4.7 Sheaves and Čech Cohomology · Homology & Cohomology

These notions were poorly received in America when they were published. It was too difficult. Mathematical Reviews asked, ‘What can this be used for?’ Henri Cartan and Jean-Pierre Serre showed what it is used for!

— Jean Leray, in M. Schmidt, Hommes de science (1990), p. 166, as quoted by Haynes Miller (2000); translated from the French

An atlas of the Earth is a book of flat maps. Every page is perfectly trustworthy, and where two pages overlap they agree. Yet no single flat page can show the whole globe without tearing or stretching it. Local truth, global impossibility.

This tension between local and global has followed us through the whole of Part IV. In §4.1 Cochains: Measuring Shapes an impossible staircase was a set of step heights that looked fine around every corner but could not come from any actual height. In §4.4 de Rham Cohomology the angle around the origin was a perfectly good function near every point of the punctured plane but not on the plane as a whole. Both times, cohomology measured exactly how the local pieces failed to fit together. Now the theme becomes a definition: describe a space by how it is covered by small pieces, describe data by how it lives on those pieces, and decide — from the overlaps alone — whether the pieces can be glued.

Where this is going

One question runs through the chapter: when do local pieces of data fit together into one global piece? Three tools answer it. The nerve of a cover turns a pattern of overlapping pieces into a simplicial complex with the same holes; it lets sensors that do not know where they are find a gap in their coverage, and it is how persistent homology (§3.8 Persistent Homology) builds shapes from data. Čech cohomology computes the cohomology groups you already know from the overlaps alone. Sheaves let the data change from place to place — heights, square roots, orientations, depths, prices — so that one kind of cohomology class can say exactly why each of them fails to glue. In the next chapter (§4.8 Curvature and Characteristic Classes) the same Čech cocycles, now with values in groups such as ${\pm 1}$ and the circle of phases, classify twisted families of vector spaces — from the Möbius band to the magnetic monopole.

Local pieces, global questions

Imagine a circular ridge trail around a crater lake, and three hikers who split it between them. Each carries an altimeter that has not been calibrated: it measures heights correctly, but from its own arbitrary zero. So each hiker can draw a perfect height profile of her own stretch of trail, but her numbers mean “metres above my zero”.

Their stretches overlap a little, so on each shared bit of trail two hikers can compare readings. “On our common stretch your numbers are always 40 higher than mine.” Since both altimeters measure differences correctly, the offset between two readings is the same all along a shared stretch: it is a single number per overlap. Now the question:

Try it with numbers. Walk once round the ring, and each time you pass from one hiker’s stretch into the next, note how much higher the new hiker’s numbers run: say $+ 40,$ then $+ 25,$ then $- 65 .$ These add up to zero, and that settles it. Fix the first hiker’s zero, and the other two are forced: the second must move her zero up by $40$ metres, the third by $65 .$ Now suppose the last offset had been $- 60 .$ Stitch the profiles together as you walk, and by the time you are back where you started, the stitched profile disagrees with itself by $5$ metres: the trail has become an Escher staircase, climbing for ever and yet returning to its start. No shifting of zeros can help. Moving one hiker’s zero raises one of her two offsets and lowers the other by the same amount, so the sum around the ring never changes. You met exactly this test in §4.1 Cochains: Measuring Shapes: a labelling of the edges of a loop is a gradient exactly when its sum around the loop is zero.

The example has three ingredients, and each will get its own section.

A space covered by overlapping pieces — the ring, covered by three stretches of trail. This is an open cover, and the pattern of overlaps is recorded by its nerve.
Data on the pieces, with a rule for comparing data on overlaps — height profiles that can be cut down to smaller stretches. This is a sheaf.
Bookkeeping that decides whether local data assemble into global data — the offsets and their sum around the ring. This is Čech cohomology.

Covers and their nerves

Open covers

Recall from §2.1 Spaces and Continuity that the open sets of a space are the sets that contain a little elbow room around each of their points: an open interval of the line, an open disk in the plane, an open arc of a circle (an arc without its two endpoints). Open sets are the natural “pieces” of a space because they have no hard edges; two of them can overlap in a set that is again open.

Some covers we will use again and again:

The circle covered by three open arcs, each a bit more than a third of the circle, so that neighbouring arcs overlap in a short arc and no point lies in all three.
The sphere covered by two caps: everything except the south pole, and everything except the north pole. They overlap in a wide band around the equator.
A region of the plane covered by disks — think of the ranges of radio sensors scattered over a field.
A manifold covered by the domains of its charts (§2.4 Manifolds and Surfaces): an atlas is an open cover with a coordinate system on each piece.

The nerve

A cover is a lot of information. Here is a way to throw almost all of it away and keep only the pattern of overlaps. Draw one dot for each piece. Join two dots by an edge if the two pieces overlap. Fill in a triangle whenever three pieces have a point in common, a solid tetrahedron whenever four do, and so on.

Why is this a simplicial complex (§2.5 Simplicial Complexes)? The one rule to check is that every face of a simplex is again a simplex. If some point lies in all of $U_{i_{0}}, \dots, U_{i_{p}},$ it certainly lies in any smaller selection of them; so every subset of a simplex spans a simplex. That is all.

A cover and its nerve Drag the disks · change their range · add or remove sensors

Nerve 7 vertices · 7 edges · 0 triangles

Its homology

b_{0} = 1, b_{1} = 1

Holes in the covered region 1

The nerve’s loops that bound nothing match the holes one for one — the nerve theorem at work.

Sensor range 62 Disks Nerve

Figure 4.7.1 Each disk is a piece of the cover. The nerve is drawn on top: a gold dot for each disk, an edge for each overlap, and a shaded triangle for each triple overlap. Rose regions are holes in the covered area. Move the disks: every time a hole opens or closes,

b_{1}

of the nerve changes with it.

Play with the figure. Pull one disk out of the ring and the hole opens out to the outside: the cycle of edges breaks and $b_{1}$ drops to zero. Push the disks together until three of them share a point and a gold triangle appears, filling in a little loop of the nerve before it can be counted as a hole. In the readout, the number of holes is computed directly from the shape of the covered region, and $b_{1}$ from the nerve alone. They always agree. That is no accident.

