Seifert-van Kampen theorem

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 ${\pi }_1(V)$.

Forthe Seifert-van Kampen theorem gives

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 $A{\ast }_{C}B$.

The free product $A\ast B$ is the amalgamated free product over the trivial group. Its nonidentity elements have unique reduced alternating normal forms.

Given group homomorphisms $f_A:A\to H$ and $f_B:B\to H$, there is exactly one group homomorphism $A\ast B\to H$ extending both. Its value on an alternating normal form is the product of the factor-map values. Thus the free product is the coproduct in the category of groups.

A free product $A\ast B$ is nontrivial as a splitting when both factors differ from the trivial group. Alternating words $(ab{)}^r$ with $a,b\ne 1$ in different factors show that every such splitting gives an infinite group. The distinct normal forms $ab$ and $ba$ also show that the group is not an abelian group.

For nontrivial groups $A,B$, the Cayley graph of $A\ast B$ with $S=(A\setminus \{1\})\cup (B\setminus \{1\})$ has the edges of the Bass-Serre tree as its vertices. Two such vertices are adjacent exactly when the tree edges share an endpoint. Mapping each group element to its tree-edge midpoint is an isometry on the vertex metrics and has image within distance $1/2$ of every tree point, hence gives a quasi-isometry of metric graphs. The normal form theorem for a free product identifies $|g{|}_S$ with reduced syllable length. The factor subgroups have diameter one in this metric.

Every element of a free product $G\ast H$ has a unique expression $u_1\cdots u_n$ in which each $u_i$ is a nonidentity element of one factor and adjacent syllables belong to different factors. The identity corresponds to the empty word.

A syllable is a nonidentity element of one factor in a free product word. A reduced normal form has successive syllables in different factors; its syllable length is the number of these elements. This length records factor elements, rather than letters in a chosen generating set.

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.