2020/05/11 by Andrew J. Blumberg, Andrew Blumberg, Michael Mandell +1 · 1 citation
Mathematics · Computer Science · #Homotopy and Cohomology in Algebraic Topology #Algebraic structures and combinatorial models #Topological and Geometric Data Analysis
paper · doi:10.1090/memo/1286
We develop a theory of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T upper H upper H"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mi>H</mml:mi> <mml:mi>H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">THH</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T upper C"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mi>C</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">TC</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of Waldhausen categories and prove the analogues of Waldhausen’s theorems for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -theory. We resolve the longstanding confusion about localization sequences in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T upper H upper H"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mi>H</mml:mi> <mml:mi>H</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">THH</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T upper C"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mi>C</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">TC</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and establish a specialized dévissage theorem. As applications, we prove conjectures of Hesselholt and Ausoni-Rognes about localization cofiber sequences surrounding <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T upper H upper H left-parenthesis k u right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mi>H</mml:mi> <mml:mi>H</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>k</mml:mi> <mml:mi>u</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">THH(ku)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and more generally establish a framework for advancing the Rognes program for studying Waldhausen’s chromatic filtration on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A left-parenthesis asterisk right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mo> ∗ </mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">A(*)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .