Results on dimensions of groups, word maps on unit quaternions, aligned presentations, right-angled Artin groups, level complexes and transport, 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.

• The comparison theorem for projective resolutions — Lemma 1.2. Chain maps from a projective augmented complex to an exact one exist over the identity and are unique up to chain homotopy. No hypothesis.
• cd <= gd — Lemma 2.2. The cellular chains of the universal cover of an n-dimensional classifying space form a free resolution of length n. Uses (CW).
• A plane subgroup — Corollary 3.3. A group containing Z^2 has cohomological dimension at least 2. No hypothesis.
• A counterexample to the Eilenberg-Ganea conjecture (OpenAI) — Theorem 4.1. There is a finitely generated residually finite group G with cd G = 2 and gd G = 3; proved in the last lesson from the named statements of the course.
• Estimates for unit quaternions — Lemma 1.1. Telescoping, the commutator estimate and invariance of the distance to 1 under inverses and conjugation. No hypothesis.
• Localized word-map degree (OpenAI) — Lemma 5.1. A word map of H^m with nonsingular exponent matrix has nonzero degree on every conjugation-invariant open set containing the identity with no root on its boundary. Uses (DEG).
• Moves that keep a basis — Lemma 2.1. Adjoining a letter with a definition relator, changing free generators and shears preserve the basis property of a presentation. No hypothesis.
• Aspherical presentations — Lemma 3.1. A group with a classifying space of dimension at most two has a presentation whose relator chains form a basis of the cycle module. Uses (HE), (CW).
• The aligned presentation (OpenAI) — Lemma 4.1. Such a group has a presentation with the basis property containing, for each word of a given cycle basis, a relator with the same chain, the other relators being controlled by letters of value 1. Uses (HE), (CW).
• Exact counts — Lemma 2.1. In an expression of a basis word as a product of conjugates of relators, the translated signed counts of the relators give exactly the basis vector of its distinguished relator. No hypothesis.
• The uniform quadratic estimate — Lemma 3.1. With S-labels within epsilon of 1 and the ordinary relators of a finite set trivial, the product along a basis word differs from that along its distinguished relator by at most C(h + epsilon)h, uniformly on the group and its finite quotients. No hypothesis.
• A forbidden value — Corollary 4.1. With epsilon <= eta, the supremum h of the distinguished products is never delta, and h <= 2C epsilon when h < delta. No hypothesis.
• Finite systems by degree — Lemma 2.1. On a finite quotient, small prescribed S-labels extend to solutions of any finite set of ordinary relators with all distinguished products within delta of 1. Uses (DEG).
• The comparison theorem (OpenAI) — Theorem 4.1. If a residually finite group with a finite cycle basis of words has a classifying space of dimension at most two, small quaternion labellings with product 1 along the basis words have product 1 along every closed path. Uses (DEG), (HE), (CW).
• Deleting a vertex — Lemma 2.1. A right-angled Artin group is the semidirect product of a free group on the cosets of the link subgroup with the group of the smaller graph. No hypothesis.
• Residual finiteness — Theorem 4.1. Right-angled Artin groups and all their subgroups are residually finite. No hypothesis.
• The cubes of the universal cover — Lemma 2.2. The cubes of the universal cover of the torus complex are embedded, meet in common faces, and are locally finite. Uses (COV), (CUBE).
• Acyclicity of the cube complex — Theorem 4.2. The cube complex of chains of a right-angled Artin group is a free resolution of Z. Uses (CUBE).
• The cube complex is contractible — Corollary 5.1. The universal cover of the torus complex is contractible, so its quotient by any subgroup is a classifying space of the same dimension. Uses (COV), (CUBE).
• Ascending and descending links — Lemma 2.1. The ascending and descending links of every vertex are barycentric subdivisions of L. No hypothesis.
• The level is acyclic — Theorem 4.1. For a nonempty acyclic flag complex L, the zero level of the cube complex is connected and acyclic. Uses (SIMP).
• The group and its cycle basis (OpenAI) — Proposition 5.1. G is finitely generated by the edge letters, residually finite, cd G = 2, gd G <= 3, and the chains of the triangle words form a basis of the cycle module. Uses (SIMP), (CUBE), (COV).
• An acyclic presentation complex — Lemma 1.1. The presentation complex of <x, y | x^2 = y^5, x^2 = (xy^-1)^3> is acyclic. Uses (CELL).
• The seed representation — Lemma 2.1. x -> u, y -> cos(pi/5) + v sin(pi/5) with u.v = 1/(2 sin(pi/5)) is a nontrivial homomorphism into the unit quaternions. Uses (PI1).
• The seed (OpenAI) — Lemma 3.1. There is a finite acyclic flag complex of dimension two with a nontrivial homomorphism of its fundamental group into the unit quaternions. Uses (BARY), (CELL), (PI1).
• Reduced traces — Lemma 2.1. Every element of a right-angled Artin group has a unique reduced trace, of length the word length. No hypothesis.
• Multiplying by a generator — Lemma 2.2. Right multiplication by a signed generator adjoins or deletes one maximal occurrence. No hypothesis.
• Direction vectors move slowly — Lemma 3.1. For g and gx different from 1, of lengths n and n + 1, the direction vectors lie in a common simplex and differ by at most 2/(n + 1) in the l^1 norm. No hypothesis.
• A slowly varying map of the level graph — Lemma 1.1. The map of the level graph into L has edge images of diameter at most 4/N and triangle boundary images of diameter at most 8/N. No hypothesis.
• Commuting-power paths — Lemma 2.1. Level paths through d a_q^-(N-j) a_r^-j map exactly onto the edge [q, r] of L, subdivided into N equal parts. No hypothesis.
• Small transport with nontrivial holonomy (OpenAI) — Proposition 4.1. For every epsilon > 0 the edges of the level complex carry quaternion labels within epsilon of 1 with product 1 around every triangle and a nontrivial product around some closed path. Uses (SC), (PI1), (COV).

Reading order

Eleven lessons. The first states the theorem; the second to fifth prove the comparison theorem (localized degree, aligned presentations, exact counts and the quadratic estimate, finite quotients and compactness); the sixth to eighth construct the group from a right-angled Artin group and prove residual finiteness, the contractible cube complex and the acyclic level with its cycle basis; the ninth builds the acyclic flag complex with its quaternion representation; the tenth and eleventh construct small labellings with nontrivial holonomy and complete the proof. Standard facts of algebraic topology are used as named statements with free locators in Hatcher; the Hurewicz, Whitehead, Sard and Tychonoff theorems come from earlier programme lessons.