The nerve theorem

For the agreement to be guaranteed, the pieces and their overlaps must themselves have no holes. Recall from §2.3 Homotopy that a space is contractible if it can be shrunk continuously to a point: disks, intervals, arcs and convex sets are contractible; circles, annuli and spheres are not.

“Reasonable” has a precise meaning, paracompact, and Hatcher (2002, Cor. 4G.3) proves the theorem in that generality. We will not prove it here; versions of it go back to Jean Leray in the 1940s and to Karol Borsuk (Borsuk 1948). But the idea is easy to believe. If every piece and every overlap is a featureless blob, then the only way the space can have a hole is for the pieces to be arranged around it — and the arrangement is precisely what the nerve records. You already met the theorem in §3.8 Persistent Homology: the Čech complex of a point cloud is the nerve of the disks around the points, and disks are convex, so every intersection of disks is convex and therefore contractible.

Holes in sensor networks

Scatter cheap radio sensors over a field. Each one can detect intruders within some range, a disk around itself, and can talk to the sensors whose disks overlap its own. Nobody knows where the sensors are. Can the network decide, from who can talk to whom, whether there is a hole in its coverage — a pocket where an intruder could hide?

The nerve theorem says that the information needed is exactly the nerve: record which pairs of disks overlap and which triples share a point, compute $b_{1}$ of the resulting simplicial complex, and you know how many holes the covered region has — without knowing a single position. In practice sensors cannot easily tell whether three of them share a point; they only know pairs. Vin de Silva and Robert Ghrist showed how to certify coverage of a whole region from pairwise information alone, using the homology of a related complex (the Vietoris–Rips complex of §3.8 Persistent Homology) together with a little extra knowledge about the sensors along the border of the region (de Silva & Ghrist 2007).

Čech cohomology: bookkeeping on overlaps

The nerve keeps the shape of a cover. Now we attach numbers to it, exactly as cochains attached numbers to the simplices of a complex in §4.1 Cochains: Measuring Shapes. The new point of view is that the numbers live on the pieces of the space and on their overlaps.

Back to the hikers

Cover the ring trail $X = S^{1}$ by three open arcs $U_{0}, U_{1}, U_{2},$ the stretches of the three hikers. Fix a group of values $G$ — the real numbers $R,$ or the integers $Z .$ Hiker $i$ may move her zero up by some amount $f_{i} \in G,$ which lowers all her readings by $f_{i} .$ On the overlap $U_{ij}$ of two stretches, with $i < j,$ let $c_{ij}$ be the offset: how much higher hiker $j ’$ s readings run than hiker $i ’$ s. Moving the zeros by $f$ removes every offset exactly when $c_{ij} = f_{j} - f_{i} on every overlap U_{ij} .$ That is a system of three equations in three unknowns — and a familiar one.

Look closely: these are word for word the formulas for simplicial cochains on the nerve (§4.2 Cohomology Groups), with “piece” for “vertex”, “overlap” for “edge”, and “triple overlap” for “triangle”. In particular $δ \circ δ = 0 :$ for instance $(δ δ f)_{ij k} = (f_{k} - f_{j}) - (f_{k} - f_{i}) + (f_{j} - f_{i}) = 0.$ So we may take cohomology, cocycles modulo coboundaries, just as before.

What do the first two groups say? A $0$ -cochain $f$ is a cocycle when $δ f = 0,$ that is $f_{i} = f_{j}$ on every overlap: the local numbers agree wherever they can be compared. A $1$ -cochain $c$ is a coboundary when $c = δ f :$ the offsets come from shifting local zeros. And a $1$ -cochain is a cocycle when $c_{ik} = c_{ij} + c_{j k}$ on every triple overlap — the offsets are consistent wherever three pieces meet, which is the least we could ask of honest data.

Čech cohomology of the circle Use − and + to set the offsets · switch between three and two arcs

Loop test

c_{01} + c_{12} - c_{02} = 2 + 1 - 3 = 0

Consistent: the numbers on the overlaps are differences of numbers on the arcs (a coboundary), so local readings glue to one global reading.

c_{01}

c_{12}

c_{02}

Figure 4.7.2 Three arcs cover the circle; the gold arcs are their overlaps, labelled with the offsets

c_{ij} .

On the right, the nerve: the figure tries

f_{0} = 0,

f_{1} = c_{01},

f_{2} = c_{02},

which fixes two overlaps, and checks the third. The loop glows green when the offsets come from a global zero and rose when a class survives in

H ˇ^{1} .

Switch to two arcs to see the overlap fall apart into two pieces.

The circle, computed

For the three arcs there are no triple overlaps, so $C ˇ^{2} = 0$ and every $1$ -cochain is automatically a cocycle. Order the overlaps as $U_{01}, U_{02}, U_{12} .$ Then $δ : G^{3} \to G^{3}$ is the matrix $δ = - 1 - 1 0 10 - 1 011, δ f_{0} f_{1} f_{2} = f_{1} - f_{0} f_{2} - f_{0} f_{2} - f_{1} .$

$H ˇ^{0} .$ The cocycles are the $f$ with $f_{0} = f_{1} = f_{2} :$ the constant choices. So $H ˇ^{0} (U; G) ≅ G$ — one free choice, because the circle is connected.
$H ˇ^{1} .$ Define the loop test, or holonomy, of a $1$ -cochain: $h (c) = c_{01} + c_{12} - c_{02} .$ It is the total offset accumulated by walking once around the circle (the minus sign because $U_{02}$ is crossed from $2$ back to $0).$ For a coboundary, $h (δ f) = (f_{1} - f_{0}) + (f_{2} - f_{1}) - (f_{2} - f_{0}) = 0 .$ Conversely, if $h (c) = 0,$ then $f = (0, c_{01}, c_{02})$ satisfies $δ f = c .$ So the coboundaries are exactly the cochains with zero holonomy, and $h$ gives an isomorphism $H ˇ^{1} (U; G) = G^{3} / im δ ≅ G, [c] \mapsto c_{01} + c_{12} - c_{02} .$

For the hikers, $c_{01} = 40,$ $c_{12} = 25$ and $c_{02} = 65 .$ The loop test gives $40 + 25 - 65 = 0,$ and $f = (0, 40, 65)$ is the shift we found by hand. Had $c_{02}$ been $60,$ the loop test would give $5,$ and the class $[c] \in H ˇ^{1} ≅ G$ would be exactly those stray $5$ metres.

