Results on assembly for the torus, the Heisenberg group and its laws, the labelled tower, Bott symbols and their sampling, the graphical group and the parity swindle, 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.

• A torsion-free counterexample to reduced Baum-Connes injectivity (OpenAI) — Theorem 1.1. There are a finitely generated torsion-free group G_inj, an injective i: Z^2 -> G_inj and c in H^2(BG_inj; Z) with <(Bi)*c, [T^2]> = 1 and i_*(beta) = 0 in K_0(C_r*(G_inj)); the induced torus Dirac class has infinite order and reduced assembly image zero. Proved in lessons 1 to 18. Uses (BG) and (CC), and for the infinite order (LB) and (IND).
• The Bott class of the course — Lemma 1.1. The class [p_k] - [e] of a scaled compact Bott projection is the rank-zero Bott generator with Chern number 1. No hypothesis.
• Assembly of the torus — Lemma 3.1. Reduced assembly for Z^2 sends the class of the torus Dirac operator to plus or minus the rank-zero Bott generator. No hypothesis.
• Induction along a subgroup — Lemma 3.2. For injective i: Z^2 -> G, the assembly of the induced torus class is the image under C_r*(Z^2) -> C_r*(G) of the assembly for Z^2. No hypothesis.
• Maps into the bar model — Lemma 3.3. Every stage of a torsion-free group maps equivariantly into the bar model EG, and any two equivariant maps from the induced plane bundle into EG are equivariantly homotopic. Uses (BG).
• Detection — Lemma 3.4. If c in H^2(BG; Z) evaluates to plus or minus 1 on the torus of i, the induced torus class has infinite order in the domain of assembly. Uses (BG), (LB) and (IND).
• Normal form of the Heisenberg group — Lemma 1.1. Every element has exactly one expression x^n y^l z^c, with (n,l,c)(n',l',c') = (n+n', l+l', c+c'-ln'), and H is isomorphic to the integer upper unitriangular group. No hypothesis.
• Commutators — Lemma 1.3. [g,h] = z^(nl'-ln') for g = (n,l,c) and h = (n',l',c'). No hypothesis.
• Coordinates of a word — Lemma 2.1. A word is an H-law exactly when every exponent sum vanishes and the quadratic polynomial Q of its third coordinate is zero. No hypothesis.
• Bicycles inside cycle rank two — Lemma 3.2. Every finite connected graph of cycle rank at least two contains a bicycle. No hypothesis.
• Laws use a bicycle four times (OpenAI) — Theorem 4.1. The support of a closed path reduced as a based path whose word is an H-law contains a bicycle every edge of which is traversed at least four times. No hypothesis.
• Sizes and cutoffs — Lemma 1.1. For every height i the active input lengths add up to L_i, |B_i| = 2^(H_i) and |A_i| = 2^(H_i + 2r - s); the active coordinates form an initial segment of the source order in one of two cutoff forms, and from height i to i + 1 exactly one active length grows, by b. No hypothesis.
• Coefficient choice (OpenAI) — Lemma 3.1. The injections into the coefficient pools can be chosen so that, within every family, extra value and side, each restricted functional on the first t input coordinates, b <= t <= s, is the same for at most 2^(1-b) n of the n names, and distinct names have distinct coefficients. For large s.
• Causality — Lemma 4.1. Every gate g_k with k > 20 reads only raw input blocks of index at most k - 5; correction offsets and table keys are determined by the label and by earlier raw blocks; marked cutoffs of one family and their identity zones do not overlap. No hypothesis.
• Inversion and uniform coordinates (OpenAI) — Lemma 1.1. For every type and height pair the auxiliary prescription has exactly one edge at every endpoint state and slot, the incidence of an extra slice is biregular with degrees d and D, and for a uniform endpoint state the free raw coordinates and head bits are independent uniform bits. No hypothesis.
• Padded-prefix recovery (OpenAI) — Lemma 3.1. Two immersed surviving paths with the same oriented label word agree in both raw streams through r + js bits at A-positions of radius at least 2j + 1 and through js bits at B-positions of radius at least 2j. No hypothesis.
• The labelled tower (OpenAI) — Proposition 5.1. The labelling is folded, the signed alphabet satisfies sqrt(|S|)/q <= 2^(-0.000025 s + O(log s)), the deleted fractions are small uniformly in the height, and an immersed path starting at height i of length at least ceil(1.002 m_i) is the only immersed path with its word. For large s and initial height.
• The conditional mean (OpenAI) — Lemma 2.2. For every history, marked cutoff and extra slice, the conditional mean incidence has norm at most sqrt(5 theta) < 3/5 off the constants, also with all extra values together. No hypothesis.
• Local lemma with conditioning — Lemma 4.1. If each event is independent of its non-neighbours and P(A_i) <= x_i prod_(j~i) (1 - x_j), all events are avoided with positive probability, P(A_i | avoidance) <= x_i, and an event B independent of the events outside J_1 has P(B | avoidance) <= P(B) prod_(J_1) (1 - x_j)^(-1). No hypothesis.
