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Reference

Notation

How to read every symbol aloud, what it means, and where it first appears.

Logic

Symbol	Read aloud	Meaning	Introduced
$P \land Q$	P and Q	both statements are true	§0.2
$P \lor Q$	P or Q	at least one is true (inclusive or)	§0.2
$\neg P$	not P	the negation of P	§0.2
$P \Rightarrow Q$	P implies Q; if P then Q	whenever P holds, Q holds	§0.2
$P \Leftrightarrow Q$	P if and only if Q	P and Q are equivalent	§0.2
$\forall x$	for all x	universal quantifier	§0.2
$\exists x$	there exists an x	existential quantifier	§0.2
$:=$	is defined to be	introduces a new name	§0.2
$■ or □$	end of proof	the argument is complete	§0.2

Sets and functions

Symbol	Read aloud	Meaning	Introduced
$x \in A$	x is an element of A	membership	§1.1
$A \subseteq B$	A is a subset of B	every element of A is in B	§1.1
$\emptyset$	the empty set	the set with no elements	§1.1
$A \cup B, A \cap B, A ∖ B$	union, intersection, difference	combining sets	§1.1
$A \times B$	A cross B	all ordered pairs (a, b)	§1.1
${x \in X ∣ P (x)}$	the set of x in X such that P(x)	set-builder notation	§1.1
$f : X \to Y$	f is a function from X to Y	X is the domain, Y the codomain	§1.1
$x \mapsto f (x)$	x maps to f of x	what f does to an element	§1.1
$g \circ f$	g after f	composition: first f, then g	§1.1
$f^{- 1} (B)$	the preimage of B under f	everything f sends into B (exists even if f has no inverse)	§1.1
$id_{X}$	the identity on X	the function that changes nothing	§1.1

Sameness and quotients

Symbol	Read aloud	Meaning	Introduced
$x \sim y$	x is equivalent to y	an equivalence relation	§1.2
$[x]$	the class of x	everything equivalent to x	§1.2
$X / \sim$	X mod tilde	the quotient set of classes	§1.2
$≅$	is isomorphic to / homeomorphic to	the same up to renaming / stretching	§1.3
$≃$	is homotopy equivalent to	the same up to deformation	§2.3

Groups and linear algebra

Symbol	Read aloud	Meaning	Introduced
$Z, Q, R, C$	integers, rationals, reals, complex numbers	the number systems	§0.2
$Z / n$	Z mod n	integers on a clock with n hours	§1.3
$ker φ, im φ$	kernel, image	what is crushed to zero; what is reached	§1.3
$G / H$	G mod H	quotient group: collapse each coset to a point	§1.4
$A \oplus B$	A direct sum B	pairs (a, b), added componentwise	§1.4
$Z^{r} \oplus Z / d_{1} \oplus \dots$	free part plus torsion	every finitely generated abelian group looks like this	§1.4
$rank A$	rank of A	the number of independent columns / the size of the free part	§1.5
$V^{*}$	V dual	the space of measurements (linear functions) on V	§1.5
$A^{T}$	A transpose	rows become columns; pulls measurements back	§1.5

Spaces

Symbol	Read aloud	Meaning	Introduced
$R^{n}$	R n	n-dimensional Euclidean space	§2.1
$S^{n}, D^{n}$	the n-sphere, the n-disk	S¹ is a circle, S² the surface of a ball	§2.1
$T^{2}, K, RP^{2}, Σ_{g}$	torus, Klein bottle, projective plane, genus-g surface	the surfaces of the book	§2.2
$X \lor Y$	X wedge Y	two spaces joined at a point	§2.2
$X / A$	X mod A	crush the subspace A to a single point	§2.2
$π_{1} (X)$	pi one of X	the fundamental group: loops up to deformation	§2.3
$χ (X)$	chi of X	Euler characteristic: V − E + F and its generalisations	§2.6

Chains and homology

Symbol	Read aloud	Meaning	Introduced
$[v_{0}, v_{1}, \dots, v_{k}]$	the oriented simplex v-zero … v-k	a k-simplex with a chosen order of vertices	§2.5
$C_{k} (K; G)$	k-chains of K with coefficients in G	formal sums of k-simplices	§3.2
$\partial_{k}$	boundary	sends a simplex to the signed sum of its faces	§3.2
$\partial \circ \partial = 0$	the boundary of a boundary is zero	the fundamental identity	§3.2
$Z_{k}, B_{k}$	k-cycles, k-boundaries	kernel of ∂ₖ, image of ∂ₖ₊₁	§3.3
$H_{k} (X) = Z_{k} / B_{k}$	the k-th homology of X	cycles modulo boundaries: k-dimensional holes	§3.3
$b_{k}$	the k-th Betti number	the rank of Hₖ: how many independent k-holes	§3.3
$H ~_{k}$	reduced homology	homology with one copy of ℤ removed from H₀	§3.3
$f_{*}$	f lower star	the map a continuous map induces on homology (pushes forward)	§3.5
$de g f$	the degree of f	how many times a map of spheres wraps, counted with signs	§3.5
$H_{k} (X, A)$	relative homology of X rel A	homology after crushing A	§3.6
$\partial_{*}$	the connecting map	Hₙ(C) → Hₙ₋₁(A) in a long exact sequence: a cycle back to the boundary it came from	§3.6