These are $H^{0} (S^{1}; G) = G$ and $H^{1} (S^{1}; G) = G,$ the cohomology of the circle from §4.2 Cohomology Groups. And the matrix is the very one you met there for the hollow triangle — of course: the nerve is a hollow triangle. This is the nerve theorem doing its job.

Two arcs: a cautionary tale

Why not cover the circle with just two arcs, $U$ (the top, a bit more than half) and $V$ (the bottom)? Look at the overlap: $U \cap V$ is not one arc but two, one on the left and one on the right. Call them $W_{1}$ and $W_{2} .$ The nerve has only two vertices and one edge, and a segment has no holes; if we trusted it, we would conclude that $H ˇ^{1} = 0 .$ Wrong!

The repair is to remember what a $1$ -cochain really is: data on the overlap. An offset is a constant on each connected stretch of overlap, and the two stretches $W_{1}$ and $W_{2}$ are free to carry different offsets $c_{1}, c_{2} .$ So $C ˇ^{1} = G^{2},$ not $G,$ and $δ (f_{U}, f_{V}) = (f_{V} - f_{U}, f_{V} - f_{U}) .$ The image is the diagonal ${(a, a)},$ so $H ˇ^{0} ≅ G$ and $H ˇ^{1} = G^{2} / {(a, a)} ≅ G$ via $(c_{1}, c_{2}) \mapsto c_{2} - c_{1} .$ The correct answer again. Try it in the figure: whenever $c_{W_{1}} \neq = c_{W_{2}}$ the two hikers’ zeros disagree on one side or the other, however they shift.

A little history Nerves and Čech

Pavel Alexandrov 1896 – 1982

Eduard Čech 1893 – 1960

Set by his teacher Nikolai Luzin to settle the continuum hypothesis, the young Pavel Alexandrov failed — as anyone would have, since it can be neither proved nor disproved — decided that he had no future in mathematics, and went off to produce plays in a provincial theatre. Back in Moscow, he introduced the nerve of a cover in the 1920s, as a way to approximate a compact space by polyhedra (Alexandroff 1928; O’Connor & Robertson 1999). In 1932 Eduard Čech used the nerves of finer and finer covers to define homology for arbitrary spaces, however wild — spaces that cannot be triangulated at all (Čech 1932). After the Germans closed the Czech universities, his topology seminar in Brno went on meeting in a student’s flat, until the Gestapo arrested the student in 1941 (O’Connor & Robertson 2000). Cohomology itself would only be invented three years after Čech’s paper; the Čech theory we use today is its dual, built on the same nerves.

Presheaves and sheaves

So far the data on every piece was a single number. Most interesting data are richer: a height profile on a stretch of trail, a temperature field on a region, a solution of a differential equation on an interval, a choice of square root of $z$ near some point. What all of these share is that data on a set can be cut down to data on any smaller set.

Data that can be restricted

A presheaf in one picture

Figure 4.7.3 Open sets shrink as you go down the page,

W \subseteq V \subseteq U;

a section

s

over

U

restricts to sections over

V

and

W .

The inclusions point up, the restrictions point down: a presheaf reverses arrows.

Here is one of the book’s four recurring ideas, reversed arrows, in its purest form. Sets get smaller along inclusions $W \to V \to U,$ but data travels the other way, $F (U) \to F (V) \to F (W) .$ That is the same reversal as cochains pulling back along maps, and it is why cohomology, not homology, is the natural home for local data.

Example Presheaves everywhere

Continuous functions. $C (U)$ = all continuous real-valued functions on $U,$ restricted in the ordinary sense. Likewise smooth functions, or polynomial functions on open subsets of $R .$
Solutions of an equation. $F (U)$ = the functions $f$ on $U \subseteq R$ with $f^{''} + f = 0 .$ The restriction of a solution is a solution.
Locally constant functions. $F (U)$ = the functions $U \to G$ that are constant near every point, that is, constant on each connected piece of $U .$ For real-valued functions these are the solutions of $f^{'} = 0,$ and they are what $H^{0}$ counted in §4.1 Cochains: Measuring Shapes.
Constant functions. $F (U)$ = the functions $U \to G$ that take a single value on all of $U .$
Bounded functions. $F (U)$ = the continuous functions $f$ on $U$ with $∣ f ∣$ bounded.
Square roots. On open subsets $U$ of the punctured plane, $F (U)$ = the continuous functions $g : U \to C$ with $g (z)^{2} = z$ for every $z$ in $U .$ (More about these soon.)

The sheaf axioms

Some of these presheaves are better behaved than others. Ask two questions of any kind of local data. If two pieces of data agree on every piece of a cover, are they the same? And: if data given piece by piece agree wherever the pieces overlap, do they come from data on the whole? For the hikers’ height profiles, once the altimeters are calibrated, both answers are yes: two height maps of the ring that agree on every stretch are the same map, and profiles that agree on the overlaps join into one. Data for which local knowledge is enough, in exactly this sense, form a sheaf.

Continuous functions form a sheaf. Locality holds because every point lies in some piece, so functions that agree on every piece agree everywhere. For gluing, define $s (x) = s_{i} (x)$ for any $i$ with $x \in U_{i};$ this does not depend on the choice of $i,$ because the pieces agree on overlaps, and $s$ is continuous because continuity can be checked near each point, where $s$ agrees with one of the continuous $s_{i} .$ The same argument works for smooth functions and for solutions of a differential equation, because being a solution is also checked point by point. The square roots form a sheaf too, and so do the locally constant functions.

Two presheaves on the list fail.

Bounded functions. Cover the line by the intervals $U_{n} = (n - 1, n + 1)$ and let $s_{n} (x) = x$ on $U_{n} .$ Each $s_{n}$ is bounded (by $∣ n ∣ + 1),$ and they agree on overlaps. The only function they could glue to is $x \mapsto x$ on the whole line, which is not bounded. Gluing fails.
Constant functions. Let $U$ be two disjoint intervals, covered by themselves. The constant $0$ on one and the constant $1$ on the other agree on the overlap — vacuously, since there is no overlap — but no constant function on $U$ restricts to both. Gluing fails again; the sheafy version is “locally constant”.

Bounded, versus bounded by 1 Add pieces · switch the property

On each piece U_n = (n−1, n+1) the function x ↦ x is bounded (by |n| + 1, the dashed lines), and neighbouring pieces agree where they overlap. Glued together they give x ↦ x on the whole union, whose largest value grows with every piece you add: the property “bounded” does not survive gluing.

