Results on Hattori-Stallings traces, path sums and Pfaffian cocycles, sparse simplicial cycles, separating filtrations, and integral traces, 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.

• Isomorphic projective modules — Lemma 1.2. eA^q and fA^r are isomorphic exactly when there are rectangular matrices a, b with ba = e and ab = f. No hypothesis.
• The Hattori-Stallings trace — Proposition 2.2. The trace of an idempotent matrix modulo commutators is a homomorphism from K_0(A) to HH_0(A), natural in A. No hypothesis.
• Traces over group rings — Lemma 3.1. HH_0(RG) is the free R-module on the conjugacy classes of G, through class sums. No hypothesis.
• The Bass trace conjecture for complex group rings (OpenAI) — Theorem 4.1. For every group G, the trace of every class in K_0(CG) vanishes at every class of elements of infinite order; proved in lessons 2 to 8. No hypothesis.
• Trace rank and idempotents for torsion-free groups (OpenAI) — Corollary 4.2. For torsion-free G, the canonical trace of an idempotent matrix over CG is the rank of its augmentation, and RG has no idempotents other than 0, 1 for every commutative domain R of characteristic zero; proved in lesson 9. No hypothesis.
• Bass's conjecture for integral group rings (OpenAI) — Corollary 4.3. For every group G, the trace of every class in K_0(ZG) is supported on the identity class; proved in lesson 9. No hypothesis.
• Reduction to finitely generated groups — Corollary 5.2. The trace theorem for finitely generated groups implies it for all groups, by subgroup locality (Lemma 5.1). No hypothesis.
• Path identities — Lemma 4.1. The path sums of an idempotent are invariant under the twisted shift, deleting an entry maps v_k to v_(k-1), and every v_k has total weight the trace coefficient at [g]. No hypothesis.
• The exterior algebra on an ordered set — Lemma 1.2. Alternating multilinear maps factor through exterior powers, linear maps induce algebra homomorphisms, and permutations act on the top vector by their sign. No hypothesis.
• The full simplex is acyclic — Lemma 2.1. The augmented exterior chain complex of a nonempty set has no homology. No hypothesis.
• Cycles from path sums — Proposition 4.1. In every even dimension the projected path sums are cycles modulo H, supported on simplices whose vertex sets are connected in a graph of bounded degree independent of the dimension. No hypothesis.
• Sparse-chain vanishing (OpenAI) — Theorem 5.1. For n >= n_0(D), every invariant cocycle vanishes on every coinvariant cycle whose top simplices have connected vertex sets in a graph of degree at most D with a free action; proved in lessons 5 to 8. No hypothesis.
• Relative exponents — Lemma 1.1. A real function beta on pairs, antisymmetric, invariant under the centralizer and shifted by s - r under left multiplication by g^r, g^s. No hypothesis.
• Pfaffians — Lemma 2.1. The Pfaffian as top coefficient of b^m/m!: the matching formula, the sign under permutations and the expansion along the last index. No hypothesis.
• A recurrence for weighted Pfaffians — Lemma 3.1. The weighted Pfaffian sums satisfy q_m = q_(m-1)/2, so q_m = 2^-m lambda. No hypothesis.
• A Pfaffian cocycle detecting the trace coefficient — Proposition 4.1. For every m >= 1 an invariant cocycle psi_m on E_2m with psi_m(z_2m) = 2^-m lambda. No hypothesis.
• Vertices of bounded polyhedra — Lemma 2.3. A bounded polyhedron has finitely many vertices, and every linear functional attains its maximum at one of them. No hypothesis.
• Branches of the transport distance — Lemma 2.4. On a closed simplex the distance to a point is the maximum of finitely many affine functions, permuted by H. No hypothesis.
• Local finiteness — Lemma 3.2. Closed balls meet finitely many simplices and are compact; stabilizers are finite; only finitely many translates of a ball meet a simplex. No hypothesis.
• Tree Euclidean structures — Lemma 4.1. Every top simplex has volume 1/n! and every distance branch has gradient of norm at most one. No hypothesis.
• The volume bound and the margin — Proposition 5.2. A ball of radius 4D^2 has total top area at most V_R, and V_R < (2D^2)^n/n! for n >= n_0(D). No hypothesis.
• A free parameter space — Lemma 1.1. Every countable group acts freely on a compact zero-dimensional space with an invariant finitely additive probability on clopen sets. No hypothesis.
• Coning a polytope — Lemma 2.2. Coning a triangulated boundary from the mean of the vertices triangulates a convex polytope. No hypothesis.
• Compatible cuts — Lemma 4.1. Equivariant, locally constant cuts along generic level sets of the transport distance, with dimension drop and bounded components. No hypothesis.
• Carriers of the zero-dimensional stratum — Lemma 5.3. Every point of the last stratum lies inside a unique i-simplex of the i-th stratum, for every i. No hypothesis.
• Few zero strata — Proposition 1.1. For every epsilon > 0 an admissible filtration whose last stratum has expected count at most C epsilon in each top simplex. No hypothesis.
• Slicing a polytope — Lemma 3.1. The integral of the slice volumes of an affine function equals the gradient norm times the volume. Uses Fubini's theorem (core course).
• Subdivision chains — Lemma 1.2. The subdivision chain of a triangulated simplex commutes with the boundary. No hypothesis.
• Surviving flags — Lemma 4.1. At most 2^n flags per point of the last stratum survive the component labelling. No hypothesis.
• Equivariant homotopy — Lemma 5.1. Two equivariant augmented chain maps into the full simplex that agree in degree -1 are equivariantly homotopic when stabilizers are finite. No hypothesis.
• Kaplansky's idempotent conjecture in characteristic zero (OpenAI) — Theorem 2.1. For torsion-free G and every commutative domain R of characteristic zero, the only idempotents of RG are 0 and 1. Uses the trace of the Kervaire course and Lemmas 3.2 and 3.3 of the direct-finiteness course.
• A congruence for traces of p-th powers — Proposition 3.2. For an integral matrix A over ZG, the class sums of tr A^(p^k) and of the p-th powers of tr A^(p^(k-1)) agree modulo p^k. No hypothesis.
• Linnell's restriction — Theorem 3.4. A nonzero trace coefficient of a projective ZG-module at [g] forces g to be conjugate to g^(s^m) for all s >= 1; nontrivial finite-order classes carry no coefficient. No hypothesis.
• Bass's conjecture for integral group rings (OpenAI) — Theorem 4.1. The trace of every class in K_0(ZG) is supported on the identity class. No hypothesis.
• Fixed points of homotopy idempotents (OpenAI) — Corollary 5.1. Homotopy idempotents of closed oriented smooth manifolds of dimension > 2 are homotopic to maps with one fixed point, and Lefschetz and L2-Lefschetz numbers agree. Uses (BCM1) and (BCM2).

Reading order

Nine lessons. The first sets up the trace and states the results; the second to fourth build the cycles and the Pfaffian cocycle and prove the trace theorem from the sparse-chain vanishing theorem; the fifth to eighth prove that theorem; the ninth draws the consequences for idempotents, integral group rings and fixed points. The trace of the Kervaire course and two lemmas of the direct-finiteness course are used; two fixed-point theorems of Berrick, Chatterji and Mislin are named with free locators.