Glossary

The boundary of a boundary is zero: each face of a face lies in exactly two faces, counted twice (mod 2) or with opposite signs (over $\mathbb{Z}).$ Consequence: every boundary is a cycle, $B_k\subseteq Z_k.$

A set of triangles in which every edge is a side of an even number of them — a closed surface, such as the four faces of a hollow tetrahedron.

In a group, $ab=ac$ implies $b=c$ (multiply on the left by $a^{-1}),$ and similarly on the right. It is why no element repeats in a row or column of a Cayley table.

The operation $\sigma ⌢\phi =\phi ([v_0,\dots ,v_p])\cdot [v_p,\dots ,v_n]$ combining an $n$-chain and a $p$-cochain into an $(n-p)$-chain: the cochain is spent on the front face and the back face survives. It satisfies $⟨\psi ,c⌢\phi ⟩=⟨\phi ⌣\psi ,c⟩.$

The size of a set. $|X|=n$ means there is a bijection $\{1,\dots ,n\}\to X;$ two sets have the same cardinality when there is a bijection between them.

$A\times B$ is the set of ordered pairs $(a,b)$ with $a\in A,$ $b\in B$ — picture a grid. For finite sets, $|A\times B|=|A|\cdot |B|.$ Example: the torus is $S^1\times S^1.$

Promoting a number or polynomial to a homology theory whose Euler characteristic recovers it, as Khovanov homology does for the Jones polynomial. Betti numbers are the original example: their alternating sum is $\chi .$

A collection of objects together with, for each pair of objects, a set of arrows between them, with an identity arrow on each object and an associative composition. Examples: sets and functions ($\mathsf{Set}),$ abelian groups and homomorphisms ($\mathsf{Ab}),$ spaces and continuous maps ($\mathsf{Top}).$

The category $\mathsf{Ch}$ whose objects are chain complexes of abelian groups and whose arrows are chain maps. Homology is a functor $H_n:\mathsf{Ch}\to \mathsf{Ab}.$

An informal name for a non-zero class in $H_2:$ a region enclosed by a 2-cycle that is not filled in, like the inside of a hollow sphere.

The multiplication table of a finite group: the entry in row $g,$ column $h$ is $gh.$ Every element appears exactly once in each row and each column (the “Sudoku property”).

The map $\delta$ on Čech cochains, $(\delta f{)}_{ij}=f_j-f_i,$ $(\delta c{)}_{ijk}=c_{jk}-c_{ik}+c_{ij},$ and in general an alternating sum of restrictions. It satisfies $\delta \circ \delta =0.$

Data attached to the pieces of a cover ($0$-cochains), to their pairwise overlaps ($1$-cochains), to triple overlaps ($2$-cochains), and so on, with indices in increasing order.

${\check{H}}^p(\mathcal{U};F):$ Čech cocycles modulo Čech coboundaries for a cover $\mathcal{U}.$ For a good cover of a nice space with constant coefficients it is the ordinary cohomology of the space.

The nerve ${\check{C}}_r(P)$ of the balls of radius $r$ around the points of a cloud: points span a simplex when their balls share a point, that is, when they fit in one ball of radius $r.$

An open $n$-dimensional disk used as a building block: a 0-cell is a point, a 1-cell an open arc, a 2-cell an open disk, a 3-cell an open ball.

$d(e_{\alpha }^n)={\sum }_{\beta }{d}_{\alpha \beta }{e}_{\beta }^{n-1},$ where $d_{\alpha \beta }$ is the degree of “attach the boundary of $e_{\alpha },$ then collapse everything except $e_{\beta }”:$ how often the boundary runs over $e_{\beta },$ with signs.

For a CW complex, $C_n^{\mathrm{CW}}=H_n(X^n,X^{n-1}),$ the free abelian group on the $n$-cells, with boundary maps built from the long exact sequences of skeleta.

The homology of the cellular chain complex, isomorphic to singular homology. Hatcher calls it “homology squared”; it computes homology from one generator per cell.

Vector spaces (stalks) on the vertices and edges of a graph — or the cells of a complex — with a linear restriction map from each vertex to each edge at it. $H^0$ is the space of global sections.

An inventory of k-simplices. Mod 2, a set of k-simplices, added by symmetric difference; with integer coefficients, a formal sum $\sum a_{\sigma }\sigma$ of oriented k-simplices, added coefficient by coefficient.

A sequence of abelian groups (or vector spaces) $C_k$ with maps ${\partial }_k:C_k\to C_{k-1}$ such that ${\partial }_{k-1}\circ {\partial }_k=0$ for every k.

$C_k(K):$ all integer k-chains of $K,$ the free abelian group on the k-simplices, $\cong {\mathbb{Z}}^{n_k}.$ Mod 2, $C_k(K;\mathbb{Z}/2)$ is a vector space over $\mathbb{Z}/2$ of dimension $n_k.$

Homomorphisms $P:C_n\to C_{n+1}^{\prime }$ with $\partial P+P\partial =\psi -\phi$ for two chain maps $\phi ,\psi .$ Chain homotopic maps induce the same map on homology.

Chain maps $f:C\to D$ and $g:D\to C$ whose composites $g\circ f$ and $f\circ g$ are chain homotopic to the identities. Chain homotopy equivalent complexes have isomorphic homology.

A family of homomorphisms ${\phi }_k:C_k\to C_k^{\prime }$ between two chain complexes that commutes with the boundary: $\partial \circ \phi =\phi \circ \partial .$ It sends cycles to cycles and boundaries to boundaries, so it induces maps on homology.

A cohomology class attached to every vector bundle, natural under pulling back along maps and zero for trivial bundles; it measures how the bundle twists.

A pair $(U,\phi )$ where $U$ is an open subset of a manifold and $\phi$ is a homeomorphism from $U$ onto an open subset of ${\mathbb{R}}^n.$ It gives each point of $U$ a list of $n$ coordinates, like one page of an atlas of the Earth.

The integer obtained by evaluating a Chern class on a closed surface — for a bundle over the sphere, the winding number $n$ of its transition function $e^{in\phi }.$

The principle that curvature forms of a bundle represent its characteristic classes in de Rham cohomology; Gauss–Bonnet is the simplest example.

$\mathbb{Z}/m\oplus \mathbb{Z}/n\cong \mathbb{Z}/mn$ exactly when $\gcd (m,n)=1.$ For example $\mathbb{Z}/2\oplus \mathbb{Z}/3\cong \mathbb{Z}/6,$ but $\mathbb{Z}/2\oplus \mathbb{Z}/2$ is not $\mathbb{Z}/4.$

A knot that is not equivalent to its mirror image, like the trefoil. A knot equal to its mirror image, like the figure-eight knot, is amphichiral. The Jones polynomial detects the trefoil’s chirality.

Angles for the points of a data set, found from an integer 1-cocycle — using $H^1(X;\mathbb{Z})\cong [X,S^1]$ — after smoothing it to its harmonic representative by least squares. Introduced by de Silva, Morozov and Vejdemo-Johansson.

The sum of an edge flow around a closed loop, each edge counted with the sign of the direction it is walked in; around a single triangle it is also called the curl. In calculus (Chapter 4.3) it is the line integral $\oint \mathbf{F}\cdot d\mathbf{r}$ of a field around a loop: how much the field pushes you along as you go round.

