Algebraic topology

Algebraic topology translates questions about topological spaces and continuous maps into questions about groups and other structures from algebra. A topological invariant compresses the enormous amount of data in a space into something one can calculate. For example, the fundamental group records which based loops can be continuously deformed into one another. A filled disk has trivial fundamental group, whereas a circle has fundamental group $\mathbb{Z}$: an integer remembers how many times a loop goes around.

This compression also respects maps. By functoriality of the fundamental group, a continuous map preserving the chosen basepoints induces a group homomorphism; successive maps induce successive group homomorphisms. Thus geometry supplies arithmetic constraints. A proposed homotopy equivalence must induce an isomorphism, so incompatible fundamental groups rule it out before anyone attempts to construct the deformation.

The beauty is that a whole annulus can retain exactly the same homotopy type as its inner circle. Its thickness disappears under a deformation retraction, while the obstruction to shrinking a loop survives. Equal fundamental groups alone do not establish a homeomorphism or a homotopy equivalence between arbitrary topological spaces.

This turns a visual question into a reusable test. Suppose a continuous map sends the inner circle of an annulus once around another circle. Could it extend over a filled disk spanning that boundary? Such an extension would supply a homotopy shrinking the image loop to a point. Its winding number would have to change from one to zero, which homotopy invariance of winding number forbids. Algebra rules out the extension even when no particular formula for it has been proposed.

A delta-complex is obtained by gluing standard simplexes along affine face maps, allowing different vertices or faces of one simplex to acquire the same image. Every simplex interior is mapped injectively, and the topology is determined simplexwise. Unlike an abstract simplicial complex, its simplexes need not be determined by their sets of global vertices. A circle built from one edge with its two endpoints identified is a basic example. Subdivisions remove these repeated-vertex and repeated-simplex obstructions.

Intersection homology is the homology of a chain complex in which chains and their boundaries satisfy codimension-dependent perversity restrictions at singular strata. Unless specified otherwise coefficients here are $\mathbb{Q}$. Ordinary homology is recovered on manifolds.

A traditional perversity is a function $\bar{p}:\{2,3,\dots \}\to \mathbb{Z}$ satisfying $\bar{p}(2)=0$ and $\bar{p}(k+1)-\bar{p}(k)\in \{0,1\}$. It specifies how far chains may meet strata of codimension $k$.

An $i$-dimensional singular simplex $\sigma :{\Delta }^i\to X$ is $\bar{p}$-allowable if ${\sigma }^{-1}(X_{N-k})$ is contained in the $(i-k+\bar{p}(k))$-skeleton of ${\Delta }^i$ for every $k\ge 2$. A negative skeleton is empty. Individual faces need not be allowable; this is why the definition of intersection chains separately imposes allowability on the total boundary.

The group $I{S}_i^{\bar{p}}(X)$ consists of finite singular chains whose nonzero simplices are allowable singular simplices and whose boundary, after cancellation, also consists of allowable simplices. Since ${\partial }^2=0$, these groups form a chain complex. Its homology is intersection homology.

The upper middle perversity is $\bar{n}(k)=\lceil (k-2)/2\rceil$. It differs from the lower middle perversity by one at odd codimensions, and is complementary to it.

The lower middle perversity is $\bar{m}(k)=\lfloor (k-2)/2\rfloor$. The complementary upper middle perversity is $\bar{n}(k)=\lceil (k-2)/2\rceil$, and $\bar{m}(k)+\bar{n}(k)=k-2$. They agree at even codimensions.

For a compact $(N-1)$-dimensional link $L$ and traditional perversity $\bar{p}$, finite-chain intersection homology of its open cone is $I{H}_i^{\bar{p}}(cL)=I{H}_i^{\bar{p}}(L)$ for $i<N-1-\bar{p}(N)$ and zero for $i\ge N-1-\bar{p}(N)$. Below the cutoff both chains and possible filling chains avoid the vertex. At and above the cutoff the cone on any allowable cycle is an allowable filling: its new vertex is allowed precisely when $0\le i+1-N+\bar{p}(N)$.

For $\dim L=2n-1$ and lower middle perversity, the suspension with two distinct vertices has $I{H}_i(\mathrm{Susp}L)=I{H}_i(L)$ for $i<n$, zero for $i=n$, and $I{H}_{i-1}(L)$ for $i>n$. Cover it by its two open cones. The Mayer–Vietoris sequence compares their truncated groups with $I{H}_{\ast }(L)$; below $n$ the overlap map is the injective signed diagonal, and above $n$ both cone groups vanish.

