Results on plane maps, free groups, Cayley graphs and van Kampen diagrams, with their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Faces and Euler's formula — Proposition 1.2. Every dart lies in one face, the boundary walks have total length 2E, a boundary walk follows a dart by its reverse exactly at a vertex of degree one, and V - E + F = 1 for the F bounded faces of a plane map. No hypothesis.
• Trees — Proposition 1.3. A plane map without bounded faces is a tree or a single vertex, and every map on a tree with an edge has one face and is a sphere map. No hypothesis.
• Deleting an edge between two faces — Proposition 2.1. If the darts of an edge lie in different faces, deleting it gives a plane map in which the two faces merge into one, cut and joined at the edge. No hypothesis.
• Deleting an edge with one face — Proposition 2.2. If both darts of an edge lie in one face, the edge is a bridge, both components are sphere maps, and the component containing the outer face is a plane map whose faces are those of D in it together with the shortened face. No hypothesis.
• Contracting an edge — Proposition 2.3. Contracting an edge that is not a loop gives a plane map whose faces are those of D with the edge removed from their boundary walks. No hypothesis.
• Pendant edges — Proposition 3.1. A new edge to a new vertex in a corner of a face gives a plane map in which that face gains the edge in both directions. No hypothesis.
• Chords — Proposition 3.2. A new edge between two corners of one face gives a plane map in which that face splits into two, each continuing along the old face. No hypothesis.
• Loops in a corner — Proposition 3.3. A new loop in a corner of a face gives a plane map with one new face of length one. No hypothesis.
• Cutting along a loop — Proposition 4.1. For a loop with one dart on the outer face, the edges with both darts on the side of the outer face form a plane map whose outer face is the old one without the loop and whose bounded faces are those of D on that side. No hypothesis.
• Normal form — Lemma 1.1. Every word is freely equal to exactly one reduced word, reached by every maximal sequence of cancellations. No hypothesis.
• The free group — Lemma 1.2. Reduced words with reduced concatenation form a group with the universal property of the free group. No hypothesis.
• Products of conjugates of relators — Lemma 2.1. The normal closure of the relators consists of the products of conjugates of words of the symmetrized set of relators. No hypothesis.
• Reduced walks — Lemma 3.1. Every walk is homotopic to exactly one walk without backtracks, and the classes of closed walks at a vertex form a group. No hypothesis.
• Closed walks in trees — Lemma 3.3. In a tree every closed walk is homotopic to the empty walk. No hypothesis.
• Determinism of the Cayley graph — Lemma 4.2. From each vertex of the Cayley graph exactly one walk has a given label; it is closed exactly when the label represents the identity. No hypothesis.
• Label-preserving maps — Lemma 4.3. A connected labelled graph maps to the Cayley graph by a label-preserving map with a prescribed image of a vertex exactly when all its closed walks at that vertex have labels representing the identity; the map is then unique. No hypothesis.
• Bouquets — Lemma 2.1. Every product of k conjugates of nonempty relators has a diagram of area k. No hypothesis.
• Cancelling a pair — Lemma 2.3. A diagram for w1 x x^(-1) w2 gives a diagram for w1 w2 whose bounded faces are bounded faces of the first. No hypothesis.
• Van Kampen's lemma: construction — Theorem 3.1. A word freely equal to a product of k conjugates of relators has a diagram with at most k bounded faces. No hypothesis.
• Van Kampen's lemma: reading the boundary — Theorem 4.1. The exterior walk of a labelled plane map is homotopic to a product of conjugates of its bounded face walks, one for each bounded face, read in either direction. No hypothesis.
• Diagrams and area — Corollary 4.2. A word represents the identity exactly when it has a diagram, and its area is the least area of a diagram for it. No hypothesis.
• Van Kampen's lemma: the map to the Cayley graph — Theorem 5.1. Every closed walk of a diagram has a label representing the identity, and the diagram maps to the Cayley graph by a unique label-preserving map with a prescribed image of the base vertex. No hypothesis.

Reading order

Read the lessons in order; the first uses the lesson Sphere maps of the course on Barnette's conjecture.