For any choice of orientations ${\epsilon }_t$ of the triangles of a closed surface, $\partial (\sum {\epsilon }_{t}t)=2c.$ The cycle $c$ of clashing edges has a class that does not depend on the choice; it is zero exactly when the surface is orientable, and otherwise has order two.

Two closed connected surfaces are homeomorphic exactly when both are orientable or both are not, and they have the same Euler characteristic.

Every closed connected surface is homeomorphic to exactly one of: the sphere; a connected sum ${\Sigma }_g$ of $g\ge 1$ tori (orientable, $\chi =2-2g);$ or a connected sum $N_k$ of $k\ge 1$ projective planes (non-orientable, $\chi =2-k).$

Every finitely generated abelian group is ${\mathbb{Z}}^r\oplus \mathbb{Z}/d_1\oplus \cdots \oplus \mathbb{Z}/d_k$ with $d_1\mid d_2\mid \cdots \mid d_k,$ and the numbers $r,d_1,\dots ,d_k$ are determined by the group. So every homology group is a list of numbers.

The group $\mathbb{Z}/n$ of an $n$-hour clock: the elements $0,1,\dots ,n-1,$ added by adding and then wrapping around past $n.$ Never written ${\mathbb{Z}}_n$ in this book; $\mathbb{Z}/2$ is also written ${\mathbb{F}}_2.$

A set that is both open and closed, such as $\varnothing$ and the whole space. A connected space has no other clopen sets.

A differential form $\omega$ with $d\omega =0.$ A closed 1-form in the plane is one with zero curl; its integral around every small loop that bounds a region is zero.

A compact manifold with empty boundary, such as a sphere, a torus or a Klein bottle. Not to be confused with a closed set: the plane is not a closed manifold, and neither is the closed disk.

A set whose complement is open; equivalently, in a metric space, a set that contains all of its limit points. A set can be open, closed, both or neither.

A route along the edges of a graph that ends where it started. Forgetting the start and the direction, and recording only which edges are used, turns a closed walk into a cycle.

The link made by joining the bottom ends of a braid back up to the top ends. Every link is a closed braid (Alexander), and Markov’s moves say when two braids have the same closure.

Two closed $n$-manifolds are cobordant if together they form the boundary of a compact $(n+1)$-manifold. Cobordism classes form groups and a generalized homology theory; for surfaces (unoriented), a closed surface bounds exactly when its Euler characteristic is even.

A cochain of the form $\delta g$ for some cochain $g$ one degree lower — for an edge labelling, a gradient. Every coboundary is a cocycle because $\delta \delta =0;$ the converse can fail around holes.

The group $B^k=\mathrm{im}{\delta }_{k-1}$ of $k$-cochains of the form $\delta g.$ Since $\delta \delta =0,$ it sits inside the cocycle group. A 1-coboundary is a gradient.

The matrix of ${\delta }_k:C^k\to C^{k+1}$ in the bases of indicator cochains. It is the transpose of the boundary matrix: ${\delta }_k={\partial }_{k+1}^{\mathsf{T}}.$

The map $\delta :C^k\to C^{k+1}$ given by $(\delta \phi )(\sigma )=\phi (\partial \sigma ):$ the discrete gradient in degree 0, the discrete curl in degree 1. It satisfies $\delta \delta =0,$ and its matrix is the transpose of a boundary matrix, ${\delta }_k={\partial }_{k+1}^{\mathsf{T}}.$

A $k$-cochain assigns a number (or an element of a group $G)$ to every $k$-simplex, and is extended to chains by adding up. Chains are places; cochains are measurements.

A sequence of groups and maps $C^0\stackrel{\delta }{\to }{C}^1\stackrel{\delta }{\to }{C}^2\to \cdots$ that raise the degree and satisfy $\delta \circ \delta =0.$ The cochains of a complex form one, because $\delta \delta =0$ follows from $\partial \partial =0.$

The group $C^k(K;G)=\mathrm{Hom}(C_k(K),G)$ of $k$-cochains: homomorphisms from $k$-chains to $G.$ Concretely, one value in $G$ on each $k$-simplex, extended to chains by adding up with multiplicities, so $C^k(K;G)\cong G^{n_k}.$

A cochain $\phi$ with $\delta \phi =0.$ For an edge labelling: its curl is zero on every filled triangle, so it passes every local test.

The group $Z^k=\ker {\delta }_k$ of $k$-cochains with zero coboundary. A 1-cocycle is an edge labelling that passes every local test.

A face of a simplex obtained by deleting exactly one vertex: a $(k-1)$-face of a k-simplex. A k-simplex has $k+1$ of them: a triangle has three edges, a tetrahedron four triangles.

The growth of huge intermediate integers during exact elimination over $\mathbb{Z}.$ Computing modulo several primes avoids it and still detects torsion.

The group $G=h_0(\mathrm{pt})$ that a homology theory assigns to a point. On cell complexes, ordinary homology with coefficients in $G$ is the only theory satisfying all five axioms with that coefficient group.

The abelian group $G$ in which cochains take their values, such as $\mathbb{Z},$ $\mathbb{R}$ or $\mathbb{Z}/2.$ It is written after a semicolon: $C^k(K;G),$ $H^k(K;G).$ Changing the coefficients changes what a measurement can see.

A simplex one dimension up that has a given simplex $\tau$ as a face. The coboundary $\delta 1_{\tau }$ is $\pm 1$ on the cofaces of $\tau$ and $0$ elsewhere: the boundary looks down to faces, the coboundary looks up to cofaces.

An orientation of every triangle of a triangulated surface such that any two triangles sharing an edge run along it in opposite directions. Possible exactly when the surface is orientable.

Two cocycles are cohomologous when they differ by a coboundary, ${\phi }^{\prime }=\phi +\delta g,$ so that they define the same cohomology class. Two 1-cocycles on a graph are cohomologous exactly when they have the same loop sums. For differential forms (Chapter 4.4) the same word means that two closed forms differ by an exact form.

The “dual” of homology: instead of collecting pieces of a shape, it attaches measurements to them, and its groups $H^n(X)$ record the obstructions to fitting local measurements together into global ones.

An element $[\phi ]=\phi +B^k$ of a cohomology group: a cocycle up to adding coboundaries.

The quotient $H^k(K;G)=Z^k/B^k=\ker {\delta }_k/\mathrm{im}{\delta }_{k-1}:$ cocycles modulo coboundaries. It collects the global obstructions in degree $k,$ with all local detail forgotten.

The direct sum $H^{\ast }(X;R)=H^0\oplus H^1\oplus H^2\oplus \cdots$ with the cup product as multiplication. It is an invariant of homotopy type, much sharper than the groups alone.

For a linear map or homomorphism $A:X\to Y,$ the quotient $Y/\mathrm{im}A:$ what is left over in the codomain once the image is collapsed.

$X/A:$ the quotient of $X$ in which all points of $A$ are glued to a single point. For example $D^2/\partial D^2\cong S^2.$

The $\mathbb{Z}/2$ algorithm that reduces the columns of a boundary matrix from left to right while recording which original columns were added. Columns that become zero give cycles. It is the engine of persistent homology.