In the Deligne-sheaf convention, $I{H}_{\bar{p}}^i(X)={\mathbb{H}}^i(X,I{C}_{\bar{p}})$, with $I{C}_{\bar{p}}$ normalized in degree zero on the regular stratum. For a compact oriented $N$-dimensional pseudomanifold over a field, $I{H}_{\bar{p}}^i(X)\cong I{H}_{N-i}^{\bar{p}}(X)$ and it is dual to $I{H}_i^{\bar{q}}(X)$, where $\bar{p}(k)+\bar{q}(k)=k-2$. Thus the dual of lower-middle singular chains uses the upper-middle Deligne sheaf. The two middle conventions agree on Witt spaces.

Let a compact oriented $(2m+1)$-manifold $M$ have collared boundary $B$, and cap $B$ by its closed cone. In the lower-middle Deligne convention the result has $I{H}^i=H^i(M)$ for $i<m$, $I{H}^m=\mathrm{im}(H^m(M,B)\to H^m(M))$, and $I{H}^i=H^i(M,B)$ for $i>m$. Excision identifies the relative intersection groups with $H^{\ast }(M,B)$; the cone neighbourhood has $H^i(B)$ only for $i<m$, so its long exact sequence gives the formula. The cap is Witt exactly when $H^m(B;\mathbb{Q})=0$.

The degree-zero normalized Deligne intersection complex extends the constant sheaf from the regular stratum by successive derived direct images and cohomological truncations at degree $\bar{p}(k)$ when codimension-$k$ strata are added. Its hypercohomology defines intersection cohomology. On a cone with vertex of codimension $k$, its local cohomology is the link intersection cohomology in degrees at most $\bar{p}(k)$ and zero above.

For an isolated singular point of an $N$-dimensional oriented pseudomanifold, the unshifted middle intersection complex restricts to $\mathbb{Q}$ on the regular stratum and has no negative cohomology. Its support condition is $H^j(i_x^{\ast }I{C}_{\bar{p}})=0$ for $j>\bar{p}(N)$; its cosupport condition is $H^j(i_x^{!}I{C}_{\bar{p}})=0$ for $j\le \bar{p}(N)+1$. These conditions, together with constructibility and normalization, characterize it. On a Witt space they are exchanged by $IC\cong D(IC)[-N]$.

A homotopy colimit of spaces is the homotopy-invariant replacement for a colimit of spaces. For a filtered diagram of CW complexes with subcomplex inclusions, the union is a model: the inclusions are cofibrations, and every sphere map and homotopy has image in a finite subcomplex. A suspension spectrum carries this construction to the corresponding homotopy colimit of spectra.

Stable homotopy theory studies topological spectra, which retain the information that persists under repeated reduced suspension. Its stable homotopy category admits invertible suspension of spectra and exact cofiber sequences of spectra.

A universe for spectra is a countably infinite-dimensional real inner product space, normally with its direct-limit topology, used to index finite-dimensional suspension coordinates. A cofinal indexing family contains arbitrarily large finite-dimensional subspaces.

Given a compact parameter space $X$ and a family $f_x:U\to U\prime$ of linear isometries, a subordinate flag for a family of linear isometries consists of nested cofinal source and target subspaces $V_i,W_i$ with $f_x(V_i)\subset W_i$ for every $x$. After enlarging the target stages, compactness makes this possible for every finite source stage. It permits simultaneous finite-dimensional Thom space constructions.

A homotopy colimit of spectra is the derived colimit of a diagram of topological spectra. For a sequential diagram $T_0\to T_1\to \cdots$, it is the cofiber of spectra of $1-s:{\bigvee }_j{T}_j\to {\bigvee }_j{T}_j$, where $s$ uses the bonding maps into the next summands.

For $D={\mathrm{hocolim}}_j{T}_j$, applying $[-,E{]}_{\mathrm{st}}$ to the telescope cofiber sequence of spectra givesHere ${\lim }^1$ is the cokernel of $1-r$ on the product of the inverse tower, where $r$ uses its bonding maps. This records the obstruction to uniqueness when compatible stage maps are realized.

The homotopy cofiber of spectra of $u:A\to B$ is the derived pushout $B{\cup }_{A}CA$. It fits into a cofiber sequence of spectra.

A cofiber sequence of spectra $A\to B\to C\to \Sigma A$ induces exact sequences after either mapping into or out of a fixed topological spectrum, and on stable homotopy groups. In particular the second map is zero precisely when the first map admits a right inverse in the stable homotopy category.

The derived smash product of spectra is the symmetric monoidal product on the stable homotopy category. It has unit sphere spectrum, preserves cofiber sequences of spectra and homotopy colimits of spectra in each variable, and satisfies ${\Sigma }^{\infty }X\wedge {\Sigma }^{\infty }Y\simeq {\Sigma }^{\infty }(X\wedge Y)$.

The internal function spectrum is characterized by $[C,F(A,E){]}_{\mathrm{st}}=[C\wedge A,E{]}_{\mathrm{st}}$. For a based space $K$, notation $F(K,E)$ abbreviates $F({\Sigma }^{\infty }K,E)$.