Pieces on each side 3

Figure 4.7.4 The line is covered by overlapping intervals

U_{n} .

Each piece of

x \mapsto x

is bounded, but the bound grows with the piece and the glued function is unbounded. A bound fixed in advance, such as

∣ f ∣ \leq 1,

can be checked point by point, so it survives gluing.

Sections and global sections

The word section comes from a picture you will see properly in §4.8 Curvature and Characteristic Classes: above each point of $X$ stands a “stalk” of possible values, and a section over $U$ slices through the stalks above $U,$ picking one value above each point. (The French word Leray chose, faisceau, is a sheaf of wheat: a bundle of stalks tied together.)

The sheaf axioms say something sharp in the language of Čech cochains. With a presheaf $F$ in place of the constant group $G,$ a Čech $0$ -cochain is a family of local sections $s_{i} \in F (U_{i}),$ and $δ s = 0$ says that they agree on all overlaps. Locality and gluing say that such families correspond exactly to global sections:

A little history Open sets, and a seminar

Henri Cartan 1904 – 2008

Élie Cartan’s son (§4.4 de Rham Cohomology) and a founding member of Bourbaki, he was perhaps the first person to understand Leray’s new theory. Leray’s sheaves gave groups to closed sets. In his Paris seminar of 1950–51 Cartan rebuilt the theory on open sets, wrote $Γ$ for sections and defined sheaf cohomology by axioms; by 1953 a sheaf was, for him, a presheaf satisfying the gluing conditions, as above, though the word “presheaf” had to wait for Grothendieck (Miller 2000). This is the work Leray had in mind in the epigraph of this chapter. The seminar’s notes, written up at the suggestion of Serre, then his student, appeared until 1964. Cartan lived to 104; in the 1970s he campaigned for the mathematician Leonid Plyushch, held in a Soviet psychiatric hospital until his release in 1976 (O’Connor & Robertson 2004).

When local solutions refuse to glue

A sheaf guarantees that compatible local sections glue. Very often, though, we can find local sections everywhere and yet cannot make them compatible. Here are three classic cases, and the cohomology class that measures each failure.

Square roots and logarithms

A complex number $z \neq = 0$ can be written in polar form as $z = r e^{i θ} :$ $r > 0$ is its distance from $0$ and $θ$ the angle it makes with the positive real axis. Multiplying complex numbers multiplies distances and adds angles, so $z$ has exactly two square roots, $\pm r e^{i θ /2} .$ On a small disk away from $0$ we can choose one of them continuously — say the one with positive real part near $z = 1 .$ So the sheaf of square roots has sections over every small disk: two of them, $g$ and $- g .$

Now walk once around the origin, following your chosen root continuously. The angle $θ$ grows from $0$ to $2 π,$ so $θ /2$ grows from $0$ to $π :$ you come back to the same point $z$ holding $r e^{iπ} = - r,$ the other square root. Walk around again and you return to where you began. This phenomenon — following a local solution around a loop and coming back to a different one — is called monodromy.

Following a branch around the puncture Drag the gold point to walk · play the walk · drag elsewhere to rotate

the value √z

walked 0°

value

z = 1.22 + 0.00 i

Back at the start after 360°, the arrow will have turned only half a turn.

start two turns

Figure 4.7.5 The walker circles the origin in the plane below; play the walk, or drag the walker around

0

yourself. Above it, its chosen value is followed continuously on the Riemann surface of the function. For

z

(height = real part) the surface has two sheets, and one turn moves you from one to the other; the dial shows the value turning only half a turn. For

lo g z

the surface is a spiral ramp, and every turn adds

2 π i .

So the sheaf of square roots on $C ∖ {0}$ has local sections everywhere but no global section: there is no continuous square root on the whole punctured plane (or even on a circle around $0).$ Čech cohomology says why, in numbers. Cover a circle around $0$ by three arcs and choose a root $g_{i}$ on each arc. On each overlap two choices are either equal or opposite: $g_{j} = ε_{ij} g_{i}$ with $ε_{ij} \in {+ 1, - 1} .$ Changing the choice on arc $i$ flips the signs of both overlaps at that arc, and therefore cannot change the product around the loop, $ε_{01} ε_{12} ε_{02}^{- 1} = - 1.$ This is the multiplicative version of the loop test: a class in $H ˇ^{1} (S^{1}; {\pm 1}),$ and it is not the trivial one.

The logarithm behaves the same way with a different group. Its values on a small disk are $ln r + i θ$ with $θ$ determined up to adding a multiple of $2 π .$ Two local choices differ by a constant $2 π ik$ with $k \in Z,$ and the loop test gives $2 π i$ times the number of times the loop winds around $0 :$ a class in $H ˇ^{1} (S^{1}; 2 π i Z) ≅ Z .$ The imaginary part of $lo g z$ is the angle $θ,$ and this class is our old friend from §4.4 de Rham Cohomology, the class of $d θ .$

Orientations of a Möbius band

Recall from §2.4 Manifolds and Surfaces that an orientation of a small patch of surface is a choice of which way is “anticlockwise” there; equivalently, by the right-hand rule, a choice of which side the normal arrow points to. Every small patch has exactly two orientations, and orientations restrict to smaller patches, so they form a sheaf: the orientation sheaf. A global section is an orientation of the whole surface, so a surface is orientable exactly when its orientation sheaf has a global section.

The orientation sheaf of the Möbius band Drag to rotate · flip patches · carry the arrow

g_{12} = + 1, g_{23} = + 1, g_{31} = - 1

g_{12} g_{23} g_{31} = - 1

Flip any patch you like: two signs change together, and the product stays −1. There is no way to choose orientations that agree everywhere.

start two laps

Figure 4.7.6 Three overlapping patches cover the band; each has its own consistent choice of normal arrow. On each overlap the two choices agree (

+ 1)

or disagree (

- 1).

Flipping a patch changes two signs at once, so the product of the three signs never changes:

- 1

on the Möbius band,

+ 1

on the cylinder. The gold arrow, carried once around, comes back upside down.