Knots

Symbol	Read aloud	Meaning	Introduced
$K$	K bar	the mirror image of the knot K	§3.7
$K_{1} # K_{2}$	K one sum K two	the connected sum: cut both open and join the ends	§3.7
$μ, λ$	mu, lambda	meridian and longitude: once around the knot’s tube, once along it	§3.7
$lk (γ, K)$	the linking number of gamma and K	how often γ winds around K, with signs: [γ] = lk(γ, K)[μ]	§3.7
$B_{n}, σ_{i}$	the braid group on n strands; sigma i	braids on n strands; σᵢ swaps the strands in places i and i + 1 once	§3.7
$g (K)$	the genus of K	the least genus of a Seifert surface for K	§3.7
$w (D)$	the writhe of D	the sum of the signs of the crossings of a diagram	§3.7
$⟨ D ⟩$	the bracket of D	the Kauffman bracket: a sum over the states of a diagram	§3.7
$V_{K} (t)$	the Jones polynomial of K	(−A³)^(−w) ⟨D⟩ with A = t^(−1/4)	§3.7
$Δ_{K} (t)$	the Alexander polynomial of K	det(V − tVᵀ) for a Seifert matrix V	§3.7
$Kh^{i, j} (K)$	Khovanov homology of K	a bigraded homology whose graded Euler characteristic is the Jones polynomial	§3.7
$u (K)$	the unknotting number of K	the fewest crossing changes that untie K	§3.7

Persistence

Symbol	Read aloud	Meaning	Introduced
$U_{r}$	U r	the union of the balls of radius r around the data points	§3.8
$C ˇ_{r} (P)$	the Čech complex of P at scale r	a simplex for every set of points whose balls share a point	§3.8
$VR_{r} (P)$	the Vietoris–Rips complex of P at scale r	a simplex for every set of points whose balls meet in pairs	§3.8
$[b, d)$	a bar from b to d	a class born at scale b that dies at scale d	§3.8
$b_{k}^{r, s}$	the persistent Betti number	how many k-classes alive at r are still alive at s	§3.8
$d_{B}$	bottleneck distance	how far apart two persistence diagrams are	§3.8

Cochains and cohomology

Symbol	Read aloud	Meaning	Introduced
$C^{k} (K; G) = Hom (C_{k} (K), G)$	k-cochains	measurements on k-simplices	§4.1
$⟨ φ, c ⟩$	phi paired with c	pairing a measurement with a place	§4.1
$δ$	coboundary	(δφ)(σ) = φ(∂σ): discrete gradient, curl, …	§4.1
$H^{k} (X; G) = Z^{k} / B^{k}$	the k-th cohomology of X	cocycles modulo coboundaries: obstructions	§4.2
$f^{*}$	f upper star	the induced map on cohomology (pulls back)	§4.2
$Ext (A, Z)$	Ext of A and Z	for finitely generated A, its torsion; it carries the torsion of Hₖ₋₁ into Hᵏ	§4.2
$α ⌣ β$	alpha cup beta	the cup product of cohomology classes	§4.5
$[M]$	the fundamental class of M	the whole closed oriented manifold as one cycle	§4.5
$c ⌢ φ$	c cap phi	the cap product: a p-cochain eats the front p-face of a chain, leaving the back face	§4.6
$H ˇ^{k} (U; G)$	Čech cohomology of the cover U	cohomology built from overlaps of an open cover	§4.7

Calculus and forms

Symbol	Read aloud	Meaning	Introduced
$Ω^{k} (M)$	k-forms on M	things you integrate over k-dimensional pieces	§4.3
$d$	exterior derivative	unifies gradient, curl and divergence; d∘d = 0	§4.3
$α \land β$	alpha wedge beta	the wedge product: oriented area/volume	§4.3
$\int_{\partial M} ω = \int_{M} d ω$	Stokes’ theorem	boundary values equal total derivative inside	§4.3
$H_{dR}^{k} (M)$	de Rham cohomology	closed forms modulo exact forms	§4.4
$d θ$	the angle form	(−y dx + x dy)/(x² + y²): closed but not exact on ℝ²∖0	§4.4

Curvature and bundles

Symbol	Read aloud	Meaning	Introduced
$K$	Gaussian curvature	the product of the two principal curvatures at a point of a surface	§4.8
$\iint_{M} K d A = 2 π χ (M)$	Gauss–Bonnet	total curvature is topology	§4.8
$w_{1} (E)$	the first Stiefel–Whitney class	in H¹(B; ℤ/2): zero exactly when E is orientable	§4.8
$e (E)$	the Euler class	the obstruction to a nowhere-zero section; e(TM) gives χ(M)	§4.8
$c_{1} (L)$	the first Chern class	in H²(B; ℤ): the twisting of a complex line bundle	§4.8

Categories

Symbol	Read aloud	Meaning	Introduced
$Set, Top, Ab$	the categories of sets, spaces, abelian groups	objects with their structure-preserving maps	§5.1
$F : C \to D$	a functor from C to D	a structure-preserving map between categories	§5.1
$C^{op}$	C op	the same category with all arrows reversed	§5.1
$η : F \Rightarrow G$	a natural transformation from F to G	a choice-free way to turn F into G	§5.1
$Ext, Tor$	Ext and Tor	derived functors that measure failures of exactness	§5.2

Horizons

Symbol	Read aloud	Meaning	Introduced
$π_{n} (X)$	the n-th homotopy group of X	maps of the n-sphere into X, up to homotopy	§5.3
$K (G, n)$	an Eilenberg–MacLane space	a space whose only homotopy group is G, in degree n	§5.3
$[X, S^{1}]$	maps from X to the circle, up to homotopy	the same as H¹(X; ℤ)	§5.3
$Δ = d d^{} + d^{} d$	the Hodge Laplacian	its kernel, the harmonic forms, picks one form in each class	§5.3
$H^{k}$	harmonic k-forms	forms with Δω = 0; isomorphic to de Rham cohomology	§5.3
$K^{0} (X)$	K-theory of X	vector bundles on X, made into a group	§5.3