If $E$ is a ring spectrum, then $F(X_{+},E)$ is a ring spectrum. Its multiplication is adjoint to evaluating the two functions on the two copies supplied by the diagonal of $X$, then multiplying in $E$. Its unit is the constant unit function. Coassociativity of the diagonal and associativity in $E$ prove the associative identity; the counit and unit identities prove both unit laws.

The stable homotopy category is obtained from topological spectra by inverting stable equivalences of spectra. Its morphisms are abelian groups; for a cellular source and fibrant target they are spectrum homotopy classes of point-set maps of topological spectra.

A map of topological spectra inducing isomorphisms on every stable homotopy group is a stable equivalence of spectra.

A sequential model of a topological spectrum consists of based spaces $E_j$ and structure maps $S^1\wedge E_j\to E_{j+1}$. Indexed models use finite-dimensional subspaces of a universe for spectra and structure maps $S^{W\ominus V}\wedge E(V)\to E(W)$. Different standard models present the same stable homotopy category.

A connective spectrum has ${\pi }_{q}E=0$ for $q<0$. This makes the homological Atiyah-Hirzebruch spectral sequence first quadrant for a based space, and prevents negative coefficient degrees from affecting its first nonzero represented homology theory.

If $E$ is connective spectrum in the sense of a connective spectrum, ${\pi }_{0}E\cong \mathbb{Z}$, and $X$ is $(n-1)$-connected with $n\ge 2$, then ${\tilde{E}}_n(X)\cong {\pi }_n(X)$. In the homological Atiyah-Hirzebruch spectral sequence, the only nonzero term in total degree $n$ is ${\tilde{H}}_n(X;{\pi }_{0}E)$. Incoming differentials have negative coefficient degree, and outgoing differentials have spatial degree below $n$. The Hurewicz theorem identifies this term with ${\pi }_{n}X$.

For a connective spectrum with ${\pi }_{0}E\cong \mathbb{Z}$, an $E$-homology isomorphism of simply connected spaces is a weak homotopy equivalence. Its simply connected homotopy mapping cone is $E$-acyclic. If it had a first nonzero integral homology group, the Hurewicz theorem and the bottom-degree generalized Hurewicz isomorphism would produce a nonzero $E$-homology group. Thus the cone is integrally acyclic and the homological Whitehead theorem applies.

A ring spectrum is a unital associative monoid object of the stable homotopy category under the smash product of spectra. It has $\mu :E\wedge E\to E$ and $\eta :\mathbb{S}\to E$, satisfying the associative and two unit identities. Structured ring spectra additionally specify coherent point-set or higher homotopies; a ring object alone does not assert those enhancements.

A commutative ring spectrum, in the homotopy-category convention, satisfies $\mu \tau =\mu$, where $\tau$ interchanges the factors of the smash product of spectra. This condition alone does not assert an $E_{\infty }$ refinement.

An indexed prespectrum over $U$ consists of based spaces $E(V)$ indexed by finite-dimensional $V\subset U$, with unital associative maps $S^{W\ominus V}\wedge E(V)\to E(W)$ for inclusions $V\subset W$. The orthogonal differences make iterated suspension coordinates compose consistently.

For compact $X$ over the space of linear isometries $U\to U\prime$, choose a subordinate flag for a family of linear isometries $V_i,W_i$. The vector bundle ${\xi }_i$ has fiber $W_i\ominus f_x(V_i)$. Form $P(W_i)=\mathrm{Th}({\xi }_i)\wedge E(V_i)$. Its structure maps use the orthogonal decompositionand the structure map of $E$. The twisted half-smash product is the spectrification $LP$, after extending from the cofinal flag. Unlike an arbitrary replacement by $X_{+}\wedge E$, this definition retains the specified isometry family at the point-set level.

For isometry families $f_x:U\to U\prime$ and $g_y:U\prime \to U\prime \prime$, the complement of $g_y{f}_x(V)$ decomposes orthogonally as the complement of $g_y(W)$ plus the image under $g_y$ of the complement of $f_x(V)$ in $W$. Their fiberwise one-point compactifications therefore give the same Thom space as the external direct sum. The resulting stage isomorphisms respect structure maps and give $Y{⋉}_g(X{⋉}_{f}E)\cong (Y\times X){⋉}_{g\circ f}E$ in the stable homotopy category.

In the genuine indexed point-set model, spectrification $L$ is the left adjoint to the forgetful functor from indexed spectra to indexed prespectra. An indexed spectrum has adjoint structure maps $E(V)\to {\Omega }^{W\ominus V}E(W)$ that are homeomorphisms. Applying $L$ to a cellular indexed prespectrum supplies the associated genuine topological spectrum.