• Cheeger bound — Lemma 5.1. A biregular bipartite multigraph whose cuts satisfy c(S) >= 0.05 vol(S) for vol(S) <= V_0/2 has normalized incidence of norm at most 1 - 1/800 off the constants. No hypothesis.
• Sequential selection (OpenAI) — Proposition 6.1. A probability law on all permutation tables makes every auxiliary incidence expand with norm at most 1 - 1/800 off the constants, with conditional joint tests of k current tables bounded by e^(eps_s k) times their uniform probability. For large s.
• Unique reconstruction and the free-bit count (OpenAI) — Lemma 3.1. For every realization of the tables, a path pattern with different junction names has at most one realizing coherent return, with at most (H_- + H_+)/2 - h free bits, and a theta pattern has at most one realizing assignment and no free bits. No hypothesis.
• Predictable accumulated rank (OpenAI) — Lemma 4.1. If tests selected at finitely many source indices have conditional product probabilities at most 2^(-R_t) and involve k_t tables, an event implying that they pass with sum R_t >= R and sum k_t <= K has probability at most e^(eps_s K) 2^(-R). For large s.
• Terminal compatibility (OpenAI) — Lemma 1.1. At a receiver with distinct keys and open terminal gates, the terminal conditions have product probability 2^(-lambda(v)) when the incident cutoffs equal that of v, and an implied event has probability at most 2^(-max(0, lambda(v) - 4b)) when they are at most b bits lower. No hypothesis.
• The first deletion gives base girth (OpenAI) — Lemma 4.2. After the first deletion every nonempty cyclically reduced closed path based at height i has more than 0.93 m_i edges. For large s and initial height.
• Expected number of thetas (OpenAI) — Lemma 4.1. The expected number of rooted bicycles with fewer than 1.52 m_i edges and root at height i is at most 2^(0.89 H_i). For large s/b and then large H_0/s.
• The pruned old graph (OpenAI) — Proposition 5.1. The two deletions can remove less than any fraction epsilon of every layer while the pruned old graph keeps folding and recovery, has every cyclically reduced closed path at height i longer than 0.93 m_i, no bicycle at height i shorter than 1.52 m_i, and no cyclically reduced closed path of at most 6.04 m_i edges whose word is an H-law. For large s and initial height.
• The compact Bott projection — Lemma 2.1. A smooth projection p on the plane, equal to diag(0,1) outside the unit disc, whose range has clutching degree -1 and Chern integral 1. No hypothesis.
• Golden separation — Lemma 4.1. ||n alpha|| >= 1/(3|n|) for alpha = (sqrt 5 - 1)/2 and every integer n != 0. No hypothesis.
• Supported spreading (OpenAI) — Lemma 5.1. Translates of the torus rectangle by powers y^j and y^k with 0 < |j - k| <= 2J_i are orthogonal, P_i = T_i E_i T_i* is a projection of trace at most 4R^2 2^(-700 H_i), and spread symbols multiply like their profiles. For large initial height.
• Fourier envelope (OpenAI) — Lemma 3.1. The coefficients of a spread symbol T_i f T_k* in the Heisenberg normal form are bounded by C N^(-1) (1 + |n|/N)^(-P) (1 + J||n alpha||)^(-3), uniformly in the height, so the symbol has bounded l^1 norm and truncations of height-independent relative size. No hypothesis.
• Finite product sampler (OpenAI) — Lemma 4.1. For every eta > 0 each spread symbol is within eta in l^1 of the mean of a finite law of a bounded matrix weight and a Heisenberg voltage, with independent coordinates n and l of atoms O(J_i/N_i) and O(1/J_i) and weights from a finite set independent of the height. For large initial height.
• Simultaneously sampled symbols (OpenAI) — Proposition 1.1. Voltages and weights from one finite set can be chosen so that the cover has no cyclically reduced closed walk of length at most 6.03 m_i at an old vertex of height i, every weighted incidence is within eta of its spread symbol in the full operator norm, and the refined alphabet satisfies log_2|A| < log_2(q^2). For large s and initial height.
• Polynomial avoidance (OpenAI) — Lemma 2.2. For a closed base walk whose word is not an H-law, the holonomy of the sampled voltages is the identity with probability at most C_b 2^(-50 H_min). No hypothesis.
• Heisenberg boxes and the regular norm (OpenAI) — Lemma 3.1. Finite boxes in H with boundary ratio C 2^(-BH) under translations with coordinates at most 2^(a_0 H) test the regular norm of finite convolution matrices up to C 2^(-BH) R. No hypothesis.
• Centered block concentration (OpenAI) — Lemma 4.2. Each centered block X_ik has norm greater than eta q_(tau,ik) with probability at most 2^(-6 H_i), uniformly in the height, under the independent law. For large s and initial height.
• Private completion (OpenAI) — Lemma 2.1. The old graph extends, with private letters and voltages, to a connected countable folded labelled graph whose covering graph is connected with deck group H, has simple cycles longer than 6.03 m(v), and in which paths with the same word and different projections avoid private edges and are shorter than L m. For large s and initial height.
