Results on systems of equations over groups, the cohomology of unitary groups, simultaneous fixed spaces and planar surfaces of several relators, 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.

• Solvability and coefficient injectivity — Lemma 1.2. A system over G is solvable in an overgroup exactly when the coefficient map G -> (G*F_n)/<<w_1, ..., w_m>> is injective. No hypothesis.
• Nonsingular systems are solvable (OpenAI) — Theorem 2.1. For every group G and words w_1, ..., w_m in G*F_n whose exponent-sum matrix has rank m over Q, the coefficient map is injective; proved in lessons 2 to 5. Uses (UC), (KU), (INT) and (REL).
• The Kervaire-Laudenbach conjecture (OpenAI) — Corollary 2.2. An equation in one unknown with nonzero exponent sum is solvable over every group. Uses Theorem 2.1.
• Infinite nonsingular systems (OpenAI) — Corollary 2.3. Any family of equations with linearly independent exponent rows is solvable over every group. Uses Theorem 2.1.
• Primitive generators of the cohomology of U(d) — Lemma 2.1. The exterior generators of H*(U(d); Q) can be chosen primitive, negated by inversion and compatible with all coordinate block inclusions. Uses (UC) and (KU).
• Pullback along block-diagonal maps — Lemma 3.1. For X -> diag(X_i^(s)) P the generator e_(a,d) pulls back to the sum over i with k_i >= a of the signed block counts times e^(i)_a. Uses (UC) and (KU).
• Incidence manifolds are orientable — Lemma 4.1. The manifold of pairs (W, E) of a unitary and an l-plane fixed by it is a compact orientable manifold of dimension d^2 - l^2. No hypothesis.
• Simultaneous fixed spaces (OpenAI) — Theorem 1.1. If the signed block counts of unitary variables of sizes k_i form a matrix of full row rank, the variables can be chosen so that the fixed spaces of the block matrices diag(X_i^(s)) P_j have total dimension at least sum k_i. Uses (UC), (KU) and (INT).
• Nested nonzero minors — Lemma 2.1. A rational matrix of full row rank r has columns j_1, ..., j_r whose leading t x t minors are nonzero for all t. No hypothesis.
• The leading term of the incidence class — Lemma 3.1. The dual class of the unitaries with an l-dimensional fixed space has leading term c e_(1,d) ... e_(l,d) with c nonzero. Uses (UC), (KU) and (INT).
• The intersection number — Lemma 4.1. The pullback of the product of incidence classes evaluates to a nonzero multiple of the product of nested minors. Uses (UC) and (KU).
• The planar boundary obstruction (OpenAI) — Proposition 1.1. On a connected disk-and-band surface in a sphere for a nonsingular system, if all boundary words but one are trivial, the last one is trivial. Uses (SURF), (UC), (KU) and (INT).
• Counting disks and boundary circles — Lemma 2.1. Row independence forces equally many positive and negative disks of each type, and a connected surface has 2 - 2K + b boundary circles. Uses (SURF).
• The mixing involution — Lemma 4.1. JSJ = S^-1, so SJ is a self-adjoint unitary of trace zero and l(SJ) = b/2. No hypothesis.
• The upper bound — Lemma 4.2. For suitable X_i, J Theta fixes 2K directions and l(S Theta) <= 3b/2 - K. Uses (UC), (KU) and (INT).
• Spaces for G and G*F_n — Lemma 1.1. A presentation complex Y of G has fundamental group G, and its wedge with n circles has fundamental group G*F_n. Uses (PI1).
• Piecewise affine circle coordinates — Lemma 3.1. The n circle coordinates of a map of a punctured disk can be approximated within 1/12 by maps affine in lifts on one triangulation, with the same boundary behaviour. No hypothesis.
• Level arcs for several unknowns — Lemma 3.2. Generic levels of the approximations consist of circles and arcs that pair the occurrences of each unknown with opposite signs, and levels of different unknowns are disjoint. No hypothesis.

Reading order

Five lessons. The first states the theorem and its consequences; the second sets up the rational cohomology of unitary groups; the third proves the simultaneous fixed-space theorem; the fourth proves the planar boundary obstruction; the fifth turns a relation into planar surfaces and proves the theorem. The trace, the spectral phase and the disk-and-band combinatorics of the Kervaire course are used; standard facts on the cohomology of unitary groups, the Künneth formula, intersection numbers and relations as maps of punctured disks are named with free locators.