes

Triangulate. Cut the surface into triangles that meet edge to edge. That every surface can be triangulated was proved by Tibor Radó in 1925; it is the hard, analytic step. For surfaces given by gluing diagrams it comes for free.
Make one polygon. Glue the triangles together one at a time, each along a single edge, so that the growing piece stays a disk, until every triangle is used. The surface is now one polygon whose edges are glued in pairs: an edge word.
Simplify the word. Cut-and-paste moves change the polygon without changing the surface: cancel a pair $a a^{- 1}$ that sits side by side (a fold), bring the two letters of a pair $\dots a \dots a \dots$ together into a cross-cap $aa,$ gather interlocked pairs into handles $ab a^{- 1} b^{- 1},$ and use Dyck’s theorem to turn handles into cross-caps when a cross-cap is present.
Read off the answer. The final word is one of the normal forms in the theorem.

John Conway found a slicker argument, the “ZIP proof”, in which a surface is assembled from disks joined by zips. He asked that it always be called by that name, “otherwise there’s a real danger that its origin would be lost, since everyone who hears it immediately regards it as the obvious proof” (Francis & Weeks 1999). The write-up by Francis and Weeks is short and full of pictures; Gallier & Xu (2013) give the classical proof in full.

A little history From Möbius to Conway

Walther von Dyck 1856 – 1934

Surfaces were classified long before the definition was settled. In the 1860s August Möbius and Camille Jordan independently classified the closed orientable surfaces; Walther von Dyck added the non-orientable ones in 1888, proving the theorem that bears his name along the way (Dyck 1888). Dyck, a painter’s son and once Klein’s assistant, had already been the first to study groups systematically by generators and relations, and later helped found the Deutsches Museum in Munich. None of these classifiers had a definition of “surface” to work with. Max Dehn and Poul Heegaard gave a far more careful proof in 1907, Henry Brahana the first fully rigorous one in 1921, for surfaces built from polygons, and Radó’s triangulation theorem closed the last gap in 1925 (Gallier & Xu 2013). Around 1992 John Conway found the ZIP proof — the name stands for “zero irrelevancy proof” (Francis & Weeks 1999).

Higher dimensions

Closed curves are easy: every connected one is a circle. Closed surfaces are decided by two pieces of data. In dimension three the story is far harder, and the first question anyone asked about it became the most famous problem in topology.

Conjecture The Poincaré conjecture

Proved in 2003 Posed by Henri Poincaré, 1904; proved by Grigori Perelman, 2003

A closed 3-dimensional manifold in which every loop can be shrunk to a point is homeomorphic to the 3-sphere $S^{3} .$

A loop that can be shrunk to a point is the subject of §2.3 Homotopy, where a space with this property is called simply connected. Among closed surfaces, only the sphere is simply connected; the conjecture says the same is true one dimension up. Perelman’s proof is part of William Thurston’s programme for describing all 3-manifolds by geometry (Perelman 2003).

How it was settled. In three papers posted in 2002 and 2003, Perelman completed Richard Hamilton’s programme of Ricci flow: let the shape of the space evolve so that its curvature evens out, and cut it open carefully where it pinches. His work proves Thurston’s geometrization conjecture, which contains Poincaré’s. He declined the Fields Medal in 2006 and the Millennium Prize in 2010.

Before and after. Poincaré first asked the question with homology in place of loops, then built a space with the homology of $S^{3}$ whose fundamental group has $120$ elements: homology alone is not enough. The analogue in higher dimensions was proved earlier, by Smale (1961) from dimension $5$ on and by Freedman (1982) in dimension $4 .$

(Poincaré 1904; Perelman 2002; Morgan & Tian 2007) Status as of October 2026 All open problems

In dimension four no complete list is possible, in a precise sense: in 1958 A. A. Markov proved that no algorithm can decide whether two given closed 4-manifolds are homeomorphic (Markov 1958). And dimension four keeps one version of Poincaré’s question open, a version that asks for more than a homeomorphism.

Conjecture The smooth Poincaré conjecture in dimension 4

Open since the 1960s Asked since the 1960s

Every 4-dimensional manifold that is homotopy equivalent to the 4-sphere $S^{4}$ is the 4-sphere not only up to homeomorphism but up to smooth deformation: $S^{4}$ has no exotic smooth structure.

A smooth structure is what lets you do calculus on a manifold: an atlas whose transition maps are smooth. Two smooth manifolds can be homeomorphic without being smoothly the same, and in dimension four this happens in unexpected ways.

What is known. Freedman proved the version with homeomorphisms in 1982. In every other dimension it is known which spheres carry exotic smooth structures; Milnor found the first, in dimension $7,$ in 1956. Dimension $4$ is the only one where nobody knows, and many proposed exotic 4-spheres have turned out, after years of work, to be the ordinary one.

