Geometry and topology

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

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

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 $S^2\times \mathbb{R}$.

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

An exact sequence has the image of each map equal to the kernel of the next.

A commutative diagram is a diagram of objects and morphisms in which every two directed paths with the same endpoints define the same morphism.

A chain complex is a sequence of abelian groups or modules and homomorphisms $d_i:C_i\to C_{i-1}$ satisfying $d_{i-1}{d}_i=0$. Its homology is $H_i(C)=\ker d_i/\mathrm{im}{d}_{i+1}$.

An $i$-chain $z$ is a cycle when $d_{i}z=0$.

An $i$-chain is a boundary when it belongs to $\mathrm{im}{d}_{i+1}$.

The coefficients of a chain are the scalars multiplying its basis simplices.

A chain map $f:C\to C^{\prime }$ is a family $f_i:C_i\to C_i^{\prime }$ satisfying $d_i^{\prime }{f}_i=f_{i-1}{d}_i$.

A chain map sends cycles to cycles and boundaries to boundaries, and therefore induces $f_{\ast }:H_i(C)\to H_i(C^{\prime })$ by $[x]\mapsto [f_i(x)]$.

Chain maps $f,g:C\to C^{\prime }$ are chain homotopic when there are maps $h_i:C_i\to C_{i+1}^{\prime }$ such thatChain-homotopic maps induce the same map on homology.

For a chain map $f:C\to C^{\prime }$, one mapping-cone convention takes $M(f{)}_i=C_{i-1}\oplus C_i^{\prime }$ with a differential combining $d,d^{\prime }$ and $f$. Its short exact sequence with $C^{\prime }$ and a shift of $C$ produces a long exact sequence in homology.

A degreewise short exact sequence $0\to A\to B\to C\to 0$ of chain complexes induces a long exact sequence of homology groups through the connecting homomorphisms.

The connecting map sends a homology class in the quotient complex to the class obtained by lifting a representative, applying the middle differential, and identifying the result in the subcomplex.

The connecting homomorphism is the degree-shifting map produced by the lift-and-boundary construction in a long exact sequence.

For an abelian coefficient group $G$, there is a split short exact sequenceFor $G=\mathbb{Q}$, the Tor term vanishes.

Rational homology satisfiesIt records the free rank of integral homology and discards torsion.

Simplicial homology computes homology from the boundary maps between free abelian groups generated by oriented simplices.

For a finite complex,

If a finite simplicial complex has $f_i$ simplices of dimension $i$, thenWriting $C_i=Z_i\oplus$ a lift of $B_{i-1}$ and $Z_i=H_i\oplus B_i$ makes the boundary dimensions cancel in the alternating sum.

The vertices of the barycentric subdivision of a simplicial complex are its nonempty faces, and its simplices are strict chains of faces.

The barycentric subdivision of a tetrahedron has f-vectorIts two-skeleton therefore has Euler characteristic $15-50+60=25$ and homology $H_0\cong \mathbb{Z}$, $H_1=0$, and $H_2\cong {\mathbb{Z}}^{24}$.

The local homology at $x\in X$ is $H_i(X,X\setminus \{x\})$. It is preserved by homeomorphisms and, in a simplicial complex, is computed from the reduced homology of the local link.

In the two-skeleton of a barycentrically subdivided tetrahedron, the vertex corresponding to the whole tetrahedron has link equal to the connected 1-skeleton of the subdivided boundary, with 14 vertices and 36 edges. Its local $H_2$ therefore has rank $36-14+1=23$. Every other point has smaller local rank, so every self-homeomorphism fixes this vertex.

A homotopy between maps $f,g:Y\to X$ is a continuous map $H:Y\times [0,1]\to X$ with $H(y,0)=f(y)$ and $H(y,1)=g(y)$.

A map is null-homotopic when it is homotopic to a constant map.

A map $f:S^{n-1}\to X$ extends to $D^n$ exactly when it is null-homotopic. An extension contracts radially through the disc; conversely, a null-homotopy descends through the cone quotient $C{S}^{n-1}\cong D^n$.

A space is contractible when its identity map is homotopic to a constant map. Every map into a contractible space is null-homotopic.

Spaces $X$ and $Y$ are homotopy equivalent when there are continuous mapssuch that $g\circ f$ is homotopic to ${\mathrm{id}}_X$ and $f\circ g$ is homotopic to ${\mathrm{id}}_Y$.

A subspace $A\subseteq X$ is a retract of $X$ when there is a continuous map $r:X\to A$ whose restriction to $A$ is the identity.

Every retract of a contractible space is contractible. If $i:A\hookrightarrow X$ is the inclusion, $r:X\to A$ is a retraction, and $H$ contracts $X$, thencontracts $A$.

A pair $(X,A)$ has the homotopy extension property when every homotopy on $A$ whose initial map extends to $X$ can itself be extended to a homotopy on $X$.

If $A\subseteq X$ is contractible and $(X,A)$ has the homotopy extension property, the quotient mapis a homotopy equivalence. Extend a contraction of $A$ to $X$; its endpoint is constant on $A$ and therefore factors through $q$ to supply a homotopy inverse.

A based loop in $(X,x_0)$ is a path $\alpha :[0,1]\to X$ whose two endpoints are $x_0$.

A based homotopy between based loops keeps both endpoints fixed at the base point throughout the deformation.

The reverse of a path $\alpha$ is $\bar{\alpha }(t)=\alpha (1-t)$. Concatenating a path with its reverse is homotopic relative to endpoints to the constant path.

The fundamental group ${\pi }_1(X,x_0)$ consists of based loops modulo based homotopy, with multiplication induced by concatenation.

