Geometry and topology

The study of convex sets, their convex hulls, supporting closed half-spaces, and inequalities for their sizes.

The Minkowski sum of subsets $A,B$ of a vector space is $A+B=\{a+b:a\in A,b\in B\}$. Minkowski addition also makes sense for nonconvex sets; translations and positive dilations commute with it.

In $\mathbb{R}$, the sets $X=\{n:n\ge 1\}$ and $Y=\{-n+1/n:n\ge 1\}$ are closed because their points have no finite accumulation point. Their Minkowski sum contains $1/n\to 0$ but does not contain zero, since $m-n+1/n=0$ has no positive-integer solution. Compactness of one summand is a useful sufficient condition missing from this counterexample.

In Euclidean space, a Minkowski sum of a compact set $X$ and a closed set $Y$ is closed. From any convergent sequence $x_j+y_j$, extract a convergent subsequence of $x_j$ using compactness. The corresponding $y_j$ then converges to the difference of the two limits, which lies in $Y$. If $Y$ is also bounded, the sum is bounded and therefore compact.

For nonempty open sets $A,B\subset {\mathbb{R}}^n$, the Lebesgue measure of their Minkowski sum satisfiesIt also holds for compact sets and in standard measurable-set formulations with the appropriate measurability qualification. Normalize both volumes to one and apply the Prékopa–Leindler inequality to indicator functions.

The geometry of affine spaces, studying incidence and parallelism without choosing an origin or a metric.

A nonempty set $A$ equipped with a group action of the additive group of a vector space $V$, such that for every $x,y\in A$ there is a unique $v\in V$ with $x+v=y$. There is no distinguished origin. Choosing one identifies $A$ with $V$; changing it gives a translation.

A map between affine spaces whose difference map is linear. After choosing origins it is $x\mapsto Ax+b$, where $A$ is its linear part of an affine map and $b$ its translation. This definition allows noninvertible maps; invertibility requires $A$ invertible.

The map $A$ associated with an affine map $x\mapsto Ax+b$; it is independent of the translation $b$. An affine Euclidean isometry has an orthogonal linear part, and an isometry between flat tori has a lift whose linear part carries one Euclidean lattice onto the other.

A closed half-space consists of a hyperplane and one of its two sides in a real affine space. Replacing the weak inequality by a strict inequality gives an open half-space. Half-spaces are convex sets, and finite intersections of closed half-spaces are linear polyhedra.

On an affine space modeled on a vector space, the map $T_v(x)=x+v$ for a fixed vector $v$. Its derivative is the identity, and $T_v{T}_w=T_{v+w}$. Changes between flat coordinates on a translation surface have this form.

A space parametrizing isomorphism classes of geometric objects. Its topology and possible stabilizers are part of the definition; a moduli space of Riemann surfaces forgets the marking retained by Teichmüller space.

The moduli space of unmarked compact genus-$g$ Riemann surfaces. For $g\ge 2$ it is the quotient ${\mathcal{T}}_g/\mathrm{Mod}(S_g)$ of Teichmüller space by the mapping class group, with its quotient topology. The thick subsets are compact by Mumford's compactness theorem.

The subset of the unmarked moduli space of Riemann surfaces with hyperbolic systole at least $\epsilon >0$ is compact for fixed genus $g\ge 2$. Use a Bers pants decomposition theorem bound, finitely many topological decomposition types and Dehn twists to place representatives in finitely many compact boxes in Fenchel–Nielsen coordinates.

A foliation decomposes a topological manifold into connected leaves which locally look like parallel slices in product coordinate charts.

A leaf is a maximal connected integral manifold of the smooth constant-rank integrable distribution defining a regular foliation. It carries its intrinsic manifold topology and is immersed in the ambient manifold. It need not be embedded or closed. Local Frobenius theorem coordinates make each leaf segment a plaque with constant transverse coordinates; global confinement to an immersed leaf follows by joining such plaques along tangent curves.