(Freedman 1982; Milnor 1956; Freedman, Gompf, Morrison & Walker 2010) Status as of October 2026 All open problems

Algebraic topology was invented to bring order to this zoo, and homology is its first and most useful tool.

Exercises

Exercise Orienting the hemisphere atlas

Take the northern chart $(x, y, z) \mapsto (x, y)$ on $z > 0,$ the eastern chart $(x, y, z) \mapsto (y, z)$ on $x > 0,$ and the chart $(x, y, z) \mapsto (x, z)$ on $y > 0 .$ Compute the Jacobian determinants of the transition maps from the northern chart to the other two. Is the six-chart atlas oriented as it stands? If not, how can you repair it?

Hint

Write $w = 1 - u^{2} - v^{2} .$ The transition maps are $(u, v) \mapsto (v, w)$ and $(u, v) \mapsto (u, w),$ and $\partial w / \partial u = - u / w,$ $\partial w / \partial v = - v / w .$

Solution

For $(u, v) \mapsto (v, w)$ the Jacobian matrix has rows $(0, 1)$ and $(- u / w, - v / w),$ with determinant $u / w,$ positive on the overlap, where $u = x > 0 .$ For $(u, v) \mapsto (u, w)$ the rows are $(1, 0)$ and $(- u / w, - v / w),$ with determinant $- v / w,$ negative on the overlap, where $v = y > 0 .$ So the atlas is not oriented as it stands. Repair it by listing each chart’s two coordinates in an order that looks anticlockwise from outside the sphere: keep $(x, y)$ on $z > 0,$ $(y, z)$ on $x > 0$ and $(x, z)$ on $y < 0,$ but use $(z, x)$ on $y > 0,$ $(y, x)$ on $z < 0$ and $(z, y)$ on $x < 0 .$ Swapping two coordinates is a reflection, which changes the sign of every Jacobian determinant involving that chart. With these choices all transition maps have positive Jacobian determinant, which shows again that the sphere is orientable.

Summary

Recap

An $n$ -manifold is a Hausdorff, second countable space in which every point has a neighbourhood homeomorphic to $R^{n}$ — or, equivalently, to an open ball or an open subset of $R^{n} .$ Its dimension is well defined (invariance of dimension).
Points that are not manifold points, such as the crossing of a figure eight or the tip of a double cone, can be detected by removing them and counting pieces. Corners are invisible to topology.
A chart $(U, φ)$ gives coordinates on an open set; an atlas covers the manifold; transition maps $ψ \circ φ^{- 1}$ convert coordinates. The sphere needs at least two charts, and stereographic projection provides two, with transition map $t \mapsto 1/ t$ for the circle and $w \mapsto w / ∣ w ∣^{2}$ for the sphere.
Manifolds with boundary are modelled on the half-space $H^{n} .$ A boundary has no boundary: $\partial (\partial M) = \emptyset .$ A closed manifold is compact with empty boundary.
A smooth manifold has smooth transition maps; level sets with nonzero gradient are smooth. The tangent space $T_{p} M$ is the vector space of velocities of curves through $p,$ and a smooth map pushes tangent vectors forward by its differential.
An orientation of a vector space is a class of ordered bases up to positive determinant; an orientation of a manifold is a continuous choice of these. A surface is non-orientable if and only if it contains a Möbius band; for one-polygon words, if and only if some letter appears twice with the same exponent. Closed surfaces in $R^{3}$ are orientable, so the Klein bottle and $RP^{2}$ only immerse there; they embed in $R^{4} .$
Connected sum: $S^{2}$ is the zero, $T^{2}$ adds a handle, $RP^{2}$ adds a cross-cap; $RP^{2} # RP^{2} ≅ K$ and $T^{2} # RP^{2} ≅ RP^{2} # RP^{2} # RP^{2} .$
Classification: every closed connected surface is exactly one of $S^{2},$ $Σ_{g}$ or $N_{k},$ told apart by orientability and Euler characteristic — and, as we will see, by homology.

Manifolds and Surfaces

Spaces that look flat up close

The small-sphere test

The definition of a manifold

Charts, atlases and transition maps

Stereographic projection: two charts are enough

Edges: manifolds with boundary

Smooth manifolds and tangent spaces

Tangent spaces: the microscope view

Orientation: which way is anticlockwise?

Orienting a surface

Reading orientability from a gluing word

One-sided surfaces, and the space around them

Connected sums: building surfaces like Lego

The classification of closed surfaces

Higher dimensions

Exercises

Summary

Further reading

References