The figure computes the Čech cocycle of this sheaf. On each patch $U_{i}$ choose an orientation $o_{i};$ on an overlap, $o_{j} = g_{ij} o_{i}$ for a sign $g_{ij} \in {\pm 1} .$ These signs satisfy the cocycle condition (on a triple overlap, comparing $i$ with $k$ directly or via $j$ gives the same answer), and changing the choice on one patch multiplies $g$ by a coboundary. The product around the core circle is $- 1$ for the Möbius band whatever you do: the orientation sheaf has no global section, and the obstruction is a nonzero class in $H ˇ^{1} (S^{1}; {\pm 1}) .$ In §4.8 Curvature and Characteristic Classes this class gets a name, the first Stiefel–Whitney class $w_{1} .$

The impossible triangle

In 1958 the psychiatrist and geneticist Lionel Penrose and his son Roger, then a young mathematician, published a drawing of an impossible object: a triangle of three square beams, each meeting the next at a right angle (Penrose & Penrose 1958). (The Swedish artist Oscar Reutersvärd had drawn a version made of cubes in 1934.) Every corner of the drawing is a perfectly ordinary corner. The whole thing cannot exist. In 1991 Roger Penrose explained why, in a short paper titled On the cohomology of impossible figures (Penrose 1992).

Cover the drawing by three overlapping pieces $U_{1}, U_{2}, U_{3},$ each containing one corner. Each piece on its own is a drawing of a real three-dimensional object: an L-shaped corner. But a drawing never tells you how far away an object is. If you double every distance from your eye, the object grows to twice the size and looks exactly the same. So each piece determines a real object only up to a scale factor $λ_{i} > 0,$ its distance from the eye.

On an overlap, say $U_{1} \cap U_{2},$ both pieces show the same stretch of beam, and the two interpretations put it at distances whose ratio is some number $d_{12} > 0 .$ Rescaling the pieces changes the ratios — rescaling piece $i$ by $λ_{i}$ multiplies $d_{ij}$ by $λ_{i} / λ_{j}$ — and the three pieces fit together into one real object exactly when we can make every ratio equal to $1,$ that is, when $d_{ij} = λ_{j} / λ_{i}$ for some choice of the $λ ’$ s. In the language of this chapter, $d$ is a Čech $1$ -cochain with values in the group $(R_{> 0}, \times)$ of positive numbers under multiplication, and the figure is realizable exactly when $d$ is a coboundary. The loop test, written multiplicatively, is $d_{12} d_{23} d_{31} = 1.$

Penrose’s cocycle Pull the pieces apart · rescale them

U_{1} \cap U_{2}

d_{12} = 1.000

U_{2} \cap U_{3}

d_{23} = 1.000

U_{3} \cap U_{1}

d_{31} = 0.712

d_{12} d_{23} d_{31} = 0.712 \neq = 1

— no rescaling of the pieces can change this product.

together apart

Rescale a piece about the eye — its drawing does not change

U_{1}

×1.00

U_{2}

×1.00

U_{3}

×1.00

Figure 4.7.7 The impossible triangle is three overlapping pieces, each a genuine corner — pull them apart to check. Rescaling a piece about the eye does not change its drawing, but it changes the depth ratios on its two overlaps. You can make any two overlaps agree; the product

d_{12} d_{23} d_{31}

stays put, so the third overlap always disagrees.

For the tribar the product is not $1 .$ Going around the triangle, each corner tells you that the next beam recedes from you, so the depth ratios compound instead of cancelling. The drawing is a nonzero class in $H ˇ^{1}$ of an annulus (the shape of the drawing) with coefficients in the positive real numbers. Penrose’s analysis gives a new meaning to the slogan of this chapter: an impossible figure is a locally consistent picture whose consistency fails to glue.

You can even see the class in three dimensions. Build the three beams for real, as an open chain, and look at it from exactly the right point: the end of the last beam then sits in front of the start of the first, and the gap closes.

Possible — from one eye only Step aside, or drag to look from elsewhere · zoom

Show U₃’s version of beam A μ = |EP₃| / |EP₀| ≈ 0.712

Figure 4.7.8 Three real beams. From the eye (where the view starts) they form a closed triangle; step aside and the gap opens. The rose line is the line of sight through the start of beam A and the end of beam C. The gold ghost is how piece

U_{3}

reads beam A, as a genuine corner: beam A shrunk towards the eye by the factor

μ .

In this model that single factor is the product

d_{12} d_{23} d_{31},

and it is why no single object fits all three pieces.

Sheaf cohomology: measuring the failure to glue

Čech cochains with values in a sheaf

Everything in the Čech complex works with a sheaf of abelian groups $F$ in place of the constant group $G :$ put sections of $F$ on the pieces and on their overlaps, and restrict before you subtract.

(The symbol $\prod$ means a list with one entry for each choice of indices — one section on each overlap of $p + 1$ pieces.) With the constant sheaf of locally constant $G$ -valued functions we recover exactly the previous section: a locally constant function on an overlap is one element of $G$ per connected piece, which is the repair we needed for the two arcs.

What the groups mean

$H ˇ^{0} (X; F) = F (X)$ : the global sections, the global solutions of whatever problem $F$ encodes.
$H ˇ^{1} (X; F)$ : the obstructions. A $1$ -cocycle is a recipe for comparing local data on overlaps that is consistent on triple overlaps; it is a coboundary when the comparisons can be absorbed by changing the local data. Nonzero classes are “twisted” versions of the trivial situation: twisted line bundles in the next chapter, impossible figures here.
Higher $H ˇ^{p}$ : obstructions to resolving the obstructions — inconsistencies on triple overlaps that cannot be absorbed by changing the data on pairs, and so on.

Here is the recipe behind every example so far, and it shows how $H ˇ^{1}$ arises from a failure to glue. Suppose every point has a neighbourhood on which some equation can be solved, and any two local solutions differ by a section of a sheaf of abelian groups $A .$ Pick local solutions $s_{i}$ on a cover; on each overlap $s_{j} - s_{i} = c_{ij}$ is a section of $A;$ these differences automatically satisfy $c_{ik} = c_{ij} + c_{j k},$ so $c$ is a cocycle; and a global solution exists exactly when $[c] = 0$ in $H ˇ^{1} (X; A) .$ (If $c = δ a,$ the corrected solutions $s_{i} - a_{i}$ agree on every overlap and glue.) The hikers, the logarithm and the depth readings are all instances.