The $k$th stable homotopy group of $E$ is ${\pi }_{k}E=[{\Sigma }^k\mathbb{S},E{]}_{\mathrm{st}}$, equivalently ${\mathrm{colim}}_j{\pi }_{k+j}{E}_j$ in a sequential model.

Suspension of spectra shifts the stable degree. Unlike reduced suspension of based spaces, it is invertible in the stable homotopy category, with inverse denoted ${\Sigma }^{-1}$.

For a based space $X$, its suspension spectrum has level $j$ equal to $S^j\wedge X$. Its derived adjunction with the infinite loop space is $[{\Sigma }^{\infty }X,E{]}_{\mathrm{st}}=[X,{\Omega }^{\infty }E{]}_{\ast }$.

The sphere spectrum is $\mathbb{S}={\Sigma }^{\infty }{S}^0$. It is the unit of the smash product of spectra. Its integer suspensions of spectra represent stable homotopy groups.

A cell spectrum is obtained by attaching stable cells, with successive cofibers of spectra wedges of suspensions of spectra of the sphere spectrum. Cellular models provide the cofibrant objects needed to compute derived maps of topological spectra. Stable cells may occur in arbitrary integer degrees.

A finite spectrum is a topological spectrum equivalent to a cell spectrum with finitely many cells. Such an object is compact in the stable homotopy category and has a Spanier-Whitehead dual. Compactness means maps out of it commute with filtered homotopy colimits of spectra.

The Spanier-Whitehead dual of a finite spectrum $A$ is $A^{\vee }=F(A,\mathbb{S})$. It is finite, and evaluation and coevaluation give the natural Spanier-Whitehead duality adjunction.

For a finite spectrum $A$, evaluation and coevaluation identify $[B,D\wedge A{]}_{\mathrm{st}}$ with $[B\wedge A^{\vee },D{]}_{\mathrm{st}}$. Cellular induction constructs the dual from the duals of stable cells and verifies these identities across cofiber sequences of spectra. In particular ${\pi }_k(D\wedge A)=[{\Sigma }^k{A}^{\vee },D{]}_{\mathrm{st}}$.

An Omega-spectrum has structure adjoints $E_j\to \Omega E_{j+1}$ that are weak homotopy equivalences. For such a model the level spaces represent the degrees of its represented cohomology theory.

A map of topological spectra is a compatible family of based maps of the level spaces: $u_{j+1}{\sigma }_j^D={\sigma }_j^E(1\wedge u_j)$. It is distinct from its class in the stable homotopy category.

A phantom map of spectra is a stable map $p:D\to E$ with $pu=0$ for every map $u:F\to D$ from a finite spectrum. Such a map can be nonzero even though it induces zero on all represented homology theories on based spaces.

For a phantom map of spectra $p:D\to E$ and a finite CW complex $K$, Spanier-Whitehead duality identifies $D_k(K)$ with maps from the finite spectrum ${\Sigma }^k({\Sigma }^{\infty }K{)}^{\vee }$. Hence $p_{\ast }=0$ there. An arbitrary CW complex is the filtered union of its finite subcomplexes, and represented homology theory commutes with that filtered union because spheres are compact. Thus $p$ induces zero on the entire represented homology theory.

Choose one representative of each finite spectrum and form $X={\bigvee }_{(F,u:F\to D)}F$. Let $e:X\to D$ evaluate each summand and let $p:D\to \mathrm{cofib}(e)$. Every map from a finite spectrum factors through its corresponding summand, so $p$ is a phantom map of spectra. If $p=0$, exactness supplies a section of $e$, making $D$ a retract of $X$. Consequently a spectrum that is not a retract of any wedge of finite spectra has a nonzero such phantom.

A hyperphantom map of spectra is a stable map $p:D\to E$ such that $pu=0$ for every $u:{\Sigma }^{-j}{\Sigma }^{\infty }X\to D$, for arbitrary based spaces $X$ and all nonnegative $j$. Equivalently it induces zero on the entire represented cohomology theory. This is a stronger condition than being a phantom map of spectra, which tests only finite spectra.

A spectrum homotopy between compatible maps $D\to E$ is a map from the cylinder spectrum $D\wedge [0,1{]}_{+}$ restricting to them at the endpoints. Thus the homotopies at all levels must themselves respect the structure maps.

A topological space with a distinguished basepoint. Maps preserve the basepoint, and based homotopies keep it fixed.

A base point is a distinguished point $x_0$ of a topological space $X$, making $(X,x_0)$ a based space. A based loop starts and ends at that point, and a map of based spaces must preserve the distinguished points.

Use the unnormalized convention: the least number of open subsets covering $X$ whose inclusions into $X$ are null-homotopic. The subsets need not themselves be contractible. This number is one for a nonempty contractible space. If $r$ positive-degree classes have nonzero product, a cover by $r$ categorical sets would make that product zero by relative cup products; hence $\mathrm{cat}(X)\ge r+1$.

