# Kasparov’s KK-theory

This draft edition contains all 23 planned numbered lessons: 01–23, plus companions on matrix positivity, noncompact induction foundations and cohomological Thom foundations. 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. Deformation asymptotic classes, direct K-theory map, tangent-groupoid closed index comparison and both Euclidean inverse E Bott products proved; four exercises solved. 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. 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. 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. Each lesson states its hypotheses and earlier proof dependencies. Lesson15 proves the assigned closed-manifold KK symbol and embedding index theorem in both Clifford degrees. Lesson16 proves wrong-way maps for possibly nonproper K-oriented maps, their composition, homotopy and projection formulas, and the global Thom/Dirac comparison. Lesson20 proves model independence, assembly naturality, finite-group and lattice isomorphisms, the all-cycle Mishchenko comparison and integral trace on the entire torsion-free assembly image. Lesson07 proves the arbitrary-source operator-homotopy and separable compact-homology extension pictures, and disproves unrestricted compact injectivity by an explicit counterexample.

The lesson text, solutions, companions and original figures use CC0 1.0. Reading references retain their authors’ terms. MathJax and fonts retain the supplied notices.

## Reading and downloads

The HTML readers, cumulative PDF and LaTeX contain the same 23 lessons and three companions. The editable source ZIP contains their Markdown, individual and cumulative LaTeX, figure sources, notices and builder. Earlier supporting programme lessons are accessible through the reader links and in the repository; they are not included in this ZIP. This is an editable source download, not an offline copy of the prerequisite programme.

## Reproduce

Run `python build_sources.py` with Pandoc installed. The builder uses the ordered lesson and supplement list in `course.json`, preserving the supplied figure assets. Run LuaLaTeX twice on `KT-KK.tex` to produce the cumulative PDF. The document uses standard packages including amsmath, amssymb, graphicx and hyperref.

The `assets` folder contains SVG diagrams and rendered PNGs. The `figures` folder contains the radial metric plot and wrong-way diagrams as PNG, SVG and Python sources, together with the cotangent and index-comparison PNGs and their Python sources; NumPy and Matplotlib are needed only to regenerate that plot.