Leray’s theorem: when one cover is enough

The true sheaf cohomology $H^{p} (X; F)$ is defined without choosing any cover, either as the limit of $H ˇ^{p} (U; F)$ over finer and finer covers, or through the machinery of derived functors that Alexander Grothendieck introduced in 1957 (Grothendieck 1957). For the spaces in this book every reasonable definition gives the same groups. To compute them, one well-chosen cover suffices.

Such a cover is called a Leray cover for $F .$ For constant coefficients it is enough that every intersection be a disjoint union of contractible pieces — which explains the cautionary tale: the overlap of the two arcs is two contractible pieces, so the two-arc cover is a Leray cover, and its Čech complex (done honestly, one number per piece) computes $H^{1} (S^{1}) = G$ correctly, even though its nerve does not look like a circle.

de Rham, revisited

Here is the bridge between this chapter and §4.4 de Rham Cohomology. Let $ω$ be a closed $1$ -form on a manifold $M,$ and take a good cover. On each contractible piece $U_{i}$ the Poincaré lemma gives a function $f_{i}$ with $d f_{i} = ω .$ On an overlap, $d (f_{j} - f_{i}) = ω - ω = 0,$ so $f_{j} - f_{i}$ is a constant $c_{ij}$ on each (connected) overlap — a Čech $1$ -cocycle with values in $R .$ Changing the local primitives by constants changes $c$ by a coboundary, and $ω$ is exact exactly when the local primitives can be chosen to agree, that is, when $[c] = 0 .$ So $[ω] \mapsto [c]$ is a well-defined, one-to-one map; with a partition of unity (a tool for blending local functions into a global one) you can show that every Čech class arises this way, and the map is an isomorphism $H_{dR}^{1} (M) ≅ H ˇ^{1} (M; R) .$ For the angle form on the circle with our three arcs, numbered anticlockwise, the local primitives are three branches of the angle, and the offsets are $c_{01} = c_{12} = 0$ and $c_{02} = 2 π :$ holonomy $- 2 π \neq = 0$ (the sign is an artefact of our conventions), the class of $d θ .$ The same argument in every degree, organised by Bott and Tu into the “tic-tac-toe” Čech–de Rham complex, proves de Rham’s theorem (Bott & Tu 1982, §8).

A little history A topologist in captivity

Jean Leray 1906 – 1998

An army officer and an expert on fluid dynamics, he was captured in 1940 and spent the next five years in Oflag XVII-A, an officers’ camp in Austria. The MacTutor biography records the reason for his change of field: “Not wishing the Germans to know that he was an expert in hydrodynamics, since he feared that if they found out he would be forced to undertake war work for them, Leray claimed to be a topologist” (O’Connor & Robertson 2001). The prisoners organised a university in captivity, with Leray as its rector. The course he gave there was sent to a journal in Paris, approved by Heinz Hopf in neutral Zürich, and published in 1945 as “a course in algebraic topology taught in captivity”; in 1946 he announced sheaves, sheaf cohomology and spectral sequences (Miller 2000). Jean-Pierre Serre made sheaves the language of algebraic geometry in 1955 (Serre 1955), and Grothendieck’s 1957 paper made sheaf cohomology a derived functor.

Sheaves on graphs: local data, global consensus

Sheaves can sound forbiddingly abstract. In applied topology they are now a practical tool, thanks to a combinatorial version that lives on graphs and cell complexes. It goes back to Allen Shepard’s 1985 thesis; Justin Curry, whose 2014 thesis turned it into a tool for data, networks and sensors, describes cellular sheaves as “finite families of vector spaces and maps parametrized by a cell complex” (Curry 2014). Here is the version on a graph.

The data at a vertex restrict to its edges just as data on an open set restrict to smaller sets. (Think of a vertex as standing for the little star-shaped open set around it, and an edge for the smaller open set around the edge’s middle.) The cochains are $C^{0} = ⨁_{v} F (v)$ and $C^{1} = ⨁_{e} F (e),$ and the coboundary measures disagreement on each edge $e$ from $u$ to $v :$ $(δ x)_{e} = F_{v ⊴ e} x_{v} - F_{u ⊴ e} x_{u},$ and as always $H^{0} (F) = ker δ = {global sections}, H^{1} (F) = C^{1} / im δ .$ With every stalk equal to $R$ and every map the identity, this is exactly the graph cohomology of §4.1 Cochains: Measuring Shapes. The point of a sheaf is that the maps need not be identities.

A sheaf of prices Change prices with − and + · let them settle · try both rate tables

€ 3.00

$ 3.20

£ 2.70

CHF 2.60

Around the triangle:

1.10 \times 0.80 \div 0.88 = 1.000

dim H^{0} = 1

(consistent price lists)

dim H^{1} = 1

The rates are consistent around the loop, so consistent price lists exist: pick the euro price and every other price is forced.

Figure 4.7.9 Each vertex holds the price of the same coffee in its own currency; each edge converts one price into the other’s currency and compares. A global section is a consistent price list. With honest rates there is a one-dimensional space of them; with an arbitrage loop, only the zero price list is consistent. “Let prices settle” runs the sheaf heat equation, which slides to the nearest consistent list.

Example Currencies as a sheaf

In the figure, the stalk at every vertex and edge is $R .$ The edge from the euro vertex to the dollar vertex compares prices in euros: the restriction map from the euro vertex is $1,$ and the one from the dollar vertex is $1/1.10,$ which converts dollars into euros. A global section is a price list such that every conversion agrees.

Count dimensions. $C^{0} = R^{4}$ (four prices) and $C^{1} = R^{4}$ (four comparisons). With honest rates — the conversion around the triangle multiplies money by $1.10 \times 0.80 \div 0.88 = 1$ — choosing the euro price forces all the others, so $dim H^{0} = 1,$ the rank of $δ$ is $3,$ and $dim H^{1} = 4 - 3 = 1 .$ Change one rate so that going around the triangle multiplies money by $1.257$ and the only consistent price list is $0 :$ $dim H^{0} = 0,$ and then $dim H^{1} = 0$ too. In both cases $dim H^{0} - dim H^{1} = dim C^{0} - dim C^{1} = 0 :$ the Euler characteristic of a sheaf on a graph does not depend on its maps (Exercise 7).

