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Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism , though usually most classify up to homotopy equivalence .
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 .
A basic example in topology is lifting a path in one topological space to a path in a covering space. [1] For example, consider mapping opposite points on a sphere to the same point, a continuous map from the sphere covering the projective plane. A path in the projective plane is a continuous map from the unit interval [0,1]. We can lift such a ...
Maunder, C. R. F. (January 1996), Algebraic Topology, Dover Publications, ISBN 0-486-69131-4 Massey, William S. (1991), A Basic Course in Algebraic Topology , Springer, ISBN 038797430X May, J. Peter (1999), A Concise Course in Algebraic Topology , ISBN 9780226511832
In algebraic geometry and algebraic topology, branches of mathematics, A 1 homotopy theory or motivic homotopy theory is a way to apply the techniques of algebraic topology, specifically homotopy, to algebraic varieties and, more generally, to schemes. The theory is due to Fabien Morel and Vladimir Voevodsky.
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent.
edit] * The base point of a based space. X + {\displaystyle X_{+}} For an unbased space X, X + is the based space obtained by adjoining a disjoint base point. A absolute neighborhood retract abstract 1. Abstract homotopy theory Adams 1. John Frank Adams. 2. The Adams spectral sequence. 3. The Adams conjecture. 4. The Adams e -invariant. 5. The Adams operations. Alexander duality Alexander ...
There are various ways of approaching the subject, each of which focuses on a slightly different flavor of Chern class. The original approach to Chern classes was via algebraic topology: the Chern classes arise via homotopy theory which provides a mapping associated with a vector bundle to a classifying space (an infinite Grassmannian in this case).