We calculate the homotopy groups of K(S) in terms of the homotopy groups of K(Z), the homotopy groups of CP∞ −1, and the homotopy groups of S. This completes the program begun by Waldhausen, who computed the rational homotopy groups (building on work of Quillen and Borel), and continued by Rognes, who calculated the groups at regular primes in terms of the homotopy groups of CP∞ −1, and the hom...

In the classical paper [A-H] Adams-Hilton constructed a free chain algebra which is an important algebraic model of a simply connected homotopy type. We show that this chain algebra (endowed with an additional structure given by a “height function”) yields actually an invariant of a proper homotopy type. For this we introduce the homotopy category of locally finite chain algebras without using ...

In this paper, first a new homotopy perturbation method for solving a fractional order nonlinear telegraph equation is introduced. By applying the proposed method, the nonlinear equation is translated to linear equations for per iteration of homotopy perturbation method. Then, the obtained problems are solved with separation method. In the examples, it is illustrated that the exact solution is ...

What is a homotopy theory T? – Examples of homotopy theories (Tpair presentation) – The homotopy category Ho(T) – Examples of homotopy categories – Model category: (C, E) with extras – Examples of model categories (C, E) – Dividends from a model category structure on (C, E) – Equivalences between homotopy theories – T(C, E) ∼ T(C′, E ′) for model categories – Examples of equivalences between ho...

While working on the completion of the proof of the Bloch-Kato conjecture I have thought a lot about what to do next. Eventually I became convinced that the most interesting and important directions in current mathematics are the ones related to the transition into a new era which will be characterized by the widespread use of automated tools for proof construction and verification. I have star...

We present a new technique for analyzing the v0-Bockstein spectral sequence studied by Shimomura and Yabe. Employing this technique, we derive a conceptually simpler presentation of the homotopy groups of the E(2)-local sphere at primes p ≥ 5. We identify and correct some errors in the original ShimomuraYabe calculation. We deduce the related K(2)-local homotopy groups, and discuss their manife...

This paper is the extended introduction of a series of three papers [11] [9] [10] about modelling T-homotopy by refinement of observation. The notion of T-homotopy equivalence is discussed. A new one is proposed and its behaviour with respect to other constructions in dihomotopy theory is explained. We also prove in appendix that the tensor product of flows is a closed symmetric monoidal struct...

A homotopy action of a group G on a space X is a homomorphism from G to the group HAUT(X) of homotopy classes of homotopy equivalences of X. George Cooke developed an 'obstruction theory to determine if a homotopy action is equivalent up to homotopy to a topological action. The question studied in this paper is: Given a diagram of spaces with homotopy actions of G and maps between them that are...

We use a new geometric construction, grope splitting, to give a sharp bound for separation of surfaces in 4{manifolds. We also describe applications of this technique in link-homotopy theory, and to the problem of locating 1{null surfaces in 4{manifolds. In our applications to link-homotopy, grope splitting serves as a geometric substitute for the Milnor group. AMS Classi cation numbers Primary...

In this paper, the Kadomtsev-Petviashvili equation is solved by using the Adomian’s decomposition method , modified Adomian’s decomposition method , variational iteration method , modified variational iteration method, homotopy perturbation method, modified homotopy perturbation method and homotopy analysis method. The existence and uniqueness of the solution and convergence of the proposed...