Consensus and data fusion

The same pattern describes many problems where local measurements must be fused into a global picture: sensors measuring one quantity in different units or coordinate frames, cameras that see overlapping parts of a scene, agents in a network trying to agree. The global sections are the consistent global pictures, $H^{0};$ edge data that cannot be explained by any vertex data are the classes in $H^{1} .$

There is even a natural way for a network to find a global section using only local communication. The sheaf Laplacian $L = δ^{T} δ$ generalises the graph Laplacian, and the heat equation $\frac{d x}{d t} = - Lx$ lets every vertex nudge its value to reduce its disagreement with its neighbours: the total disagreement $∥ δ x ∥^{2}$ only ever goes down. The part of $x$ that is already a global section (it lies in $ker δ = ker L)$ never moves, and everything else decays, so the flow settles on the global section closest to the starting data — consensus, when every map is the identity. Jakob Hansen and Robert Ghrist call the study of such Laplacians “spectral sheaf theory — an extension of spectral graph theory to cellular sheaves” (Hansen & Ghrist 2019). Press “Let prices settle” in the figure to watch it work: with honest rates the prices converge to a consistent list; with an arbitrage loop they drain away to zero, the only consistent list there is.

Exercises

Exercise Four arcs

Cover the circle by four open arcs $U_{0}, U_{1}, U_{2}, U_{3},$ arranged in order around the circle, each overlapping only its two neighbours. Draw the nerve. Write down the Čech coboundary $δ : G^{4} \to G^{4}$ and find a loop test $h$ with $H ˇ^{1} ≅ G .$

Hint

The overlaps are $U_{01}, U_{12}, U_{23}$ and $U_{03} .$ Walk around the circle $0 \to 1 \to 2 \to 3 \to 0 .$

Solution

The nerve has four vertices and the four edges $01, 12, 23, 03,$ and no triangles: a square, which is a circle. $δ f = (f_{1} - f_{0}, f_{2} - f_{1}, f_{3} - f_{2}, f_{3} - f_{0}) .$ Walking around, the offsets accumulate to $h (c) = c_{01} + c_{12} + c_{23} - c_{03},$ with a minus sign on $c_{03}$ because that overlap is crossed from $3$ back to $0 .$ For a coboundary the sum telescopes to $0;$ conversely if $h (c) = 0,$ then $f = (0, c_{01}, c_{01} + c_{12}, c_{03})$ satisfies $δ f = c .$ So $im δ = ker h,$ $h$ is onto $G,$ and $H ˇ^{1} ≅ G .$ Also $ker δ$ is the constants, so $H ˇ^{0} ≅ G .$

Exercise Covering the sphere

(a) Cover the sphere by two caps, each missing one pole. What is the nerve, and why does the nerve theorem not apply? (b) Draw a tetrahedron on the sphere (four vertices, six great-circle edges, four triangular faces) and let $U_{i}$ be a slightly enlarged copy of face $i .$ What is the nerve now? What does the nerve theorem say?

Hint

In (b), which collections of faces share a point? Faces that share an edge; three faces at a vertex; and all four?

Solution

(a) Two vertices and one edge: a segment, which is contractible. The overlap of the caps is a band around the equator, which is not contractible (it contains a loop that cannot be shrunk), so the cover is not good and the theorem does not apply; indeed the sphere is not contractible.

(b) Every two faces of a tetrahedron share an edge and every three share a vertex, so enlarged slightly, every pair and every triple of pieces overlaps; but no point is close to all four faces, so there is no $3$ -simplex. The nerve is the boundary of a tetrahedron: four vertices, six edges, four triangles. Each piece is a disk, each overlap of two pieces is a thin strip along an edge, and each overlap of three is a small disk around a corner — all contractible — so the cover is good and the nerve theorem says the sphere is homotopy equivalent to the hollow tetrahedron — which has $H^{0} = Z,$ $H^{1} = 0,$ $H^{2} = Z,$ the cohomology of $S^{2} .$

Exercise Sheaf or not?

Which of these presheaves on $R$ are sheaves? (a) Continuous functions with $f (x) \geq 0$ for all $x .$ (b) Continuous functions with $∣ f (x) ∣ \leq 1$ for all $x .$ (c) Bounded continuous functions. (d) Constant functions. (e) Differentiable functions with $f^{'} = f .$ (f) Continuous functions with at most one zero in $U .$ (g) A trap: continuous functions with $\int_{U} f = 0$ (for bounded $U).$

Hint

Ask whether the defining property can be checked near each point separately. For (g), check first that restriction makes sense at all.

Solution

(a), (b) and (e) are sheaves: each property is checked point by point, so glued functions still have it (for (e), the glued function is differentiable near each point, with $f^{'} = f$ there). (c) is not (the pieces of $x \mapsto x$ on $(n - 1, n + 1)$ glue to an unbounded function). (d) is not (two different constants on two disjoint intervals). (f) is a presheaf — cutting a function down cannot create zeros — but not a sheaf: the zeros of $sin x$ are $π$ apart, so on each interval $(n - 1, n + 1),$ of length $2 < π,$ it has at most one zero, yet the glued function $sin x$ on $R$ has infinitely many.

(g) is not even a presheaf. Take $f (x) = sin (π x)$ on $U = (0, 2) :$ its integral is $0,$ so $f$ is a section over $U .$ Its restriction to $V = (0, 1)$ has integral $2/ π \neq = 0,$ so it is not a section over $V :$ there is no restriction map $F (U) \to F (V) .$ Before asking about gluing, always check that the data can be restricted.

Exercise Cube roots

Every $z \neq = 0$ has three cube roots. Describe what happens to a continuously chosen cube root as you walk once, twice and three times around the origin. Which group do the transition factors $g_{j} = ε_{ij} g_{i}$ live in, and what is their product around the loop?

Solution

The cube roots of $r e^{i θ}$ are $r^{1/3} e^{i θ /3}$ times $1,$ $ω = e^{2 π i /3}$ and $ω^{2} .$ Following one continuously, the angle $θ /3$ grows by $2 π /3$ per turn, so one walk multiplies the root by $ω,$ two by $ω^{2},$ and three bring it back. The factors $ε_{ij}$ are cube roots of unity, a group ${1, ω, ω^{2}} ≅ Z /3$ under multiplication. Multiplying the root on one arc by a cube root of unity $ζ$ multiplies the product $ε_{01} ε_{12} ε_{02}^{- 1}$ by $ζ$ once and by $ζ^{- 1}$ once, so the product does not change. With the arcs numbered anticlockwise it is $ω^{- 1} = ω^{2}$ (numbered clockwise, it is $ω)$ — never $1,$ whatever local choices are made: a nonzero class in $H ˇ^{1} (S^{1}; Z /3) .$

Exercise The Möbius band with two patches

Cover the Möbius band by two patches $U$ and $V$ whose overlap consists of two pieces $W_{1}$ and $W_{2}$ (like the two arcs of the circle). Choose an orientation on $U$ and one on $V,$ and let $g_{1}, g_{2} \in {\pm 1}$ record whether they agree on $W_{1}$ and on $W_{2} .$ Show that $g_{1} g_{2} = - 1$ for every choice, and that $g_{1} g_{2} = + 1$ for the cylinder.

