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Since its introduction by Friedhelm Waldhausen in the 1970s, the algebraic K-theory of spaces has been recognized as the main tool for studying parametrized phenomena in the theory of manifolds. However, a full proof of the equivalence relating the two areas has not appeared until now. This book presents such a proof, essentially completing Waldhausen's program from more than thirty years ago. The main result is a stable parametrized h-cobordism theorem, derived from a homotopy equivalence between a space of PL h-cobordisms on a space X and the classifying space of a category of simple maps of spaces having X as deformation retract. The smooth and topological results then follow by smoothing and triangulation theory. The proof has two main parts. The essence of the first part is a \"desingularization,\" improving arbitrary finite simplicial sets to polyhedra. The second part compares polyhedra with PL manifolds by a thickening procedure. Many of the techniques and results developed should be useful in other connections.

The canonical map from the Kan subdivision of a product of finite simplicial sets to the product of the Kan subdivisions is a simple map, in the sense that its geometric realization has contractible point inverses.

The author introduces the notion of a Galois extension of commutative $S$-algebras ($E \\infty$ ring spectra), often localized with respect to a fixed homology theory. There are numerous examples, including some involving Eilenberg-Mac Lane spectra of commutative rings, real and complex topological $K$-theory, Lubin-Tate spectra and cochain $S$-algebras. He establishes the main theorem of Galois theory in this generality. Its proof involves the notions of separable and etale extensions of commutative $S$-algebras, and the Goerss-Hopkins-Miller theory for $E \\infty$ mapping spaces. He shows that the global sphere spectrum $S$ is separably closed, using Minkowski's discriminant theorem, and he estimates the separable closure of its localization with respect to each of the Morava $K$-theories. He also defines Hopf-Galois extensions of commutative $S$-algebras and studies the complex cobordism spectrum $MU$ as a common integral model for all of the local Lubin-Tate Galois extensions.

We specify exterior generators for \\(\\pi_* THH(MU) = \\pi_*(MU) \\otimes E(\\lambda'_n \\mid n\\ge1)\\) and \\(\\pi_* THH(BP) = \\pi_*(BP) \\otimes E(\\lambda_n \\mid n\\ge1)\\), and calculate the action of the \\(\\sigma\\)-operator on these graded rings. In particular, \\(\\sigma(\\lambda'_n) = 0\\) and \\(\\sigma(\\lambda_n) = 0\\), while the actions on \\(\\pi_*(MU)\\) and \\(\\pi_*(BP)\\) are expressed in terms of the right units \\(\\eta_R\\) in the Hopf algebroids \\((\\pi_*(MU), \\pi_*(MU \\wedge MU))\\) and \\((\\pi_*(BP), \\pi_*(BP \\wedge BP))\\), respectively.

Given a compact Lie group \\(G\\) and a commutative orthogonal ring spectrum \\(R\\) such that \\(R[G]_* = \\pi_*(R \\wedge G_+)\\) is finitely generated and projective over \\(\\pi_*(R)\\), we construct a multiplicative \\(G\\)-Tate spectral sequence for each \\(R\\)-module \\(X\\) in orthogonal \\(G\\)-spectra, with \\(E^2\\)-page given by the Hopf algebra Tate cohomology of \\(R[G]_*\\) with coefficients in \\(\\pi_*(X)\\). Under mild hypotheses, such as \\(X\\) being bounded below and the derived page \\(RE^\\infty\\) vanishing, this spectral sequence converges strongly to the homotopy \\(\\pi_*(X^{tG})\\) of the \\(G\\)-Tate construction \\(X^{tG} = [\\widetilde{EG} \\wedge F(EG_+, X)]^G\\).

We compare the weight and stable rank filtrations of algebraic K-theory, and relate the Beilinson-Soulé vanishing conjecture to the author's connectivity conjecture.

A minimal resolution of the mod 2 Steenrod algebra in the range \\(0 \\leq s \\leq 128\\), \\(0 \\leq t \\leq 200\\), together with chain maps for each cocycle in that range and for the squaring operation \\(Sq^0\\) in the cohomology of the Steenrod algebra.

We establish motivic versions of the theorems of Lin and Gunawardena, thereby confirming the motivic Segal conjecture for the algebraic group \\(\\mu_\\ell\\) of \\(\\ell\\)-th roots of unity, where \\(\\ell\\) is any prime. To achieve this we develop motivic Singer constructions associated to the symmetric group \\(S_\\ell\\) and to \\(\\mu_\\ell\\), and introduce a delayed limit Adams spectral sequence.

We discuss proofs of local cohomology theorems for topological modular forms, based on Mahowald-Rezk duality and on Gorenstein duality, and then make the associated local cohomology spectral sequences explicit, including their differential patterns and hidden extensions.