Algebraic K-Theory and Manifold Topology

Lecture notes.Course Website Familiarity with the machinery of modern algebraic topology (simplicial sets,spectra, ...). Several other topics (such as piecewise linear topology, microbundles, im-mersion theory, the language of quasi-categories) will receive a cursory review as we needthem. A high level of mathematical sophistication will be assumed.Possible Topics• Wall’s Finiteness Obstruction• Simple homotopy equivalences and Whitehead torsion.• Polyhedra and simple maps; “higher” simple homotopy theory• Waldhausen’s algebraic K-theory of spaces• Assembly; the “parametrized index theorem” of Dwyer-Weiss-Williams.• Regular neighborhood theory.• Piecewise linear manifolds, microbundles, and immersion theory• The parametrized (stable) s-cobordism theorem• Concordance spaces and the Hatcher spectral sequence• Hilbert cube manifolds and infinite-dimensional topology 0 Course Syllabus 1 Overview 2 The Wall Finiteness Obstruction 3 Whitehead Torsion 4 Whitehead Torsion, Part II 5 Cell-Like Maps 6 Concordance of Polyhedra 7 Higher Simple Homotopy Theory 8 Fibrations of Polyhedra 9 Fibrations of Simplicial Sets 10 Combinatorial Models for M 11 Equivalence of the Combinatorial Definition 12 Some Loose Ends 13 Homotopy Types and Simple Homotopy Types 14 (Lower) K-theory of ∞-categories 15 The Wall Finiteness Obstruction Revisited 16 Higher K-Theory of ∞-Categories 17 The Additivity Theorem 18 Additive K-Theory 19 Algebraic K-Theory of Ring Spectra 20 Lower K-Groups of Ring Spectra 21 The Algebraic K-Theory of Spaces 22 Constructible Sheaves 23 Universal Local Acyclicity 24 The Assembly Map 25 The Assembly Map, Part II 26 The Assembly Map, Part III 27 Higher Torsion 28 Another Model of the Assembly Map 29 Another Model of the Assembly Map II 30 The Whitehead Space 31 The Whitehead Space II 32 A PullBack Square 33 Digression: Microbundles 34 Overview of Part 3 35 The Setup 36 Combinatorics I 37 Combinatorics II 38 Thickenings of a Point