A picture of sets (dots) and functions (arrows) in which any two paths of arrows with the same start and end give the same composite: “all roads lead to the same place.”

A ring in which $ab=ba$ for all elements. $\mathbb{Z},$ $\mathbb{Z}/2,$ $\mathbb{Q}$ and $\mathbb{R}$ are commutative rings, and serve as coefficients for cohomology rings.

The element $[a,b]=ab{a}^{-1}{b}^{-1}$ of a group; it is the identity exactly when $a$ and $b$ commute. On the figure eight it is a loop that cannot be shrunk although it goes round each circle zero times net.

Every open cover has a finite subcover. In ${\mathbb{R}}^n,$ compact means closed and bounded (Heine–Borel): no escape routes, to infinity or to a missing edge.

Inside a fixed universe $U,$ the complement of $A$ is $A^c=U\setminus A.$ De Morgan: $(A\cup B{)}^c=A^c\cap B^c$ and $(A\cap B{)}^c=A^c\cup B^c.$

${\mathbb{CP}}^n:$ the complex lines through the origin in ${\mathbb{C}}^{n+1}.$ It has one cell in each even dimension $0,2,\dots ,2n,$ so $H_{2k}\cong \mathbb{Z}$ and odd groups vanish. Its cohomology ring is $\mathbb{Z}[x]/(x^{n+1})$ with $x$ in degree $2$ (Chapter 4.5).

For $f:X\to Y$ and $g:Y\to Z,$ $g\circ f:X\to Z$ is $(g\circ f)(x)=g(f(x)),$ read “$g$ after $f”:$ first $f,$ then $g.$ Associative, but order matters.

Doing one path and then another, each at double speed: $(\alpha \cdot \beta )(s)=\alpha (2s)$ for $s\le \frac{1}{2}$ and $\beta (2s-1)$ for $s\ge \frac{1}{2}.$ It is the multiplication of the fundamental group.

$CX=(X\times [0,1])/(X\times \{1\}):$ the cylinder over $X$ with its top collapsed to a point. The cone on a circle is a disk.

$x\equiv y(\mathrm{mod}n)$ means that $n$ divides $x-y,$ i.e. $x$ and $y$ leave the same remainder on division by $n.$ Example: $15\equiv 3(\mathrm{mod}12).$

A precise statement believed to be true but not yet proved. A conjecture may end up proved (Fermat’s last theorem), disproved (Euler’s sum of powers conjecture), shown to be undecidable (the continuum hypothesis), or stay open (Goldbach’s conjecture).

A space is connected if it cannot be split into two disjoint, non-empty open sets. The maximal connected pieces of a space are its connected components.

The surface $M\#N$ made by cutting a small open disk out of each of $M$ and $N$ and gluing the two boundary circles together. The sphere acts as zero: $S^2\#M\cong M.$

The knot $K_1\#K_2$ made by cutting each knot open at a point and joining the loose ends in pairs. The granny knot is $3_1\#3_1$ and the square knot $3_1\#\overline{3_1}.$ Genus adds up under it.

The map ${\partial }_{\ast }:H_n(C)\to H_{n-1}(A)$ in a long exact sequence: lift a cycle of $C$ to $B,$ take its boundary, and pull the result back to $A.$ For a pair it literally takes the boundary of a relative cycle.

A vector field that is the gradient of a function. Its line integrals depend only on the endpoints, and its integral around every closed loop is zero.

A function $F:X\to Y$ is constant on classes if $x\sim x^{\prime }$ implies $F(x)=F(x^{\prime }).$ Then $F=f\circ q$ for exactly one $f:X/\sim \to Y:$ $F$ “factors through” the quotient.

A function $f:X\to Y$ such that the preimage $f^{-1}(V)$ of every open set $V$ is open. It may glue points together but never tears.

A space homotopy equivalent to a point; equivalently, its identity map is homotopic to a constant map. ${\mathbb{R}}^n,$ disks and trees are contractible; circles and spheres are not.

The contrapositive of $P\Rightarrow Q$ is $\neg Q\Rightarrow \neg P.$ It always says the same thing as the original — which is how invariants prove that two shapes are different.

Said of a rule that turns maps into homomorphisms going in the opposite direction, as cohomology does: $f:X\to Y$ gives $f^{\ast }:H^k(Y)\to H^k(X).$ Preimages and transposes are contravariant too.

A functor that reverses arrows: $f:A\to B$ goes to $F(f):F(B)\to F(A),$ and $F(g\circ f)=F(f)\circ F(g).$ Equivalently, a functor ${\mathcal{C}}^{\mathrm{op}}\to \mathcal{D}.$ Cohomology, dual spaces, $\mathrm{Hom}(-,G)$ and preimages are contravariant.

The converse of $P\Rightarrow Q$ is $Q\Rightarrow P.$ It is a different statement and may be false when the original is true.

A solid is convex if the straight segment between any two of its points stays inside it — it has no dents or tunnels. Every convex polyhedron has $V-E+F=2.$

A subset of ${\mathbb{R}}^n$ that contains the straight segment between any two of its points, like a disk or a ball. Convex sets are contractible.

The dual of a product: an object $Q$ with arrows from $A$ and from $B$ such that every pair of arrows $A\to X,$ $B\to X$ comes from exactly one arrow $Q\to X.$ The disjoint union for sets and spaces; the direct sum $A\oplus B$ for abelian groups, where it is also the product.

A set of corners of a gluing diagram that end up as one point of the glued surface. The torus square has one corner class, the sphere square three.

A shifted copy $a+H=\{a+h:h\in H\}$ of a subgroup $H.$ Two cosets are either equal or disjoint, and $a+H=b+H$ exactly when $a-b\in H.$ The cosets of $3\mathbb{Z}$ in $\mathbb{Z}$ are $3\mathbb{Z},$ $1+3\mathbb{Z}$ and $2+3\mathbb{Z}.$

A set is countable if it is finite or can be listed as a sequence (a bijection with $\mathbb{N}).$ $\mathbb{Z}$ and $\mathbb{Q}$ are countable; $\mathbb{R}$ is uncountable (Cantor).

A single case in which a “for all” statement fails. One counterexample disproves it; no number of examples proves it.

Said of a rule that turns spaces into groups and maps into homomorphisms going in the same direction, as homology does: $f:X\to Y$ gives $f_{\ast }:H_k(X)\to H_k(Y).$

A linear map $\phi :V\to F$ from a vector space to its field of scalars: a linear measurement. On ${\mathbb{R}}^2$ it has the form $ax+by$ and is pictured as a stack of evenly spaced parallel lines.

For classes $a$ on $X$ and $b$ on $Y,$ the class $a\times b=p_1^{\ast }a⌣p_2^{\ast }b$ on $X\times Y,$ where $p_1,p_2$ are the projections.

A model of the projective plane in ${\mathbb{R}}^3:$ a hemisphere whose opposite rim points are zipped together along a segment ending in two pinch points.