A codimension-one foliation of a closed three-manifold is taut if every leaf meets a closed transverse curve. In particular, the fibers of a bundle over $S^1$ form a taut foliation.

A closed transversal is a closed curve everywhere transverse to the leaves of a foliation. A single such curve meeting every leaf proves that the foliation is a taut foliation.

For a compact topological surface with cone-point orders ${\alpha }_1,\dots ,{\alpha }_k$, the orbifold Euler characteristic is ${\chi }_{\mathrm{orb}}=\chi (\text{underlying surface})-{\sum }_i(1-1/{\alpha }_i)$. An orbifold covering multiplies this quantity by its degree.

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 ${\mathbb{Z}}^2$.

Every continuous map $f:S^n\to {\mathbb{R}}^n$ takes the same value at some antipodal pair. Equivalently, every continuous antipodal map $S^n\to {\mathbb{R}}^n$ has a zero, and there is no antipodal map $S^n\to S^{n-1}$.

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.

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 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.

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 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.

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 $X$.

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.

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

For based spaces, the smash product iswhere the wedge $X\vee Y$ consists of pairs having at least one basepoint coordinate.

Real projective space is the antipodal quotient ${\mathbb{RP}}^n=S^n/(x\sim -x)$.

This is the CW complex obtained as the increasing union of finite Real projective spaces under their standard inclusions. It has one cell in each nonnegative dimension. Its cellular chain complex has boundary multiplication by two in positive even dimensions and zero in odd dimensions. Consequently its integral cohomology is $\mathbb{Z}$ in degree zero, $\mathbb{Z}/2$ in positive even degrees and zero in odd degrees. Its mod-two cohomology is ${\mathbb{Z}}_2$ in every nonnegative degree.

Collapse the indicated standard skeleton to a point. Besides its basepoint, the quotient has one cell in dimensions $k+1$ through $m$. Its integral cellular boundary is two in even dimensions and zero in odd dimensions. Steenrod operations on its relative cohomology are inherited from real projective space.

For this quotient, homotopy groups through degree six and in degree eight vanish. The degree-seven and degree-nine groups are $\mathbb{Z}/2$. A map to $K(\mathbb{Z}/2,7)$ induces an isomorphism on degree-nine integral homology because the relevant square of the bottom mod-two class is nonzero. Two relative Hurewicz steps then compute degrees eight and nine.

For the degree-one mod-two generator, ${\mathrm{Sq}}^{1}x=x^2$ and higher squares of $x$ vanish by instability. The Cartan formula gives the displayed binomial identity. Naturality applies it to relative classes of a stunted real projective space.

The $n+1$ manifold charts $U_i=\{[X]:X_i\ne 0\}$ have coordinates $X_k/X_i$, with $k\ne i$. Their inverse inserts $1$ as the $i$th homogeneous coordinate. On an overlap, changing the denominator from $X_i$ to $X_j$ divides each remaining coordinate by the nonzero coordinate $X_j/X_i$. These rational smooth transition maps give Real projective space its standard smooth manifold structure.

For $m\ge 0$, the cohomology ring isandThe generator $a$ reduces to the square of the degree-one generator in the mod-two cohomology ring of real projective space; $b$ is the integral orientation class in odd top dimension.

The cellular cochain complex of finite Real projective space consists of alternating maps zero and two. Its groups can be tensored using the Künneth theorem; the integral Künneth torsion polynomial efficiently counts all tensor and Tor functor summands. A product of factors of dimensions two, three and four has free-rank polynomial $1+t^3$ and torsion polynomial $3t^2+3t^3+5t^4+6t^5+5t^6+4t^7+2t^8+t^9$.

Real projective space has one cell in every dimension from zero through $n$. With integral coefficients its cellular differential $C_k\to C_{k-1}$ is multiplication by $1+(-1{)}^k$, hence is zero for odd $k$ and multiplication by two for even $k$. With coefficients in ${\mathbb{F}}_2$, every cellular differential vanishes.