Every smooth real-valued function on a closed connected smooth manifold has at least its unnormalized Lusternik–Schnirelmann category many distinct critical points. No nondegeneracy hypothesis is required. If there are infinitely many critical points the bound is immediate; in the finite case, the sublevel-set deformation argument shows that category can increase only at critical levels, by at most the number of critical points on that level. In conjunction with cup length, this gives at least $r+1$ critical points whenever there is a nonzero $r$-fold positive-degree cup product. This theorem is distinct from the antipodal covering theorem also sometimes named the Lusternik–Schnirelmann theorem. A primary account of the critical-point formulation is Conley pairs in geometry—Lusternik–Schnirelmann theory and more.

The theta graph ${\Theta }_n$ has two vertices joined by $n$ edges. It has first Betti number $n-1$ and occurs as the link of a point on the spine of a book with $n$ two-dimensional pages.

A dumbbell graph consists of two cycles joined by a path. Its connecting path contains bridges, unlike a Theta graph, although both the two-loop dumbbell and ${\Theta }_3$ have first homology ${\mathbb{Z}}^2$.

The Borsuk-Ulam theorem says that recording $n$ continuous real-valued measurements on an $n$-dimensional unit sphere must give identical records somewhere on opposite sides. The measurements can vary wildly; continuity alone forces a coincidence at an antipodal pair.

Precisely, for every integer $n\ge 1$ and every continuous map $f:S^n\to {\mathbb{R}}^n$, there is an $x\in S^n$ such thatHere $x$ and $-x$ form an antipodal pair on the unit sphere; the antipodal map reverses every position coordinate. The target is the Euclidean space ${\mathbb{R}}^n$. Both points have the same entire output, so every one of the $n$ measurements agrees at this common pair. See Hatcher, Algebraic Topology, Corollary 2B.7, p. 176.

Circle example. Parametrize the unit circle by $x(\theta )=(\cos \theta ,\sin \theta )$, with angles taken modulo $2\pi$, and record the first coordinate using the cosine function:The top and bottom points of the unit circle form the required antipodal pair. Their first coordinates agree, while their second coordinates are $1$ and $-1$.

Sphere example. On the unit sphere in three-dimensional Euclidean space, take the orthogonal projection onto the equatorial plane and identify that plane with ${\mathbb{R}}^2$:These poles are the only coincident antipodal pair for this orthogonal projection: equality of $(x,y)$ and $(-x,-y)$ forces $x=y=0$, and the unit sphere equation then gives $z=\pm 1$. The Borsuk-Ulam theorem guarantees an antipodal pair for every continuous map into ${\mathbb{R}}^2$, even when no convenient projection formula is available.

Equal output need not mean equal position coordinates. Keeping all three coordinates gives a continuous map $i:S^2\to {\mathbb{R}}^3$ defined by $i(x,y,z)=(x,y,z)$, the restriction of the identity map of ${\mathbb{R}}^3$. Its outputs at an antipodal pair are always distinct, since $u=-u$ would force $u=0$, which is outside the unit sphere. Thus replacing the target ${\mathbb{R}}^2$ by ${\mathbb{R}}^3$ destroys this guarantee.

Why the circle case follows from the intermediate value theorem. For an arbitrary continuous map $f:S^1\to \mathbb{R}$, write $F(\theta )=f(x(\theta ))$ using the unit circle parametrization above. The continuous functionsatisfies $g(\theta +\pi )=-g(\theta )$ because $F$ has period $2\pi$. If $g(0)=0$, the first antipodal pair already works. Otherwise $g(0)$ and $g(\pi )$ have opposite signs, and the intermediate value theorem supplies a zero between them. At that zero, the two antipodal values of $f$ agree. In the cosine function example, $g(\theta )=2\cos \theta$. This short argument handles only $n=1$. See Hatcher, Algebraic Topology, discussion of Theorem 1.10, p. 32.

An ideal weather interpretation. Model the surface of Earth as a sphere. At one fixed instant, suppose temperature $T$ and pressure $P$ are globally defined continuous scalar measurements on that sphere. Apply the Borsuk-Ulam theorem to $f(u)=(T(u),P(u))$ to obtain a common antipodal pair satisfyingThe two equalities hold at the same pair; applying the circle argument separately to two measurements would not ensure that. This is a conclusion about the ideal continuous model, with both measurements taken at the same instant, rather than an assertion that an exactly matching pair has been empirically measured. Matoušek gives this motivating interpretation and explicitly cautions that physical temperature need not be continuous.^[ref]

Equivalent formulations. A continuous map $h:S^n\to {\mathbb{R}}^n$ is odd when $h(-x)=-h(x)$, extending the symmetry of an odd function. The Borsuk-Ulam theorem equivalently says every such map has a zero. For a general $f$, the difference $h(x)=f(x)-f(-x)$ has this symmetry; conversely, equal antipodal outputs of an odd map must be zero. Another equivalent formulation is that no odd map between spheres $S^n\to S^{n-1}$ exists. Normalizing a nonvanishing odd map by its Euclidean norm would produce such an odd map between spheres. These formulations are useful starting points for algebraic topology; the general proof is beyond the circle argument here.^[ref]

For further study, Jiří Matoušek's Using the Borsuk-Ulam Theorem (2003) develops the theorem and its applications to combinatorics and Euclidean geometry. An early publication is Karol Borsuk's 1933 paper Drei Sätze über die n-dimensionale euklidische Sphäre.