The smallest number of crossings in any diagram of a knot: $0$ for the unknot, $3$ for the trefoil, $4$ for the figure-eight knot. It is the number in a knot’s table name, as in $5_2.$

A polyhedron with 7 vertices, 21 edges and 14 flat triangular faces, found by Ákos Császár in 1949: the seven-vertex torus built in space without self-crossings. It has no diagonals — every pair of corners is an edge.

The product of cochains, “front face times back face”: $(\phi ⌣\psi )([v_0,\dots ,v_{p+q}])=\phi ([v_0,\dots ,v_p])\psi ([v_p,\dots ,v_{p+q}]).$ It passes to cohomology and makes $H^{\ast }(X)$ a ring.

For a surface, the table of products $⟨x_i⌣x_j,[M]⟩$ of a basis of degree-1 classes. Mod 2 the torus has $\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ and the Klein bottle $\left(\begin{array}{cc}0& 1\\ 1& 1\end{array}\right).$

The product $\alpha ⌣\alpha$ of a class with itself. Over $\mathbb{Z}$ odd-degree classes on orientable surfaces square to zero; over $\mathbb{Z}/2$ squares can survive, as on ${\mathbb{RP}}^2$ and the Klein bottle.

For an edge flow, its circulation around one filled triangle; a gradient has curl zero on every triangle. In calculus (Chapter 4.3) it is circulation per unit area around a tiny loop; in the plane, $\mathrm{curl}\mathbf{F}=\partial Q/\partial x-\partial P/\partial y,$ and a paddle wheel in the flow spins at half this rate.

The rate $\kappa =d\theta /ds$ at which the direction of a curve turns per unit length; $1/r$ for a circle of radius $r.$

A point whose removal disconnects a connected space. Homeomorphisms send cut points to cut points, which tells a circle from an interval and T from X.

A space built in stages: first some points (0-cells), then arcs glued at their ends (1-cells), then disks glued along their boundary circles (2-cells), and so on. The sphere is one point plus one disk; the torus is one point, two loops and one disk.

In a graph (mod 2): a set of edges in which every vertex has even degree — no loose ends. More generally, a k-cycle is a collection of k-dimensional pieces with no loose $(k-1)$-dimensional pieces; in symbols, a chain with zero boundary.

The number $b_1=E-V+c$ of independent cycles of a graph, also called its cyclomatic number or first Betti number.

The set of all cycles of a graph, added by symmetric difference. It is a vector space over $\mathbb{Z}/2$ whose dimension is $b_1=E-V+c.$

$Z_k=\ker {\partial }_k:$ the k-chains with zero boundary. Since ${\partial }_0=0,$ every 0-chain is a cycle.

A group generated by a single element: $G=⟨a⟩.$ Every cyclic group is isomorphic to $\mathbb{Z}$ or to some $\mathbb{Z}/n.$

The square with one pair of opposite edges glued straight across, $(0,y)\sim (1,y);$ also the product $S^1\times [0,1].$ It has two boundary circles.

A space glued from simplices more freely than a simplicial complex allows: faces of one simplex may be glued together, and several simplices may share the same vertices. The torus is a Δ-complex with 1 vertex, 3 edges and 2 triangles.

$f$ is continuous at $a$ if for every $\epsilon >0$ there is a $\delta >0$ such that $\left|x-a\right|<\delta$ implies $\left|f(x)-f(a)\right|<\epsilon :$ close enough inputs give close outputs.

An additive category in which every arrow has a kernel and a cokernel and the first isomorphism theorem holds. Abelian groups, vector spaces, chain complexes and sheaves of abelian groups are examples; exact sequences, homology and all of homological algebra make sense in any of them.

A group in which order never matters: $ab=ba$ for all $a,b.$ Named after Niels Henrik Abel. Abelian groups are usually written additively ($+,$ $0,$ $-a);$ all the groups of homology are abelian.

The quotient $G^{\mathrm{ab}}=G/[G,G]$ of a group by the subgroup generated by its commutators: the group made commutative in the most economical way. For a free group it keeps only the exponent sum of each letter, and $H_1(X)$ is the abelianized ${\pi }_1(X)$ (Hurewicz).

A finite set of vertices with a collection of non-empty subsets (the simplices) that contains every single vertex and every non-empty subset of each of its sets. A shape stored as a list of vertex sets.

A category in which arrows between any two objects form an abelian group, composition distributes over addition, there is a zero object, and finite direct sums exist (each both a product and a coproduct).

The convention of writing an abelian group operation as $a+b,$ the identity as $0,$ the inverse as $-a,$ and $a+\cdots +a$ ($k$ times) as $ka.$ A plus sign promises that the order of terms does not matter.

A quantum particle travelling around a solenoid acquires a phase $q\Phi /\hslash$ per turn, although the magnetic field vanishes along its path. The phase is set by the integral of the vector potential around the solenoid, which depends only on its de Rham class.

For a compact, locally contractible, nonempty proper subspace $K$ of the sphere $S^n,$ ${\tilde{H}}_i(S^n\setminus K)\cong {\tilde{H}}^{n-i-1}(K):$ the homology of the space around $K$ depends only on the cohomology of $K.$ It implies the Jordan curve theorem, and that every knot complement has $H_1\cong \mathbb{Z}.$

A Laurent polynomial ${\Delta }_K(t)$ attached to a knot, playing the role of the characteristic polynomial of the shift acting on $H_1$ of the infinite cyclic cover of the complement; $\Delta =\det (V-t{V}^{\mathsf{T}})$ for a Seifert matrix $V.$ For the trefoil, $t-1+t^{-1}.$

A continuous family of homeomorphisms $h_t$ of space, starting at the identity, that carries one knot to another: the string moves and drags the space along, never passing through itself. Two knots are equivalent when one exists.

For a steady current $I,$ the circulation of the magnetic field around a loop is ${\mu }_{0}I$ times the number of times the loop links the current. Outside a wire the field is closed but not exact.

$P\wedge Q$ holds when both hold; $P\vee Q$ when at least one holds (“or” is inclusive); $\neg P$ when $P$ fails. De Morgan: $\neg (P\wedge Q)\equiv \neg P\vee \neg Q.$

At a corner of a polyhedron, the amount by which the angles of the faces meeting there fall short of a full turn, $360°$ (or $2\pi$ in radians). A cube has defect $90°$ at each corner; a corner can also have negative defect. The defects of a closed polyhedron add up to $360°\cdot \chi ,$ that is $2\pi \chi :$ the polyhedral Gauss–Bonnet theorem of Chapter 4.8.

For a geodesic triangle, $\alpha +\beta +\gamma -\pi .$ Gauss proved it equals the total curvature ${\iint }_{T}KdA$ inside the triangle.

The 1-form $d\theta =(-ydx+xdy)/(x^2+y^2)$ on the punctured plane. It is closed but not exact: its integral around any loop is $2\pi$ times the loop’s winding number about the origin.

For a set $U$ of covectors, the subspace $U^{\perp }$ of vectors that every covector in $U$ sends to 0. Duality theorem: $\mathrm{im}A=(\ker A^{\mathsf{T}}{)}^{\perp },$ so a target is unreachable exactly when some measurement certifies it.

The map $x\mapsto -x$ of a sphere, sending each point to the opposite point. On $S^n$ it has degree $(-1{)}^{n+1}:$ orientation-preserving on odd spheres, reversing on even ones.