The mod-two cohomology ring of real projective space is

The complement of an open three-ball in ${\mathbb{RP}}^3$ is the mapping cylinder of the antipodal covering $S^2\to {\mathbb{RP}}^2$. It deformation retracts to ${\mathbb{RP}}^2$ and has boundary $S^2$.

The integral homology groups are

Doubling ${\mathbb{RP}}^3$ minus an open ball along its spherical boundary gives ${\mathbb{RP}}^3\#{\mathbb{RP}}^3$. Its fundamental group isand its universal cover is $S^2\times \mathbb{R}$.

The mapping torus of $f:F\to F$ isIf $f$ is a homeomorphism, the projection to the circle $S^1$ is a fiber bundle with fiber $F$. For a general continuous map, the mapping torus need not be a fiber bundle, but its homology still has the Wang sequence.

This mapping torus fibers over the circle with fiber $T^{n-1}$ and inversion as monodromy. The Wang sequence computes its additive cohomology. Coordinate projections to the Klein bottle expose its mod-two cup products.

The base class $t$ and coordinate classes $x_i$ all have degree one. Pullback of the mod-two intersection pairing of the Klein bottle gives $x_i^2=t{x}_i$. The Leray-Hirsch theorem gives the basis consisting of square-free products $x_I$ and $t{x}_I$. Thus the additive dimensions match those of a torus, but the degree-one squares differ for $n\ge 2$.

For $r=n-1$, the Wang sequence gives an additive splittingThe kernel is free, so the splitting exists as groups. The map in even degree is zero; the map in odd degree is multiplication by minus two.

Choose the base class $a$, degree-two torsion classes ${\tau }_i=\hat{\beta }(x_i)$, and degree-two free lifts $b_{ij}$ reducing to $x_i{x}_j$. A generator $\kappa$ of the top $\mathbb{Z}/2$ reduces to $t{x}_1{x}_2{x}_3$. The mod-two cohomology ring of an inversion mapping torus determines products in top integral degree, since reduction there is injective. The products of distinct $b_{12},b_{13},b_{23}$ equal $\kappa$; so does $b_{ij}{\tau }_k$ when $i,j,k$ are all distinct. Their other degree-four products vanish. Also $a{\tau }_i=0$ because degree-three integral cohomology is torsion-free.

A fiber monodromy is the return homeomorphism of the fiber of a bundle over $S^1$. Its mapping torus reconstructs the total space.

Homological monodromy is the induced automorphism of the homology of a fiber. For a fibered knot, its action on the first homology group presents the Alexander module.

The quotient of the real Heisenberg group by the Integer Heisenberg group is the mapping torus of the torus automorphismIts universal cover is the contractible real Heisenberg group, so its fundamental group is the integer Heisenberg group.

For the mapping torus of $f:F\to F$, the Wang exact sequence contains

For a continuous self-map $f:X\to X$, the mapping torus has the Wang sequence even when $f$ is not a homeomorphism. Cover the time circle by a seam neighborhood and an interval away from the seam. Their inverse images retract to $X$: the seam neighborhood is a mapping cylinder, and the other is a product. Their intersection retracts to two copies of $X$. The Mayer–Vietoris sequence map on homology is, after consistent choices of signs,Invertible changes of coordinates reduce it to the direct sum of $\mathrm{id}-f_{\ast }$ and an isomorphism. Canceling the isomorphism summands gives the Wang sequence. This proof does not assume a fiber bundle structure for a noninvertible $f$.

Homology assigns abelian groups $H_i(X)$ to a space, measuring cycles modulo boundaries in each dimension.

A positive-degree class whose reduced homology diagonal vanishes. In a torsion-free homology Hopf algebra, powers of an even primitive class have binomial diagonals. Dualizing that diagonal produces the multiplication of a divided power algebra.

A nonempty space is acyclic with the chosen coefficients if its reduced homology vanishes in every degree. Equivalently it has the homology of a point. It need not be contractible.