A cover of $S^d$ by $d+1$ closed sets has a member containing an antipodal pair. The equivalent version for open sets follows by shrinking finite open covers on a compact metric space; the reverse implication follows by enlarging antipodal-pair-free closed sets slightly. This is a covering formulation of the Borsuk-Ulam theorem.

Let $v_0,\dots ,v_{n+1}$ be the vertices of a regular simplex centered at zero in ${\mathbb{R}}^{n+1}$. The displayed spherical Voronoi cells are a closed cover by $n+2$ sets. If $x$ and $-x$ belonged to the same cell, all their scalar products with the vertices would coincide. Their sum is zero and the vertices span the ambient vector space, so this would force $x=0$, a contradiction. Thus each cell contains no antipodal pair. Together with the Lusternik-Schnirelmann-Borsuk theorem, this shows $n+2$ is the minimum number of antipodal-free members of a finite closed cover.

For a finite open cover $U_0,\dots ,U_d$ of $S^d$, take continuous nonnegative functions ${\varphi }_i$ summing to one and positive only in $U_i$. Such a partition of unity comes from normalizing the distance to a set functions $d(x,S^d\setminus U_i)$. The Borsuk-Ulam theorem applied to $({\varphi }_1,\dots ,{\varphi }_d)$ gives equal values at an antipodal pair. The sum condition makes ${\varphi }_0$ equal too, and some common positive value places both points in one $U_i$.

A CW complex is built inductively by attaching discs along maps from their boundary spheres. Its $n$-skeleton contains all cells of dimension at most $n$.

A bouquet of circles identifies one point of each of a family of circles to a common point, using the quotient topology. As a CW complex it has one vertex and one graph loop for each circle. The empty family means a single point. Its fundamental group is the free group on the family, as described by the fundamental group of a bouquet of circles.

A compact subset of a CW complex is contained in a finite CW complex that is a subcomplex of the original complex. Otherwise choose one point in each of infinitely many cells met by the subset. The closure-finiteness axiom makes each closed cell meet the chosen set in finitely many points. The weak topology axiom then makes the chosen set, and each of its subsets, closed. It is consequently an infinite closed discrete subspace of the compact subset, a contradiction. Only finitely many cells are met; their closures form a finite subcomplex containing the subset.

The $q$-skeleton of a CW complex is the subcomplex consisting of its topological cells of dimension at most $q$. A cellular map preserves these skeleta. The cellular approximation theorem lets continuous maps respect this filtration up to homotopy.

An open $q$-cell of a CW complex is a copy of an open $q$-dimensional ball. Its characteristic map is a map from a closed ball whose interior parametrizes that cell and whose boundary lands in the preceding skeleton. The characteristic map need not be injective on the boundary. Each oriented cell gives one generator of the cellular chain complex.

A CW pair consists of a CW complex $X$ and a subcomplex $A$. Its inclusion is a cofibration, so homotopies on $A$ extend across $X$. The relative lifting property of a Serre fibration for such pairs allows specified boundary lifts to be retained while lifting a homotopy.

Attaching a $q$-cell along $\varphi :S^{q-1}\to X$ forms the quotient of $X\bigsqcup D^q$ identifying each boundary point with its image under $\varphi$. A constant attaching map gives $X\vee S^q$. An attaching map can affect the cohomology ring even when it does not change the additive cohomology groups.

Attaching cells of dimension $i+1$ to a connected CW complex, for $i\ge 2$, preserves all homotopy groups below degree $i$. In degree $i$ it takes the quotient by the subgroup generated by the attaching maps and their translates under the fundamental group action. This follows from the relative homotopy sequence and the generation of the first nonzero relative group by the new cells. In particular, a single attaching map representing a generator of ${\pi }_i\cong \mathbb{Z}$ kills that group even when the fundamental group is nontrivial. Higher attachments can create higher homotopy groups, which may then be killed at later stages.

The weak topology of a CW complex is determined by its closed cells: a subset is closed if and only if its inverse image under every characteristic map from a closed cell is closed. A map out of the complex is continuous when its restrictions along those characteristic maps are continuous. This is distinct from the weak topology of a vector space. Taking the product with a locally finite CW complex, such as a closed interval with its finite cell structure, retains the corresponding CW weak topology.