A loop of currency trades that ends with more money than it began with. Using logarithms of exchange rates as an edge labelling, a market is free of arbitrage exactly when the labelling is a gradient.

The rule $(a\ast b)\ast c=a\ast (b\ast c):$ brackets do not matter, so a long product can be written without them. Subtraction is not associative: $(5-3)-1\ne 5-(3-1).$

A collection of charts whose domains cover the whole manifold. The sphere needs at least two charts; the six hemispheres, or the two stereographic projections, form atlases.

The continuous map that says where the boundary of a new $n$-cell is glued onto the part of a CW complex already built. The torus’s 2-cell is attached along the loop $ab{a}^{-1}{b}^{-1}.$

The homomorphism $\epsilon :C_0\to \mathbb{Z}$ that adds up the coefficients of a 0-chain. It vanishes on boundaries, since $\epsilon ([v]-[u])=0.$

For an ordered simplex $[v_0,\dots ,v_{p+q}],$ the face $[v_p,\dots ,v_{p+q}]$ made of its last vertices. It shares exactly one vertex, $v_p,$ with the front face. The right factor of a cup product is read on it.

The collection of intervals $[b,d),$ one for each class in a suitable basis, recording when the homology classes of a filtration are born and die, drawn as horizontal bars. The number of $k$-bars containing $r$ is $b_k$ at $r.$

The centre of a simplex: the point whose barycentric coordinates are all equal, $\frac{1}{n+1}.$ The barycentre of an edge is its midpoint.

Numbers $t_0,\dots ,t_n\ge 0$ with $t_0+\cdots +t_n=1$ describing the point $t_0{v}_0+\cdots +t_n{v}_n$ of a simplex: the balance point of weights $t_i$ hung at the corners. The non-zero coordinates name the face the point lies in.

The finer complex $\mathrm{sd}K$ with a vertex at the barycentre of every simplex of $K,$ and a simplex for every chain of faces ${\sigma }_0\subset {\sigma }_1\subset \dots .$ A triangle becomes 6 triangles; an $n$-simplex becomes $(n+1)!$ pieces.

A space with a chosen point (its basepoint). Based maps send basepoint to basepoint; based spaces and based maps form the category ${\mathsf{Top}}_{\ast },$ on which the fundamental group is a functor.

A chosen point $x_0$ of a space at which all loops start and end, needed to define the fundamental group ${\pi }_1(X,x_0).$

A set of elements such that every element is a combination of them with integer coefficients in exactly one way. In ${\mathbb{Z}}^2,$ $\{(1,0),(1,1)\}$ is a basis but $\{(2,0),(0,1)\}$ is not, because we may not divide by 2.

A linearly independent list of vectors that spans the space. Every vector is then a combination of the basis vectors in exactly one way; the coefficients are its coordinates.

The geometric phase acquired by a quantum system whose parameters are carried slowly around a loop; it is the holonomy of a line bundle over the parameter space, and its curvature integrates to Chern numbers.

The rank $b_k$ of $H_k:$ the number of independent $k$-dimensional holes. $b_0$ counts pieces, $b_1$ independent loops, $b_2$ cavities. Over a field $F$ it is $\dim H_k(K;F).$ Often written ${\beta }_k.$

Both injective and surjective: a perfect matching between domain and codomain. Exactly the functions that have an inverse.

A rule that combines two elements $a,b$ of a set, in order, into one element $a\ast b.$ Examples: $+$ on numbers, composition $\circ$ of moves.

In a filtration, a homology class is born at the first parameter value where it exists without coming from an earlier class, and dies when it becomes zero or merges into an older class.

Complex K-theory repeats with period two: $K^n(\mathrm{pt})$ is $\mathbb{Z}$ for even $n$ and $0$ for odd $n.$ Proved by Raoul Bott in 1959.

The distance between two persistence diagrams: the smallest $\delta$ such that their points can be matched one-to-one, each moving at most $\delta$ in each coordinate, where a point may also be matched to the diagonal at cost $(d-b)/2.$

$B_k=\mathrm{im}{\partial }_{k+1}:$ the k-chains of the form $\partial c.$ Because $\partial \partial =0,$ $B_k\subseteq Z_k;$ the gap between them is homology.

A cycle that is the rim of something filled in: in dimension 1, the rim of a set of filled triangles; in dimension k, the rim of a collection of $(k+1)$-dimensional pieces. Every boundary is a cycle; the converse fails exactly where there are holes.

The set $\partial M$ of points that a chart sends to the edge of the half-space. It is a manifold of one dimension less, with no boundary of its own: $\partial (\partial M)=\varnothing .$ For example $\partial D^2=S^1.$

$\partial [v_0,\dots ,v_k]={\sum }_{i=0}^k(-1{)}^i[v_0,\dots ,{\hat{v}}_i,\dots ,v_k]:$ the alternating sum of the faces, the i-th leaving out the i-th vertex. For an edge, $\partial [v_0,v_1]=v_1-v_0$ (head minus tail).

The matrix of ${\partial }_k:$ one row per $(k-1)$-simplex, one column per k-simplex; column j lists the coefficients of the boundary of the j-th simplex. Consecutive boundary matrices multiply to zero.

The map ${\partial }_k:C_k\to C_{k-1}$ sending a simplex to the (signed) sum of its codimension-one faces and extended to chains by linearity. Mod 2, $\partial c$ is the set of faces occurring an odd number of times: the loose ends of a set of edges, the rim of a set of triangles.

The orientation a boundary inherits from the region it bounds: “outward first”. In the plane the boundary is walked with the region on the left, so the outer edge goes counterclockwise.

A smooth immersion of the projective plane in ${\mathbb{R}}^3,$ found by Werner Boy in 1901, with three-fold symmetry and a single triple point.

The group $B_n$ of braids on $n$ strands, multiplied by stacking. Artin showed it is generated by the crossings ${\sigma }_1,\dots ,{\sigma }_{n-1}$ subject to ${\sigma }_i{\sigma }_j={\sigma }_j{\sigma }_i$ for $|i-j|\ge 2$ and ${\sigma }_i{\sigma }_{i+1}{\sigma }_i={\sigma }_{i+1}{\sigma }_i{\sigma }_{i+1}.$

Euler’s 1736 puzzle: a round trip crossing each of seven bridges once is impossible, because a closed walk must touch every land mass an even number of times. The first instance of the parity test for cycles.

Every continuous map from the closed disk $D^n$ to itself has a fixed point. Proof: otherwise the rays from $f(x)$ through $x$ would define a retraction of the disk onto its boundary.

Applying the exterior derivative twice always gives zero. It packages $\mathrm{curl}\mathrm{grad}=0$ and $\mathrm{div}\mathrm{curl}=0,$ and mirrors $\partial \circ \partial =0$ for chains.

The vector space $H_{\mathrm{dR}}^k(M)$ of closed $k$-forms on $M$ modulo exact ones: it measures how far closed forms are from being exact, which happens exactly when $M$ has holes.

For every smooth manifold $M,$ integration over chains gives an isomorphism $H_{\mathrm{dR}}^k(M)\cong H^k(M;\mathbb{R}):$ the smooth and the combinatorial cohomology agree (with real coefficients).

