I start by telling a bit about stable homotopy theory, to give a flavor of the world we’ll be working in today. One of the key ideas in algebraic topology is to use algebraic invariants to distinguish topological spaces. This leads to H∗, H ∗, and π∗. The first two are relatively easy to compute, but they are coarse invariants: they don’t capture enough of the information of the space. The last...

Contractible transformations of graphs consist of contractible gluing and deleting of vertices and edges of graphs. They partition all graphs into the family of homotopy classes. Contractible transformations do not change the Euler characteristic and the homology groups of graphs. In this paper we describe the minimal representatives of some homotopy classes and find the formula for computing t...

Let G be a finite group given in one of the forms listed in the title with period 2d and X(n) an n-dimensional CW -complex with the homotopy type of an n-sphere. We study the automorphism group Aut (G) to compute the number of distinct homotopy types of orbit spaces X(2dn − 1)/μ with respect to free and cellular G-actions μ on all CW complexes X(2dn−1). At the end, the groups E(X(2dn−1)/μ) of s...

Through an investigation of properties of Chern classes, Quillen discovered a connection between stable homotopy theory and 1-dimensional formal group laws [31]. After almost 40 years, the impacts of this connection are still being felt. The stratification of formal group laws in finite characteristic gives rise to the chromatic filtration in stable homotopy theory [32], and has definite calcul...

The unstable homotopy groups of spheres can be approached by the EHP spectral sequence. There are computations of the low dimensional portion of the EHP sequence by Toda [1, 2] for the 2,3-primary part, and Behrens [3], Harper [4] for the 5-primary part. There are certain stable phenomenon in the EHP sequence. In fact, there is one portion in the E1-term which are in the stable range, which mea...

Let K(n) be the nth Morava K-theory spectrum. Let En be the Lubin-Tate spectrum, which plays a central role in understanding LK(n)(S 0), the K(n)-local sphere. For any spectrum X, define E∨(X) to be the spectrum LK(n)(En ∧ X). Let G be a closed subgroup of the profinite group Gn, the group of ring spectrum automorphisms of En in the stable homotopy category. We show that E∨(X) is a continuous G...

In telescopic homotopy theory, a space or spectrum X is approximated by a tower of localizations LnX, n ≥ 0, taking account of vn-periodic homotopy groups for progressively higher n. For each n ≥ 1, we construct a telescopic Kuhn functor Φn carrying a space to a spectrum with the same vn-periodic homotopy groups, and we construct a new functor Θn left adjoint to Φn. Using these functors, we sho...

In the 1960’s and 1970’s, the Adams Conjecture figured prominently both in homotopy theory and in geometric topology. Quillen sketched one way to attack the conjecture and then proved it with an entirely different line of argument. Both of his approaches led to spectacular and beautiful new mathematics. 1. Background on the Adams Conjecture For a finite CW -complex X, let KO(X) be the Grothendi...