A topological graph is a CW complex with only zero-cells and one-cells, called vertices and edges. Edge endpoints attach to vertices; loops, multiple edges and infinitely many edges are allowed. A subgraph in this setting is a CW subcomplex. A connected acyclic graph is a tree and is a contractible space.

A CW subcomplex is a union of cells of a CW complex containing the attaching boundary of every included cell. In a combinatorial 2-complex, including a two-cell therefore includes every edge in its boundary. A connected CW subcomplex has a connected 1-skeleton, whose intrinsic path metric may differ from the ambient graph metric.

The one-skeleton $X^{(1)}$ of a CW complex is the union of its zero-cells and one-cells. It is a graph. The unit-edge path metric measures the infimum of lengths of paths in this graph; it is different from metrics that allow paths through two-cell interiors.

A two-dimensional CW complex has cells only in dimensions zero, one and two. In a combinatorial two-complex the two-cells are polygons attached by combinatorial edge paths. Its 1-skeleton carries the path metric assigning length one to each edge. Compact combinatorial complexes have finitely many cells, so their cell-boundary lengths have a finite maximum.

A disc diagram over a combinatorial 2-complex is a finite contractible planar combinatorial complex together with a combinatorial map to that complex. Its exterior boundary circuit records a null-homotopic path. Tree portions and cut vertices are allowed; the diagram need not be an embedded disk or map injectively. Its area is its number of two-cells.

A ladder is a disc diagram whose blocks form a linear chain, each block a two-cell or a connecting edge, with cells attached successively along interior arcs. Its boundary consists of two opposite side paths between its ends. Every cell meets both side paths. If cell perimeters are bounded by $L$, each point of one side is within $L/2$ of the other by traveling around a cell boundary; connecting edge paths coincide on the two sides.

A spur is an exposed degree-one vertex and its incident edge in a disc diagram. The exterior boundary runs out along that edge and immediately back. Thus a spur in the interior of a boundary side contradicts that side being a reduced combinatorial path or a metric geodesic. A diagram with no two-cells may be a tree and have spurs rather than shells.

A shell is an exposed two-cell in a disc diagram, with perimeter split as an exterior boundary arc $Q$ and an interior path $P$. An $i$-shell has $P$ consisting of $i$ interior pieces. Under $C^{\prime }(1/6)$, an $i$-shell with $i\le 3$ has $|P|<|\partial R|/2<|Q|$, so removal replaces a long exterior path by a strictly shorter path.

A closed combinatorial path in a 2-complex is null-homotopic exactly when it is the boundary path of a disc diagram over that complex. A finite cellular null-homotopy can be arranged into such a planar diagram; conversely the contractible diagram supplies a null-homotopy. In a group presentation, this is the geometric form of expressing a trivial word as a product of conjugates of relators.

A disc diagram is reduced if it has no adjacent pair of two-cells that can be folded together across their common boundary and removed while preserving the outside boundary. A minimum-area diagram is reduced. Among diagrams of minimum area, minimizing the number of edges removes unnecessary tree portions. Internal common arcs in a reduced diagram satisfy the applicable small cancellation theory piece bounds.

A cellular map $f:X\to Y$ between CW complexes satisfies $f(X^q)\subseteq Y^q$ for every skeleton. It induces a chain map $f_{\#}:C_{\ast }^{\mathrm{cell}}(X)\to C_{\ast }^{\mathrm{cell}}(Y)$.

A continuous map between CW complexes is homotopic to a cellular map. If the map is already cellular on a subcomplex, the homotopy can be taken relative to that subcomplex. In particular, a map from a $d$-dimensional CW complex can be deformed into the $d$-skeleton of the target. This permits finite cellular chain complexes to compute induced homology maps, and turns maps to Complex projective space into maps to an appropriate finite-dimensional projective skeleton.

A map from the $n$-sphere to a CW complex can be homotoped into its $n$-skeleton. By compact subsets of CW complexes lie in finite subcomplexes, its image lies in a finite subcomplex. For a top-dimensional cell of dimension $d>n$, use smooth approximation of maps into Euclidean space in its interior, with a cutoff supported away from the cell boundary. The Sard theorem supplies a point in a smaller interior ball omitted by the smoothed map; the cutoff is chosen so the transition region also avoids that point. Radial retraction from the omitted point pushes the image into the cell boundary and fixes all other cells. Repeating over the finitely many higher-dimensional cells proves the assertion, without assuming the initial continuous image has measure zero.

A finite CW complex has finitely many cells. Its cellular chain complex consists of finitely generated free abelian groups and vanishes outside finitely many degrees, so its integral homology groups are finitely generated.