Suppose $X={\bigcup }_a{X}_a$ is directed by inclusion and every compact subset of $X$ lies in some $X_a$. Every finite singular chain then lies in one $X_a$, so $C_{\ast }(X)={\underrightarrow{\lim }}_a{C}_{\ast }(X_a)$. Exactness of filtered colimits gives

Every open subset $U\subseteq {\mathbb{R}}^N$ is the directed union of finite unions of closed rational cubes contained in $U$. Every compact subset of $U$ lies in one such finite polyhedron, whose homology is finitely generated. The homology of a directed union therefore expresses each $H_i(U;\mathbb{Z})$ as a direct limit over a countable family of countable groups, so it is countable.

Reduced homology modifies degree zero so that a one-point space has zero homology in every degree. For a nonempty space, ${\tilde{H}}_k(X)=H_k(X)$ when $k>0$, while $H_0(X)\cong {\tilde{H}}_0(X)\oplus \mathbb{Z}$.

A homology group is one of the abelian groups produced by a homology theory. Its elements are homology classes, represented in singular homology by cycles modulo boundaries.

The zeroth homology group is the free abelian group on the path components of $X$. In particular, if $X$ has finitely many path components, then

A homology class is an equivalence class of chain cycles modulo chain boundaries in a chain complex.

A continuous map $f:X\to Y$ induces homomorphisms $f_{\ast }:H_n(X)\to H_n(Y)$ satisfying $(g\circ f{)}_{\ast }=g_{\ast }\circ f_{\ast }$ and $({\mathrm{id}}_X{)}_{\ast }={\mathrm{id}}_{H_n(X)}$.

Homotopic maps induce the same homomorphism on every homology group. Consequently, homotopy equivalent spaces have isomorphic homology groups.

The first homology group $H_1(X)$ is the group of one-dimensional cycles modulo boundaries. For a path-connected space it is the abelianization of the fundamental group.

Singular homology forms a chain complex from formal sums of continuous maps from standard simplices into a topological space. Its homology groups are invariant under homotopy.

A Betti number is the dimension of a real singular homology group; on a smooth manifold it is also the dimension of the corresponding de Rham cohomology group. For a finite-dimensional cohomology group on a closed Riemannian manifold, it equals the dimension of its space of harmonic representatives. Betti numbers can increase under a finite cover, so equality is a separate hypothesis, not a general covering-space theorem.

A singular $n$-chain is a finite formal integer linear combination of singular simplices. Its support is contained in a compact subset of the ambient space.

The singular chain group is the free abelian group on all continuous maps ${\Delta }^q\to X$, the singular simplices. In particular it is free even when the set of simplices is infinite. This freeness makes applying $\mathrm{Hom}(C_q(X),-)$ to a short exact coefficient sequence exact.

A singular $n$-simplex in a space $X$ is a continuous map $\sigma :{\Delta }^n\to X$ from the standard $n$-simplex. Its alternating restrictions to the faces form its boundary.

For an oriented compact manifold of dimension $4k$, the intersection form is the bilinear pairing on middle-dimensional homology obtained by representing classes by transverse cycles and counting their signed intersection points. On an oriented four-manifold it is equivalently the cup-product pairing on $H^2$ evaluated on the fundamental class.

An integral intersection form on a finite-rank free abelian group $L$ is unimodular if its adjoint $x\mapsto \lambda (x,-)$ is an isomorphism to the integral dual. In an integral basis, this is equivalent to the pairing matrix having determinant $1$ or $-1$, not merely nonzero determinant. For example, projective lines generate a unimodular intersection form on the Complex projective plane, with matrix $(1)$ in its complex orientation.

When compact oriented four-manifolds are glued along an entire closed boundary three-manifold, the signature of the result is the sum of their intersection form signatures. It applies to $X_0{\cup }_{\partial X_0}(-X_1)$ with the difference of the two signatures. Gluing along only part of a boundary can require a correction; that is a different situation.

An intersection pairing counts signed intersections of transverse representatives of complementary homology classes, using one relative representative when the manifold has boundary. With ${\mathbb{F}}_2$ coefficients it is defined without orientation choices. Poincare-Lefschetz duality makes this a perfect pairing for compact manifolds.