$A:=B$ means “$A$ is defined to be $B”:$ not a claim but the introduction of a new name.

A homomorphism out of a free abelian group can be defined by choosing, freely, where each basis element goes; it then extends uniquely: $\phi (\sum n_{s}s)=\sum n_s\phi (s).$ The boundary map of homology is defined this way.

A precise membership test that introduces a new word. By an almost-universal convention, definitions are written with “if” but mean “if and only if”.

A homotopy that starts at the identity of $X,$ ends with everything inside a subspace $A,$ and never moves the points of $A.$ Then $A\simeq X.$ Example: the punctured plane onto a circle.

A class in $H^p$ has degree $p,$ written $|\alpha |=p.$ Degrees add under the cup product.

The number of edges of the set that touch the vertex.

For $f:S^n\to S^n,$ the integer $d$ such that $f_{\ast }:H_n(S^n)\to H_n(S^n)$ is multiplication by $d.$ It counts preimages of a point with signs; for $n=1$ it is the winding number.

The rate of change of a function: the slope $F^{\prime }(x)$ of its graph at $x.$ For a small step $\Delta x$ it predicts the change $\Delta F\approx F^{\prime }(x)\Delta x.$

The functors obtained by applying a functor that is only partly exact to a resolution and taking homology: $\mathrm{Tor}$ comes from $\otimes ,$ $\mathrm{Ext}$ from $\mathrm{Hom},$ and sheaf cohomology from taking global sections. They record, degree by degree, how exactness fails.

The angle defects of a convex polyhedron add up to exactly $720°.$ More generally, for any closed polyhedral surface they add up to $360°\times \chi$ — a discrete version of the Gauss–Bonnet theorem.

For a $2\times 2$ matrix, $ad-bc:$ the signed factor by which the map multiplies areas. It is zero exactly when the columns are dependent, so that the map flattens the plane.

The absolute value of the determinant of the colouring equations with one row and one column struck out; it equals $|\Delta (-1)|.$ A knot has non-constant $p$-colourings exactly when $p$ divides it.

The map $X\to X\times X,$ $x\mapsto (x,x).$ Every space has one; pulling measurements back along it is what lets cohomology multiply.

A proof that follows a single element around a commutative diagram, using commutativity and exactness at each step. The snake lemma is the classic example.

Another name for the maps ${\partial }_k$ of a chain complex, borrowed from the $d$ of calculus, which also satisfies $d\circ d=0.$

For a smooth map $f:M\to N,$ the linear map $d{f}_p:T_{p}M\to T_{f(p)}N$ that sends the velocity of a curve $\gamma$ to the velocity of $f\circ \gamma :$ the best linear approximation to $f$ near $p.$ It pushes vectors forward and is also written $f_{\ast }.$

A covector at every point, varying smoothly: $\omega =Pdx+Qdy.$ It eats a small step and returns a number, so it can be integrated along oriented curves.

A smoothly varying measuring device for tiny oriented $k$-dimensional parallelograms, such as $gdx\wedge dy.$ A $k$-form is integrated over oriented $k$-dimensional pieces; the space of them is written ${\Omega }^k(M).$

The six symmetries of an equilateral triangle: three rotations $e,r,r^2$ and three flips $f_1,f_2,f_3.$ It is the smallest group that is not abelian: $r{f}_1\ne f_{1}r.$

The number of vectors in any basis of a vector space: its number of degrees of freedom. For example $\dim {\mathbb{R}}^n=n,$ and $(\mathbb{Z}/2{)}^n$ has dimension $n$ and $2^n$ elements.

The Eilenberg–Steenrod axiom saying that a point has homology only in degree 0. Dropping it gives generalized homology and cohomology theories such as K-theory and cobordism.

For a charge $q$ and a monopole of strength $g,$ $2qg/\hslash$ must be an integer (units with $c=1);$ so a single monopole would force all electric charges to be whole multiples of one unit.

$G\oplus H,$ the pairs $(g,h)$ added coordinate by coordinate: two groups side by side, like two independent dials. Examples: ${\mathbb{Z}}^2=\mathbb{Z}\oplus \mathbb{Z},$ and $\mathbb{Z}/2\oplus \mathbb{Z}/2,$ the symmetries of a rectangle.

The sum of an edge labelling around the boundary of a filled triangle $[a,b,c]:$ $\psi (ab)+\psi (bc)-\psi (ac).$ It is the coboundary $\delta \psi$ evaluated on the triangle, and the curl of a gradient is always zero.

The edge labelling $\delta f$ made from a potential $f:$ on an edge from $u$ to $v,$ $(\delta f)(u\to v)=f(v)-f(u),$ “head minus tail”. Edge labellings of this form are called gradients.

A systematic way of pairing off simplices that can be collapsed, leaving a much smaller complex (or chain complex) with the same homology.

$⟨\delta \phi ,c⟩=⟨\phi ,\partial c⟩$ for every cochain $\phi$ and chain $c:$ measuring the change of $\phi$ over a region equals measuring $\phi$ on the region’s boundary. In degree 0 it is telescoping.

The topology in which every subset is open: each point is isolated from all the others. It comes from the discrete metric.

Two spaces side by side, not touching: $X\bigsqcup Y.$

$D^n:$ the points at distance at most $1$ from the origin in ${\mathbb{R}}^n.$ Its boundary $\partial D^n$ is the sphere $S^{n-1}.$

Outward flux per unit area (or volume) out of a tiny region; in the plane, $\mathrm{div}\mathbf{F}=\partial P/\partial x+\partial Q/\partial y.$ Positive at sources, negative at sinks.

The flux of a field out through the boundary surface of a solid equals the total divergence inside: ${\iint }_{\partial V}\mathbf{F}\cdot \mathbf{n}dA={\iiint }_V\mathrm{div}\mathbf{F}dV.$ Also called Gauss’s theorem.

$n\mid m,$ read “$n$ divides $m”,$ means $m=nk$ for some integer $k.$ It is a statement (true or false), not a fraction.

For $f:X\to Y,$ the domain $X$ is where inputs come from and the codomain $Y$ is where outputs must land. Changing either one changes the function.

The set where an immersed surface passes through itself — drawn in rose on the Klein bottle and Boy’s surface.

The dual of the dual, $V^{\ast \ast }.$ For finite-dimensional $V,$ evaluation $v\mapsto (\phi \mapsto \phi (v))$ is an isomorphism $V\to V^{\ast \ast }$ that needs no choice of basis — the standard example of a natural isomorphism.

To show two sets are equal, show every element of the first is in the second and every element of the second is in the first: $A\subseteq B$ and $B\subseteq A.$

For a basis ${\mathbf{e}}_1,\dots ,{\mathbf{e}}_n,$ the covectors ${\mathbf{e}}_i^{\ast }$ that read off coordinates: ${\mathbf{e}}_i^{\ast }({\mathbf{e}}_j)$ is 1 when $i=j$ and 0 otherwise. Each depends on the whole basis.

For a $k$-simplex $\sigma$ of a triangulated $n$-manifold, the $(n-k)$-dimensional cell ${\sigma }^{\ast }$ built from the barycentric subdivision, running outwards from the barycentre of $\sigma .$ It crosses $\sigma$ in exactly one point: a vertex has a dual face, an edge a dual edge, a triangle of a surface a dual vertex.

