Sources and component terms

The twenty-three available lessons develop extensions, Fredholm modules, graded correspondences, Kasparov products, unbounded modules, Thom classes and equivariant descent. Each lesson names the mathematical sources it uses and gives the precise hypotheses and prerequisites for its arguments.

Human mathematical sources

Bruce Blackadar: K-Theory for Operator Algebras, second edition (1998).
John M. Erdman: Functional Analysis and Operator Algebras: An Introduction, October 4, 2015.
Thierry Fack and Georges Skandalis: Connes–Thom isomorphism reference.
Heath Emerson and Ralf Meyer: Dualities in equivariant Kasparov theory (2010).

The references in each lesson give the other human mathematical credits. Reading references retain their authors’ terms and are not dedicated to the public domain by this course.

Authorship

Written by GPT-6.1 Sol (OpenAI).

The noncompact induction companion contains the topology, Haar and integration proofs used in equivariant induction. Its CC0 source is included in the PDF, LaTeX and source ZIP. Tao’s free Haar notes, Hunter’s free Bochner appendix and Fremlin’s free topology appendix are linked as reading.

Component licences

The independent numbered lessons and original figures use CC0 1.0.

The matrix-positivity companion gives complete finite-dimensional proofs and solved exercises under CC0 1.0. Erdman’s freely accessible author edition is further reading. Its Markdown and LaTeX are included.

MathJax retains Apache 2.0 and its font terms. Figure font notices: DejaVu, STIX, BaKoMa. Supporting readings retain their identified authorship and terms.

Read and reproduce

PDF of the complete available text · LaTeX · Editable source package · Reproduction instructions.

Supporting proof readings

The polynomial and contour foundation preserves independently written finite algebra, topology and contour proofs under CC0 1.0, with its structured source and PDF.

The analytic foundation readings contain independent bounded trace-ideal, Banach, positive spectral and cyclic proofs. Their component record identifies the retained sources and notices. The compact-group and Lie companions prove Haar measure, integration, strict Hilbert-module averaging and the closed-subgroup theorem used by the numbered lessons. Cited external works retain their rights.

The cohomological Thom companion proves the exact ordinary-cohomology and relative-character inputs used in Lesson 13. Its CC0 source is included in all course downloads, with Hatcher’s freely posted texts linked as reading.

Unbounded Kasparov modules and spectral triples proves the bounded transform, unbounded representatives and the elliptic and analytic estimates used in its examples. Its general smooth correspondence statement cites Mesland’s freely available preprints; the elliptic comparison uses Grubb’s freely posted lecture notes. Exact references and source terms appear in the lesson.

Pictures of KK proves the scalar Fredholm pictures, coefficient Toeplitz and Bott inputs, and separable extension comparison. Its unrestricted compact-homology comparison is disproved by a complete counterexample; the arbitrary-source operator-homotopy and separable compact-homology comparisons are proved. Its mathematical references are freely accessible author editions and preprints.

The Baum–Connes assembly map and the conjecture proves the cutoff and naturality constructions, finite-group assembly and the free covering Dirac/Mishchenko comparison. Its universal-model construction, lattice isomorphism, all-cycle Mishchenko comparison and torsion-free assembly trace integrality are proved in full. Its references are freely available preprints.

Wrong-way maps for K-oriented maps proves the manifold construction, composition, Thom and Dirac comparisons, homotopy and projection formulas, including noncompact manifolds and nonproper maps. Its reading references are freely accessible author editions and the freely readable Connes–Skandalis paper. Deformation, correspondence and foliation formulations are stated separately and are not premises of these proofs.

Dirac classes and the cotangent Dolbeault element proves the assigned closed-manifold KK symbol and embedding index comparison, including both graded Clifford degrees. Its complete pseudodifferential proofs are in the freely sourced unbounded lesson; its readings are the free author editions and freely readable Connes–Skandalis paper. Full duality, open and boundary results are separate unused statements.

Asymptotic morphisms and E-theory. Asymptotic composition, scalar comparison, KK-to-E functor, both Bott products, simultaneous suspension, half-exactness and both six-term sequences proved; all four exercises solved. The stronger universal property and nuclear comparison are unused stated results.

Deformations and the analytic index in E-theory. Deformation asymptotic classes, direct K-theory map, tangent-groupoid closed index comparison and both Euclidean inverse E Bott products proved; four exercises solved.

The longitudinal index theorem and assembly for foliations. Embedding independence, zero-foliation and full Clifford-source one-leaf comparisons, compact smooth geometric assembly descent and four exercises proved; general index and proper-model comparisons stated with exact free sources.

The exact extension comparison: arbitrary-source compact injectivity is false under the stated countably generated conventions; the counterexample and corrected operator-homotopy theorem are proved in full.

Compact-group separation is proved by continuous convolution and finite spectral detection for every compact Hausdorff group. This supplies the small compact-normal kernels used by the Lie-quotient proof.

Proper actions, universal proper spaces and equivariant K-homology. Proper-space cutoffs, positive Hilbert universal models, CAT(0) barycenters, stagewise equivariant K-homology, finite-group calculation and enlarged buildings for arbitrary connected reductive groups over local fields proved; all four exercises solved.

The index pairing between K-theory and K-homology. Even and odd index pairings, relative nonunital classes, boundary sign and analytic twisted Dirac comparison, with a complete bounded Fredholm-character proof using free Schatten and trace foundations; all four exercises solved.