Concatenating two based homotopies proves that loop concatenation depends only on based-homotopy classes. Endpoint-fixing reparametrizations provide associativity and the identity laws on classes.

A continuous based map $f:(X,x_0)\to (Y,y_0)$ induces a homomorphism $f_{\ast }:{\pi }_1(X,x_0)\to {\pi }_1(Y,y_0)$ by composition of loops with $f$.

For a suitably path-connected open cover $X=U\cup V$, the fundamental group of $X$ is the pushout of the homomorphisms from ${\pi }_1(U\cap V)$ to ${\pi }_1(U)$ and ${\pi }_1(V)$.

Given homomorphisms $i_A:C\to A$ and $i_B:C\to B$, the amalgamated free product $A{\ast }_{C}B$ comes with homomorphisms from $A$ and $B$ that agree on $C$. It is universal with this property: every compatible pair of homomorphisms $A\to H$ and $B\to H$ factors uniquely through $A{\ast }_{C}B$.

If $X=U_1\cup U_2$, the sets $U_1,U_2$, and their intersection are path-connected, and the base point lies in the intersection, then ${\pi }_1(X)$ is generated by the images of ${\pi }_1(U_1)$ and ${\pi }_1(U_2)$. Subdivide each loop into arcs lying in cover members and join every transition point to the base point within the intersection; the inserted paths telescope.

A Möbius band retracts to its core circle, and its boundary wraps twice around the core. Attaching it to $X$ along a loop representing $w\in {\pi }_1(X)$ therefore adjoins a generator $x$ with relation $x^2=w$.

Attach Möbius bands to a torus along the classes $ab$ and $a^2{b}^3$. If their core generators are $x,y$, the resulting presentation isBecause the exponent vectors $(1,1)$ and $(2,3)$ form a unimodular basis of ${\mathbb{Z}}^2$, eliminating $a,b$ givesThis group maps onto $S_3$ by $x\mapsto (12)$ and $y\mapsto (23)$, so it is nonabelian.

Attaching a disc to $X$ along a based loop $\alpha$ kills precisely the normal closure of its homotopy class:

A group presentation gives a two-dimensional cell complex with one vertex, one oriented loop for each generator, and one 2-cell attached along each relator. Its fundamental group is the presented group.

Attach three 2-cells to $S^1\vee S^1$ along loops representing $a^2$, $b^3$, and $(ab{)}^2$. The resulting group isIndeed, $a\mapsto (12)$ and $b\mapsto (123)$ gives a surjection to $S_3$, while $aba=b^{-1}$ reduces every word to one of $b^i$ or $a{b}^i$, so the presented group has at most six elements.

For the closed orientable surface of genus $g$,In particular, the sphere has trivial fundamental group and the torus has fundamental group ${\mathbb{Z}}^2$.

A covering map $p:\tilde{X}\to X$ is locally a disjoint union of homeomorphisms onto the same evenly covered open neighbourhood.

A universal cover is a simply connected covering space. The universal cover of a cylinder is the plane, with the angular coordinate unwrapped to a real coordinate.

For a path-connected locally path-connected base, any two based universal covers lift to one another. The two composites and the respective identity maps are based lifts of the same covering maps, so uniqueness of lifts makes the composites identities. The lift is therefore a homeomorphism.

Let $X$ have contractible universal cover, let $K$ be a path-connected simplicial complex, and let $i:K^1\hookrightarrow K$. A map $f:|K^1|\to X$ extends to $|K|$ exactly when its induced fundamental-group map factors through $i_{\ast }$. The factorization kills every 2-simplex boundary; all higher-dimensional boundary maps lift to the contractible universal cover and are null-homotopic.

Under the usual local hypotheses, based connected coverings of $X$ correspond to subgroups of ${\pi }_1(X)$; changing the point above the base point conjugates the subgroup. Normal subgroups correspond to regular coverings.

For a connected covering corresponding to $H\le {\pi }_1(X)$, every fibre has cardinality $[{\pi }_1(X):H]$.

If the base of a covering is path-connected, a path from $x_0$ to $x_1$ gives a bijectionby sending each initial lift point to the endpoint of its unique lifted path. Lifting the reversed path gives the inverse bijection.

For a presentation complex, the covering associated with a subgroup has one vertex for each coset. A generator gives directed edges according to its coset action, and every relator has one lifted 2-cell beginning at each vertex.

For a connected regular covering corresponding to $H\le {\pi }_1(X,x_0)$, the subgroup $H$ is normal andThe quotient acts on a fibre by endpoints of lifted loops, and regularity makes this action transitive with kernel $H$.

A connected covering is normal, or regular, when its deck transformations act transitively on each fibre. Under the subgroup classification, this is equivalent to the associated subgroup of the fundamental group being normal.

A universal covering has simply connected total space, so its associated subgroup of the base fundamental group is trivial. The trivial subgroup is normal; hence every universal covering map is a normal covering map.

For a connected degree-$n$ cover of a closed orientable surface ${\Sigma }_g$, normality is forced when $n=1$, when $n=2$, or when $g=1$. For $g=0$, existence itself forces $n=1$. For every $g\ge 2$ and $n\ge 3$, nonnormal connected degree-$n$ covers exist.

For $g\ge 2$ and $n\ge 3$, sendwhere $\sigma =(12\cdots n)$ and $\tau =(12)$ in $S_n$, and send all remaining surface generators to the identity. The two commutators cancel, and $\sigma ,\tau$ generate $S_n$. The inverse image of a point stabilizer is a nonnormal subgroup of index $n$, hence defines a connected nonnormal degree-$n$ cover.