Higher cohomology operations and R–completion

Higher homotopy operations and the cohomology of diagrams

Algebraic topology tries to answer problems of homotopy theory by translating them into algebra, using invariants such as the homotopy or cohomology groups of a space. These invariants, even when endowed with extra algebraic structure such as Whitehead products or Steenrod operations, rarely reflect fully the original homotopical information, and the additional data needed for a full invariant ...

Cohomology operations and algebraic geometry

This manuscript is based on a ten hours series of seminars I delivered in August of 2003 at the Nagoya Institute of Technology as part of the workshop on homotopy theory organized by Norihiko Minami and following the Kinosaki conference in honor of Goro Nishida. One of the most striking applications of homotopy theory in “exotic” contexes is Voevodsky’s proof of the Milnor Conjecture. This conj...

Geometric Objects and Cohomology Operations

Cohomology operations (including the cohomology ring) of a geometric object are finer algebraic invariants than the homology of it. In the literature, there exist various algorithms for computing the homology groups of simplicial complexes ([Mun84], [DE95,ELZ00], [DG98]), but concerning the algorithmic treatment of cohomology operations, very little is known. In this paper, we establish a versi...

Robot Motion Planning, Weights of Cohomology Classes, and Cohomology Operations

The complexity of algorithms solving the motion planning problem is measured by a homotopy invariant TC(X) of the configuration space X of the system. Previously known lower bounds for TC(X) use the structure of the cohomology algebra of X. In this paper we show how cohomology operations can be used to sharpen these lower bounds for TC(X). As an application of this technique we calculate explic...

Stable Operations in Generalized Cohomology