Suppose a compact skeleton $Y$ embeds as $e(Y)\subset {\mathbb{R}}^m$. Attach a disk by $a:S^{k-1}\to Y$ and extend its boundary coordinate map radially by $G(ru)=re(a(u))$. Map the disk to $(G(x),(1-|x|)x,1-|x|)$ and $Y$ to $(e(y),0,0)$. The boundary identifications agree; a positive last coordinate recovers each interior point. This defines a continuous injection of the compact quotient into Euclidean space, hence an embedding. Repeating proves Hausdorffness and Euclidean embeddability for every finite CW complex, without assuming triangulability of arbitrary attaching maps.

A CW approximation of a space $X$ is a CW complex $X_{\mathrm{cw}}$ with a weak homotopy equivalence $X_{\mathrm{cw}}\to X$. The realization of the singular simplicial set of $X$ supplies one functorially.

For simply connected $X$ with finitely generated integral homology in every degree, the Hurewicz theorem modulo a Serre class gives finitely generated homotopy groups. Realize finite generating sets of successive relative homotopy groups by attaching finitely many cells at each stage. The resulting weak equivalence has finitely many cells in each dimension.

Start with a finite $n$-dimensional $n$-connected approximation. Its degree-$n$ homology kernel is finite free. Realize a basis via the Relative Hurewicz theorem and attach $(n+1)$-cells; their boundary is injective, so they kill precisely that kernel without adding top homology. The homological Whitehead theorem gives a finite weak equivalence of dimension at most $n+1$.

A CW filtration builds a CW complex by its skeletawhere each quotient $X^n/X^{n-1}$ is a wedge of $n$-spheres, one for each $n$-cell.

For an abelian group $G$ and an integer $m\ge 1$, a Moore space $M(G,m)$ is a CW complex whose reduced homology is $G$ in degree $m$ and zero in every other degree.

Attach an $(i+1)$-cell to $S^i$ by a degree-$n$ map, with $i\ge 1$ and $n\ge 2$. The integral cellular differential is multiplication by $n$, while its modulo-$n$ reduction is zero. Lifting the degree-$i$ generator modulo $n^2$ and applying the differential gives $n$, so the Bockstein homomorphism sends that generator to the degree-$(i+1)$ generator. This supplies a nonzero example in every positive degree.

Attaching a two-cell to $S^1$ by a map of degree $n$ gives the Moore space $M(\mathbb{Z}/n,1)$. Its positive-dimensional cellular chain complex isso $H_1\cong \mathbb{Z}/n$ and all its other reduced integral homology groups vanish.

The mapping cylinder of $f:A\to X$ isIt deformation retracts onto the copy of $X$.

Use the mapping cylinder convention $(x,1)\sim f(x)$ for $f:X\to Y$. Its bottom copy of $X$ is a retract precisely when $f$ has a left homotopy inverse $g:Y\to X$. A retraction restricted to $Y$ supplies $g$, and its values along the cylinder supply the homotopy from ${\mathrm{id}}_X$ to $gf$. Conversely use that homotopy on the cylinder and $g$ on $Y$; agreement at the attached end makes the map descend by the universal property of the quotient topology.

The topological mapping cone of $f:X\to Y$ is $Y{\cup }_{f}CX$, where $CX$ is the cone on $X$ and its base is identified with $Y$ through $f$. For nonempty $X$, the cone vertex supplies a distinguished basepoint. This construction is related to, but distinct from, the mapping cone of a chain map.

Successive mapping cones of a based map give this sequence up to homotopy. Applying based maps into a space gives an exact sequence of pointed sets in the opposite direction. The exactness statement at $[X,Y]$ says that a map whose restriction to $A$ is null-homotopic extends over $C_i$ using the chosen null-homotopy.

For a cellular map between nonempty CW complexes, the reduced cellular chain complex of its topological mapping cone has differentialHere $C_{-1}(X)=0$ and the degree-zero group identifies a vertex $y$ with the reduced chain $y-v$, where $v$ is the cone vertex. Thus this is the mapping cone of $f_{\#}$, in the displayed ordering of the summands.

For nonempty $X$, the topological mapping cone has this long exact sequence in homology, with ordinary homology for $X,Y$ and reduced homology for $C_f$. In degree zero, the augmentation of $X$ is accounted for by the cone vertex. The sequence follows from the Mayer–Vietoris theorem applied to neighborhoods of the cone and $Y$.

The mapping telescope of $X_0\stackrel{f_0}{\to }{X}_1\stackrel{f_1}{\to }\cdots$ glues the mapping cylinders of the $f_i$ end to end. Its homology is naturally the direct limit ${\underrightarrow{\lim }}_i{H}_{\ast }(X_i)$.

List the primes as $p_0,p_1,\dots$ and form the mapping telescope of degree-$p_i$ maps $S^n\to S^n$. Its reduced homology vanishes outside degree $n$, while