The cell decomposition $K^{\ast }$ of a triangulated closed manifold formed by all the dual cells. Matching each simplex with its dual cell turns cochains of $K$ into chains of $K^{\ast }$ and the coboundary into the boundary — the idea behind Poincaré duality.

For a network drawn on a sphere or in the plane: the graph with one vertex in each face, and one edge crossing each edge of the original. Used in the "two trees" proof of Euler’s formula.

The vector space $V^{\ast }$ of all covectors on $V.$ For finite-dimensional $V,$ $\dim V^{\ast }=\dim V;$ vectors are columns and covectors are rows.

For a closed oriented $n$-manifold, the map $D(\phi )=[M]⌢\phi$ from $H^p(M)$ to $H_{n-p}(M).$ The modern statement of Poincaré duality is that it is an isomorphism.

The space obtained from a triangle by gluing its sides according to the word $aa{a}^{-1}.$ It is contractible, although no triangulation of it can be collapsed face by face to a point.

$T^2\#{\mathbb{RP}}^2\cong {\mathbb{RP}}^2\#{\mathbb{RP}}^2\#{\mathbb{RP}}^2$ (Walther von Dyck, 1888): in the presence of a cross-cap, a handle can be traded for two cross-caps. So mixing handles and cross-caps produces nothing new.

A choice of a number (or of an element of a group) on every oriented edge — the same thing as a 1-cochain. Reversing the arrow on an edge changes the sign of its number. Also called an edge flow.

The letters read once around a gluing diagram (here anticlockwise from the bottom-left corner), written $x$ along an arrow and $x^{-1}$ against it; e.g. $ab{a}^{-1}{b}^{-1}$ for the torus.

A space $K(G,n)$ whose only nonzero homotopy group is ${\pi }_n=G.$ It represents cohomology: $H^n(X;G)\cong [X,K(G,n)]$ for cell complexes $X.$ Examples: $S^1=K(\mathbb{Z},1)$ and ${\mathbb{CP}}^{\infty }=K(\mathbb{Z},2).$

Five axioms for a homology theory on pairs of spaces: homotopy invariance, a long exact sequence for every pair, excision, additivity, and the dimension axiom (a point has homology only in degree 0). On cell complexes they determine homology from the coefficient group.

When two classes merge in a filtration, the older one (born earlier) survives and the younger one dies. In a Rips filtration all points are born at once, and ties are broken by labels.

$x\in X,$ read “$x$ is an element of $X”,$ means that $x$ is one of the things in the set $X;$ $x\notin X$ means it is not.

Removing a simplex together with a free face, a face that belongs to no other simplex. It does not change homology, and is used to shrink complexes before computing.

A one-to-one continuous map that is a homeomorphism onto its image: a faithful copy of one space inside another.

The unique set with no elements, $\varnothing .$ It is a subset of every set. Note that $\{\varnothing \}$ is not empty: it has one element.

The parameter value $f(\sigma )$ at which a simplex enters a filtration; a face never enters after the simplices containing it. For Vietoris–Rips it is half the largest distance between the simplex’s points.

The class of $x$ is $[x]=\{y\in X:y\sim x\},$ everything equivalent to $x.$ Two classes are either equal or disjoint, and $[x]=[y]$ exactly when $x\sim y.$

Functors $F:\mathcal{C}\to \mathcal{D}$ and $G:\mathcal{D}\to \mathcal{C}$ whose round trips $G\circ F$ and $F\circ G$ are naturally isomorphic to the identity functors. Finite-dimensional vector spaces are equivalent to the category of matrices.

A relation that is reflexive, symmetric and transitive: the precise meaning of “declaring things the same.” Example: congruence modulo $n$ on $\mathbb{Z}.$

A homology class that never dies in a filtration, giving an infinite bar $[b,\infty ).$ In the full Rips filtration of a point cloud there is exactly one: the class of the single final piece in $H_0.$

A cohomology theory built by Grothendieck and his school for shapes defined by polynomial equations over any field, including finite fields. With it Pierre Deligne proved the last of the Weil conjectures in 1974.

The alternating count $\chi =n_0-n_1+n_2-\cdots$ of the vertices, edges, faces, … of a complex — for a polyhedron, $V-E+F.$ It depends only on the space, not on how it is cut up: $\chi (S^2)=2,$ $\chi (T^2)=0,$ $\chi ({\mathbb{RP}}^2)=1.$

Every closed manifold of odd dimension has $\chi =0:$ in ${\sum }_k(-1{)}^k{b}_k,$ the terms for $k$ and $n-k$ have equal Betti numbers but opposite signs, so they cancel.

$\chi (A\#B)=\chi (A)+\chi (B)-2$ for closed surfaces: cutting out a disk lowers each $\chi$ by 1, and gluing along the circles changes nothing. Adding a handle lowers $\chi$ by 2.

$\chi =V-E,$ which equals the number of connected pieces minus the number of independent loops. A tree has $\chi =1;$ a circle has $\chi =0.$

The characteristic class $e(E)\in H^n(B;\mathbb{Z})$ of an oriented rank-$n$ bundle: the obstruction to a nowhere-zero section, the signed count of zeros of a generic section. For a tangent bundle, $e(TM)=\chi (M).$

$\chi (K)=\sum (-1{)}^k{n}_k=\sum (-1{)}^k{b}_k:$ the alternating count of simplices equals the alternating sum of Betti numbers, over any field.

$V-E+F=2$ for every polyhedron whose surface is a sphere in disguise (for instance every convex polyhedron). Announced by Euler in 1750, published in 1758.

An ordering of labels is even or odd according to whether it takes an even or odd number of swaps to reach it from the increasing order — equivalently, whether the number of pairs out of order is even or odd. $[1,2,0]$ is even; $[1,0,2]$ is odd.

A differential form that is the exterior derivative of another, $\omega =d\eta .$ Exact forms are closed, because $d\circ d=0;$ an exact 1-form has zero integral around every closed loop.

A chain of homomorphisms $\cdots \to A\stackrel{f}{\to }B\stackrel{g}{\to }C\to \dots$ in which, at every inner group, the image of the incoming map equals the kernel of the outgoing one: what $f$ produces is exactly what $g$ kills.

If $Z\subseteq A\subseteq X$ and the closure of $Z$ lies in the interior of $A,$ then $H_n(X\setminus Z,A\setminus Z)\cong H_n(X,A):$ what happens deep inside $A$ does not matter. It makes homology local.

The number of times a letter occurs in a word, counting $a$ as $+1$ and $a^{-1}$ as $-1.$ For a disk glued along a word, it is the coefficient of that edge in the cellular boundary of the face.