Solution

Flipping the orientation on $U$ (or on $V)$ flips both $g_{1}$ and $g_{2},$ so $g_{1} g_{2}$ does not depend on the choices. To compute it, carry an orientation from $W_{1}$ through $U$ to $W_{2},$ and back through $V$ to $W_{1} :$ this is carrying it once around the core circle. On the Möbius band it comes back reversed, which means the two comparisons differ: $g_{1} g_{2} = - 1 .$ On the cylinder it comes back unchanged, so $g_{1} g_{2} = + 1 .$ This is the class in $H ˇ^{1} (S^{1}; {\pm 1}) ≅ Z /2$ computed with the two-piece overlap, just as for the two arcs.

Exercise Sheaves on graphs: trees, loops and the Euler characteristic

Consider a cellular sheaf on a graph in which every stalk is $R$ and every restriction map is multiplication by a nonzero number. (a) Show that if the graph is a tree, then $dim H^{0} = 1$ and $dim H^{1} = 0 .$ (b) Show that for any graph with $V$ vertices and $E$ edges, $dim H^{0} - dim H^{1} = V - E .$ (c) For a triangle, show that $dim H^{0} = 1$ if the product of the “conversion factors” around the loop is $1,$ and $0$ otherwise.

Hint

For (b), use rank–nullity for $δ : R^{V} \to R^{E} .$

Solution

(a) Choose $x$ at one vertex. Each edge condition $F_{u ⊴ e} x_{u} = F_{v ⊴ e} x_{v}$ determines the value at the far end from the near end, and in a tree each vertex is reached along exactly one path, so every section is determined by one number and every number works: $dim H^{0} = 1 .$ Then $dim H^{1} = E - rank δ = (V - 1) - (V - 1) = 0 .$

(b) By rank–nullity, $dim H^{0} = dim ker δ = V - rank δ$ and $dim H^{1} = E - rank δ;$ subtract.

(c) Going around the triangle, the edge conditions force $x_{1} = a x_{0},$ $x_{2} = b x_{1}$ and $x_{0} = c x_{2}$ for the three conversion factors $a, b, c;$ so $x_{0} = ab c x_{0} .$ If $ab c = 1$ any $x_{0}$ works and $dim H^{0} = 1;$ otherwise $x_{0} = 0,$ hence everything is $0,$ and $dim H^{0} = 0 .$ By (b), $dim H^{1}$ is then $1$ or $0$ respectively.

Exercise Refining the two arcs

Cut the bottom arc $V$ of the two-arc cover into two overlapping arcs $V_{1}, V_{2},$ so that $U, V_{1}, V_{2}$ is a three-arc cover. Explain how a Čech $1$ -cocycle $(c_{1}, c_{2})$ for the two-arc cover gives one for the three-arc cover, and check that its loop test equals $c_{2} - c_{1}$ (up to sign).

Solution

Label the new pieces $U_{0} = U,$ $U_{1} = V_{1}$ (containing $W_{1}),$ $U_{2} = V_{2}$ (containing $W_{2}).$ On $U_{01} = W_{1}$ use $c_{1};$ on $U_{02} = W_{2}$ use $c_{2};$ and on $U_{12},$ which lies inside $V$ where the old cochain has a single zero, use $0 .$ The loop test is $c_{01} + c_{12} - c_{02} = c_{1} + 0 - c_{2} = - (c_{2} - c_{1}) .$ So the two covers give the same group $G,$ and the classes match up to the orientation of the loop: refining a Leray cover does not change Čech cohomology.

Summary

Recap

An open cover splits a space into overlapping pieces; its nerve records which pieces overlap. For a good cover (contractible overlaps), the nerve has the same shape as the space — the nerve theorem, used to find holes in sensor networks.
Čech cochains put data on pieces and overlaps; the coboundary $(δ f)_{ij} = f_{j} - f_{i}$ is the simplicial coboundary of the nerve. For three arcs on a circle, $H ˇ^{0} = H ˇ^{1} = G,$ detected by the loop test $c_{01} + c_{12} - c_{02} .$
With two arcs the overlap has two pieces: the Čech complex (one value per piece) still gives the right answer, but the one-edge nerve does not. Leray’s theorem says when a cover computes true cohomology.
A presheaf is data on open sets with restriction maps (arrows reversed); a sheaf also satisfies locality and gluing. Continuous functions and solutions of equations form sheaves; bounded and constant functions do not. For a sheaf, $H ˇ^{0} = F (X),$ the global sections.
Monodromy of $z$ and $lo g z,$ the orientation sheaf of the Möbius band, and Penrose’s impossible triangle are all nonzero classes in $H ˇ^{1} :$ local solutions exist, but their comparisons on overlaps do not come from a global choice.
Cellular sheaves on graphs bring all this to data: $H^{0}$ is the space of consistent global pictures, and the sheaf Laplacian’s heat flow finds them by local averaging.

Sheaves and Čech Cohomology

Local pieces, global questions

Covers and their nerves

Open covers

The nerve

The nerve theorem

Holes in sensor networks

Čech cohomology: bookkeeping on overlaps

Back to the hikers

The circle, computed

Two arcs: a cautionary tale

Presheaves and sheaves

Data that can be restricted

The sheaf axioms

Sections and global sections

When local solutions refuse to glue

Square roots and logarithms

Orientations of a Möbius band

The impossible triangle

Sheaf cohomology: measuring the failure to glue

Čech cochains with values in a sheaf

What the groups mean

Leray’s theorem: when one cover is enough

de Rham, revisited

Sheaves on graphs: local data, global consensus

Consensus and data fusion

Exercises

Summary

Further reading

References