• Closure of null-labelled paths (OpenAI) — Lemma 4.1. A path in the covering graph whose word is trivial in Gamma is closed. No hypothesis.
• Graphical realization (OpenAI) — Proposition 5.1. The covering graph embeds label-preservingly in the Cayley graph of Gamma, the Heisenberg group embeds as a subgroup, and the sheets are indexed by Gamma/H. No hypothesis.
• The exterior-arc bound (OpenAI) — Lemma 1.1. For a polygon of paths through k >= 2 distinct sheets there are a finite diagram and paths with immersed sides, at least seven arc incidences on every bounded face and at most 4k - 6 on the unbounded face. No hypothesis.
• Finite peeling (OpenAI) — Lemma 3.1. Every nonempty finite family of sheets has a sheet whose contacts with the others are empty, in an old-centered ball or a private vertex, so the family can be ordered with allowed removed sets. No hypothesis.
• Direct cycle coordinates (OpenAI) — Lemma 1.1. For every commutative ring the cycle space of the Cayley graph is the direct sum of the cycle spaces of the sheets. No hypothesis.
• Simultaneous disjoint domains (OpenAI) — Proposition 2.1. There are removed sets R_Q, each empty, an old-centered ball or a private singleton, whose complementary domains D_Q are pairwise disjoint, such that the Cayley edges outside the sheets' own domains form a forest. No hypothesis.
• Exact cone chains (OpenAI) — Lemma 1.1. The augmented cellular chain complex of the cone complex is exact over every commutative ring. No hypothesis.
• A primitive Heisenberg detector (OpenAI) — Lemma 3.1. alpha gamma' + binom(alpha,2) beta' is an integral 2-cocycle on H whose class evaluates to -1 on the torus (z,x). Uses (BG) and (CC).
• Integral restriction is onto in degree two (OpenAI) — Lemma 4.1. Restriction H^2(B Gamma; Z) -> H^2(BH; Z) is onto. Uses (BG) and (CC).
• Torsion-freeness and the integral detector (OpenAI) — Proposition 5.1. The graphical group is torsion-free, Z^2 -> Gamma via (z,x) is injective, and some c in H^2(B Gamma; Z) evaluates to -1 on the torus. Uses (BG) and (CC).
• Synthesis on the active carriers (OpenAI) — Lemma 3.2. The embedding of the model space into the sheets differs from its compression to the disjoint domains by at most C 2^(-344.5 H_0)/(1 - 2^(-344.5 b)) in norm. For large initial height.
• Constant extraction — Proposition 4.2. For every band profile f, the route R_k(f), a finite word in the mean incidence blocks, is within C_f (rho^(2k) + k sqrt(eps_del)) of the model profile. No hypothesis.
• The residual forest — Lemma 2.1. The part of a Cayley polynomial along the edges outside the domains has norm at most C sqrt(|S_wt|)/q, which tends to zero. No hypothesis.
• Finite formulas for a profile family (OpenAI) — Lemma 3.1. Every prescribed band profile, compressed to the domains and placed in the regular representation, is within eta of a finite matrix element of the group algebra, simultaneously over heights and sheets. For large s and initial height.
• Finite additivity — Lemma 3.1. For self-adjoint a', a'' with a'a'' = a''a' = 0 and e + a', e + a'' projections, [e + a' + a''] - [e] = ([e + a'] - [e]) + ([e + a''] - [e]), through an explicit path of projections. No hypothesis.
• Supported Bott transport (OpenAI) — Lemma 4.1. For a scalar projection e, e + a a projection and v with v*va = a, [e + a] - [e] = [e + vav*] - [e] in K_0(B), through an explicit path of projections. No hypothesis.
• Eventual coordinate equality (OpenAI) — Lemma 5.1. Equal classes of projections of fixed size in K_0 of a norm sequence quotient give equal classes in all late coordinates. No hypothesis.
• Reduced Bott vanishing (OpenAI) — Proposition 5.2. For all large constructions in the compatible sequence the rank-zero Bott class of the torus (z,x) vanishes in K_0 of the reduced group C*-algebra. No hypothesis.
• A finite witness (OpenAI) — Lemma 5.3. An equality of K_0 classes of projections over C_r*(H) that holds in C_r*(Gamma) holds already over a finitely generated subgroup containing H. No hypothesis.

Reading order

Eighteen lessons. The first computes the assembly of the torus class and deduces the theorem from the two assertions proved in the rest of the course; lessons 2 to 8 build and prune the labelled tower; lessons 9 to 11 sample Heisenberg voltages and matrix weights that realize spread Bott symbols; lessons 12 to 15 define the graphical group, embed the cover, choose disjoint domains and prove torsion-freeness and the integral detector; lessons 16 to 18 carry the Bott symbols into the reduced group algebra and kill the Bott class. Van Kampen diagrams and plane maps come from the course Van Kampen diagrams; the bar model, cellular cohomology, the classification of line bundles and the index theorem on the torus are used as named statements with free locators.