$\mathrm{Ext}(A,G)$ is the cohomology in degree 1 of $\mathrm{Hom}(F_{\bullet },G),$ for a free resolution $F_{\bullet }$ of $A.$ It measures how $\mathrm{Hom}(-,G)$ fails to be exact and classifies extensions: $\mathrm{Ext}(\mathbb{Z}/n,G)\cong G/nG$ and $\mathrm{Ext}(\mathbb{Z},G)=0.$

A group $\mathrm{Ext}(A,G)$ built from abelian groups $A$ and $G,$ defined properly in the chapter on homological algebra. For finitely generated groups: $\mathrm{Ext}(\mathbb{Z},G)=0,$ $\mathrm{Ext}(\mathbb{Z}/n,G)\cong G/nG,$ and $\mathrm{Ext}$ of a direct sum is the direct sum.

Defining a map on all chains from its values on single simplices by $\partial (\sum a_{\sigma }\sigma )=\sum a_{\sigma }\partial \sigma .$ The result respects sums: $\partial (c+c^{\prime })=\partial c+\partial c^{\prime }.$

A short exact sequence $0\to A\to B\to C\to 0,$ viewed as a way of building $B$ from a subgroup $A$ and a quotient $C.$ For instance $\mathbb{Z}/4$ and $\mathbb{Z}/2\oplus \mathbb{Z}/2$ are both extensions of $\mathbb{Z}/2$ by $\mathbb{Z}/2.$

The ring generated by elements that anticommute and square to zero, like $dx,dy.$ The cohomology ring of the torus is the exterior algebra $\Lambda [\alpha ,\beta ];$ that of $T^n$ is the exterior algebra on $n$ generators.

The operator $d:{\Omega }^k\to {\Omega }^{k+1}$ with $d(fd{x}_I)=df\wedge d{x}_I.$ In ${\mathbb{R}}^3$ it is the gradient, curl and divergence; pictorially it marks where the sheets of a form end.

The list $(n_0,n_1,n_2,\dots )$ of the numbers of vertices, edges, triangles, … of a simplicial (or cell) complex. The hollow tetrahedron has f-vector $(4,6,4).$

A simplex spanned by some (at least one) of the vertices of a given simplex. An $n$-simplex has $\left(\genfrac{}{}{0ex}{}{n+1}{k+1}\right)$ faces of dimension $k$ — row $n+1$ of Pascal’s triangle — and $2^{n+1}-1$ faces in all.

A simplex of a complex that is not a face of any larger simplex in it. Listing the facets describes the whole complex: the hollow triangle has facets $\{0,1\},\{0,2\},\{1,2\}.$

A picture of a 1-cocycle on a surface: curves crossing the edges, as many times (with sign) as the cocycle’s value on each edge. The value of the cocycle on a closed loop is the number of times the loop crosses the fence, counted with sign. A fence coming from a coboundary looks like the level curves of a function.

The vector space $E_b={\pi }^{-1}(b)$ of a vector bundle $\pi :E\to B$ sitting over a point $b$ of the base.

A number system in which you can add, subtract, multiply and divide by anything except zero, with the usual rules of arithmetic. Examples: $\mathbb{Q},$ $\mathbb{R}$ and $\mathbb{Z}/2.$ The integers $\mathbb{Z}$ are not a field, because $2$ has no integer reciprocal.

A chain complex $C$ with a nested sequence of subcomplexes $F_{0}C\subseteq F_{1}C\subseteq \cdots \subseteq C$ — for example the chains of the skeleta of a cell complex. Its spectral sequence computes $H(C)$ from the pieces $F_{p}C/F_{p-1}C.$

A nested family of subcomplexes $K_r\subseteq K_s$ (for $r\le s)$ of a simplicial complex — a “film” of growing complexes, such as the Vietoris–Rips complexes of a point cloud as the radius grows.

A group is finitely generated if finitely many of its elements generate it. ${\mathbb{Z}}^3$ and every finite group are; $\mathbb{Q}$ is not. Homology groups of finite complexes are finitely generated.

The characteristic class $c_1(L)\in H^2(B;\mathbb{Z})$ of a complex line bundle. Complex line bundles are classified by it; over the sphere it is a single integer, the Chern number.

For a homomorphism $\phi :G\to H,$ the rule $a+\ker \phi \mapsto \phi (a)$ is a well-defined isomorphism $G/\ker \phi \cong \mathrm{im}\phi :$ collapse what $\phi$ cannot see, and what remains is exactly what it reaches.

A point $x$ with $f(x)=x:$ a point the map does not move.

A sign (or integer) attached to an isolated fixed point, measuring how the map turns around it; for a nondegenerate fixed point it is the sign of $\det (I-Df).$ The indices of all fixed points add up to the Lefschetz number.

A simplicial complex determined by its edges: a set of vertices spans a simplex exactly when every two of them are joined by an edge. Also called a clique complex.

The flat network obtained by removing one face of a polyhedron and stretching the rest out on the plane; the removed face becomes the outside region. Used in Cauchy’s proof of Euler’s formula.

The net amount of a flow crossing a curve or surface, $\int \mathbf{F}\cdot \mathbf{n},$ counted positively in the direction of the chosen normal $\mathbf{n}.$

$\forall x,$ “for every $x”.$ To prove a ∀-statement, argue for an arbitrary $x;$ to disprove it, one counterexample is enough.

A functor that forgets structure, such as $\mathsf{Grp}\to \mathsf{Set},$ which sends a group to its underlying set and a homomorphism to its underlying function.

An integer combination of named pieces, such as $2a-b+3c,$ kept as an inventory rather than evaluated. Formal sums are added coefficient by coefficient. Chains in homology are formal sums of edges, triangles, and so on.

A labelling of the arcs of a diagram by elements of $\mathbb{Z}/p$ such that at every crossing twice the over-arc equals the sum of the two under-arcs. Their number is a knot invariant: $9$ mod $3$ for the trefoil, $3$ for the unknot.

The group $\mathbb{Z}[S]$ of formal sums of the elements of a set $S$ with integer coefficients. If $S$ has $k$ elements it is a copy of ${\mathbb{Z}}^k.$ The chain groups of homology are free abelian groups.

The group of reduced words in some letters and their inverses, multiplied by joining words and cancelling. The fundamental group of the figure eight is the free group on two letters $a,b;$ it is not abelian.

An exact sequence $\cdots \to F_2\to F_1\to F_0\to A\to 0$ in which every $F_i$ is free abelian. Every abelian group has one of length one, $0\to F_1\to F_0\to A\to 0:$ generators and the relations among them.

For an ordered simplex $[v_0,\dots ,v_{p+q}],$ the face $[v_0,\dots ,v_p]$ made of its first vertices. In $\phi ⌣\psi$ the left factor $\phi$ is read on the front face.

A rule $f:X\to Y$ assigning to each element $x$ of the domain $X$ exactly one element $f(x)$ of the codomain $Y.$ Written $x\mapsto f(x).$

A map between categories, $F:\mathcal{C}\to \mathcal{D}:$ it sends objects to objects and arrows to arrows, preserving identities and composition, $F(g\circ f)=F(g)\circ F(f).$ Homology $H_n:\mathsf{Top}\to \mathsf{Ab}$ is the motivating example; the word first appears for it in Chapter 3.5.

The rules $(g\circ f{)}_{\ast }=g_{\ast }\circ f_{\ast }$ and $(\mathrm{id}{)}_{\ast }=\mathrm{id}:$ homology turns spaces into groups and maps into homomorphisms, respecting composition. This is what it means for homology to be a functor.