scholarly journals Algebraic K-theory and abstract homotopy theory

2011 ◽  
Vol 226 (4) ◽  
pp. 3760-3812 ◽  
Author(s):  
Andrew J. Blumberg ◽  
Michael A. Mandell
Keyword(s):  
Homotopy Theory ◽  
K Theory ◽  
Abstract Homotopy Theory

scholarly journals Homotopical and operator algebraic twisted K-theory

2020 ◽  
Vol 378 (3-4) ◽  
pp. 1021-1059
Author(s):  
Fabian Hebestreit ◽  
Steffen Sagave
Keyword(s):  
Homotopy Theory ◽  
Complex Case ◽  
Joint Work ◽  
The Real ◽  
Comparison Results ◽  
Algebraic Constructions ◽  
K Theory ◽  
Multiplicative Structures

Abstract Using the framework for multiplicative parametrized homotopy theory introduced in joint work with C. Schlichtkrull, we produce a multiplicative comparison between the homotopical and operator algebraic constructions of twisted K-theory, both in the real and complex case. We also improve several comparison results about twisted K-theory of $$C^*$$ C ∗ -algebras to include multiplicative structures. Our results can also be interpreted in the $$\infty $$ ∞ -categorical setup for parametrized spectra.

Stable operations in mod p K-theory

1967 ◽  
Vol 63 (3) ◽  
pp. 631-646 ◽  
Author(s):  
C. R. F. Maunder
Keyword(s):  
Homotopy Theory ◽  
Cohomology Theory ◽  
Stable Homotopy ◽  
Stable Homotopy Theory ◽  
The Subject ◽  
Cohomology Operations ◽  
K Theory ◽  
Mod P

There comes a time in the development of a cohomology theory when a discussion of cohomology operations becomes necessary. In the case of complex K-theory, the subject of the present paper, such operations have of course already been investigated by Adams (see (2)), so that any further discussion might appear superfluous. Powerful as Adams's results are, however, the situation still leaves something to be desired: it is not known just what other operations can be defined in K-theory, and it is an inconvenience from the standpoint of stable homotopy theory that Adams's operations are not themselves stable.

Daniel Quillen, the father of abstract homotopy theory

2013 ◽  
Vol 11 (3) ◽  
pp. 479-491
Author(s):  
Jean-Claude Thomas ◽  
Micheline Vigué-Poirrier
Keyword(s):  
Homotopy Theory ◽  
Short Paper ◽  
Rational Homotopy Theory ◽  
Rational Homotopy ◽  
Abstract Homotopy Theory

AbstractIn this short paper we try to describe the fundamental contribution of Quillenin the development of abstract homotopy theory and we explain how he uses this theory to lay the foundations of rational homotopy theory.

scholarly journals On the $K$-theory and homotopy theory of the Klein bottle group

2011 ◽  
Vol 13 (2) ◽  
pp. 63-72 ◽  
Author(s):  
Jens Harlander ◽  
Andrew Misseldine
Keyword(s):  
Homotopy Theory ◽  
Klein Bottle ◽  
K Theory

scholarly journals Chern–Weil forms and abstract homotopy theory

2013 ◽  
Vol 50 (3) ◽  
pp. 431-468 ◽  
Author(s):  
Daniel S. Freed ◽  
Michael J. Hopkins
Keyword(s):  
Homotopy Theory ◽  
Abstract Homotopy Theory
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