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Algebraic topology is a branch of mathematics that ... edition, Mathematics Lecture Note ... to algebraic topology. Higgins, Philip J. (1971), Notes on ...
A mapping : between total spaces of two fibrations : and : with the same base space is a fibration homomorphism if the following diagram commutes: . The mapping is a fiber homotopy equivalence if in addition a fibration homomorphism : exists, such that the mappings and are homotopic, by fibration homomorphisms, to the identities and . [2]: 405-406
In mathematics, the Tor functors are the derived functors of the tensor product of modules over a ring.Along with the Ext functor, Tor is one of the central concepts of homological algebra, in which ideas from algebraic topology are used to construct invariants of algebraic structures.
This terminology is often used in the case of the algebraic topology on the set of discrete, faithful representations of a Kleinian group into PSL(2,C). Another topology, the geometric topology (also called the Chabauty topology ), can be put on the set of images of the representations, and its closure can include extra Kleinian groups that are ...
In the above example, a connection with classical Galois theory can be seen by regarding ^ as the profinite Galois group Gal(F /F) of the algebraic closure F of any finite field F, over F. That is, the automorphisms of F fixing F are described by the inverse limit, as we take larger and larger finite splitting fields over F .
Call a cohomology theory even periodic if = for i odd and there is an invertible element .These theories possess a complex orientation, which gives a formal group law.A particularly rich source for formal group laws are elliptic curves.
"Lecture Notes on Homotopy Theory and Applications" (PDF). www.math.wisc.edu. Determination of the Second Homology and Cohomology Groups of a Space by Means of Homotopy Invariants - gives accessible examples of Postnikov invariants; Hatcher, Allen (2002). Algebraic topology. Cambridge University Press. ISBN 978-0-521-79540-1. Zhang.
A three-dimensional model of a figure-eight knot.The figure-eight knot is a prime knot and has an Alexander–Briggs notation of 4 1.. Topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling ...