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Chromatic Homotopy Theory
In mathematics, chromatic homotopy theory is a subfield of stable homotopy theory that studies complex-oriented cohomology theory, complex-oriented cohomology theories from the "chromatic" point of view, which is based on Daniel Quillen, Quillen's work relating cohomology theories to formal groups. In this picture, theories are classified in terms of their "chromatic levels"; i.e., the heights of the formal groups that define the theories via the Landweber exact functor theorem. Typical theories it studies include: complex K-theory, elliptic cohomology, Morava K-theory and Topological modular forms, tmf. Chromatic convergence theorem In algebraic topology, the chromatic convergence theorem states the homotopy limit of the chromatic tower (defined below) of a finite local spectrum, ''p''-local spectrum X is X itself. The theorem was proved by Hopkins and Ravenel. Statement Let L_ denotes the Bousfield localization with respect to the Morava E-theory and let X be a finite, p-loca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stable Homotopy Theory
In mathematics, stable homotopy theory is the part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor. A founding result was the Freudenthal suspension theorem, which states that given any pointed space X, the homotopy groups \pi_(\Sigma^n X) stabilize for n sufficiently large. In particular, the homotopy groups of spheres \pi_(S^n) stabilize for n\ge k + 2. For example, :\langle \text_\rangle = \Z = \pi_1(S^1)\cong \pi_2(S^2)\cong \pi_3(S^3)\cong\cdots :\langle \eta \rangle = \Z = \pi_3(S^2)\to \pi_4(S^3)\cong \pi_5(S^4)\cong\cdots In the two examples above all the maps between homotopy groups are applications of the suspension functor. The first example is a standard corollary of the Hurewicz theorem, that \pi_n(S^n)\cong \Z. In the second example the Hopf map, \eta, is mapped to its suspension \Sigma\eta, which generates \pi_4(S^3)\cong \Z/2. One of the mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morava E-theory
Morava may refer to: Rivers * Great Morava (''Velika Morava''; or simply Morava), a river in central Serbia, and its tributaries: ** South Morava (''Južna Morava'') *** Binač Morava (''Binačka Morava'') ** West Morava (''Zapadna Morava'') * Morava (river), a river in the Czech Republic, Austria and Slovakia Places * , a village in the Svishtov Municipality, Bulgaria * Morava (Kočevje), a village in the municipality of Kočevje, Slovenia * Morava (Serbian Cyrillic: Морава), the old name for Gnjilane (Albanian: ''Gjilan'') * Suva Morava ("Dry Morava"), a village in the municipality of Vladičin Han, Serbia * Dolní Morava ("Lower Morava"), a municipality and village in the Ústí nad Orlicí District, Czech Republic * Malá Morava ("Little Morava"), a municipality and village in the Šumperk District, Czech Republic * , a mountain in southeast Albania, near Korçë * Morava Banovina, a province of the Kingdom of Yugoslavia between 1929 and 1941 * Donja Morava ("Lower Mor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |