Results on equations over groups, the regular trace and spectral phase, unitaries with fixed spaces, disk-and-band surfaces and the planar boundary theorem, 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. An equation w(t)=1 over A is solvable in an overgroup exactly when the coefficient map A -> (A*<t>)/<<w>> is injective. No hypothesis.
• Coefficient injectivity for unimodular words (OpenAI) — Theorem 2.1. For every group A and every w in A*<t> with exponent sum +-1 the coefficient map is injective; proved in the last lesson from the named statements of the course.
• The Kervaire conjecture (OpenAI) — Corollary 2.2. If A is nontrivial, then (A*<t>)/<<w>> is nontrivial for every w. Uses Theorem 2.1.
• The regular trace — Lemma 1.2. tau(T) = <T delta_1, delta_1> is tracial, positive and faithful on the commutant of the right translations of any group. No hypothesis.
• Matrices over the commutant — Lemma 1.3. The trace tau_b on b x b matrices over the commutant is tracial, bounded, positive and faithful. No hypothesis.
• Functional calculus in the commutant — Lemma 2.1. Bounded Borel functions of a unitary in M_b(M) lie in M_b(M), their traces converge under bounded pointwise convergence, and they transform correctly under unitary conjugation, adjoints and exponentials. No hypothesis.
• Spectral phase — Lemma 3.1. l(U) = tau_b(h(U)) lies in [0,b], vanishes only at the identity, is invariant under conjugation and is the sum of the eigenvalue phases for scalar matrices. No hypothesis.
• Subadditivity of spectral phase (OpenAI) — Lemma 4.1. l(UV) <= l(U) + l(V) for all unitaries U, V in M_b(M), for every group. No hypothesis.
• Unitary groups — Lemma 1.1. U(m) is a compact connected smooth manifold of dimension m^2, with exponential charts. No hypothesis.
• Graph charts — Lemma 1.2. k-planes near E_0 are graphs of linear maps, and e^(K_Z) E_0 is the graph of Z tan(sqrt(Z*Z))/sqrt(Z*Z). Uses the inverse function theorem.
• The incidence manifold — Lemma 2.1. Pairs (W, E) of a unitary and a k-plane that it fixes form a compact smooth manifold of dimension m^2 - k^2. Uses the inverse function theorem.
• The unique preimage — Lemma 4.1. The model map G_(F_0) has exactly one preimage of the identity, at which its derivative is an isomorphism. No hypothesis.
• A unitary with a large fixed space (OpenAI) — Lemma 5.1. For signs with sum +-1 and every unitary P of size nk, some X in U(k) makes diag(X^(s_1), ..., X^(s_n)) P fix a subspace of dimension k. Uses degree modulo two from the programme.
• The planar boundary theorem (OpenAI) — Theorem 2.1. On a connected disk-and-band surface in a sphere for a unimodular word, if all boundary words but one are trivial, so is the last one; proved in the next lesson. Uses (SURF).
• Boundary circles — Lemma 3.1. The boundary circles are the cycles of sigma theta, and the boundary words are products of corner labels. Uses (SURF).
• Counting boundary circles — Lemma 4.1. A connected surface has k positive and k negative disks, nk bands and nk - 2k + 2 boundary circles. Uses (SURF).
• Regions of a connected surface — Lemma 4.2. Each complementary region of a connected disk-and-band surface in a sphere is a disk with one boundary circle. Uses (SURF).
• Flows with prescribed circulation — Lemma 5.1. Real phases on the boundary circles with sum zero are realized by antisymmetric numbers on the slots. No hypothesis.
• The mixing involution — Lemma 2.1. JSJ = S^-1, so SJ is a self-adjoint unitary of trace zero and l(SJ) = nk/2. No hypothesis.
• The upper bound — Lemma 2.2. For a suitable X, J Theta fixes 2k directions and l(S Theta) <= 3nk/2 - k. No hypothesis.
• Phase of a cycle — Lemma 3.1. l(T_z) = (|z| - 1)/2 + tau(h(e^(2 pi i eta_z) L(r_z^-1))). No hypothesis.
• Spaces for A and A*<t> — Lemma 1.1. A presentation complex Y of A has fundamental group A, and its wedge with a circle has fundamental group A*<t>. Uses (PI1).
• A piecewise affine circle coordinate — Lemma 3.1. The circle coordinate of a map of a punctured disk can be approximated within 1/12 by a map affine in lifted angles on a triangulation, with the same boundary behaviour. No hypothesis.
• Level arcs — Lemma 3.2. A generic level of the approximation consists of circles and arcs that pair the occurrences of t with opposite exponents, in a region where the coefficient coordinate is constant. No hypothesis.

Reading order

Six lessons. The first states the theorem and deduces the conjecture; the second builds the trace on matrices over the group von Neumann algebra and proves subadditivity of spectral phase; the third proves the fixed-space lemma by degree modulo two; the fourth sets up disk-and-band surfaces and their boundary words; the fifth proves the planar boundary theorem; the sixth turns a relation into such surfaces and proves the theorem. The functional calculus, degree modulo two and the inverse function theorem come from earlier programme lessons; three standard topological statements are named with free locators.