Spanier Algebraic Topology Djvu To Pdf
Algebraic topology refers to the application of methods of algebra to problems in topology. More specifically, the method of algebraic topology is to assign homeomorphism/homotopy-invariants to topological spaces, or more systematically, to the construction and applications of functors from some category of topological objects (e.g. Hausdorff spaces, topological fibre bundles) to some algebraic category (e.g. abelian groups, modules over the Steenrod algebra). Landing in an algebraic category aids to the computability, but typically loses some information (say getting from a topological spaces with a continuum or more points to rather discrete algebraic structures).
spanier algebraic topology djvu to pdf
The archetypical example is the classification of surfaces via their Euler characteristic. But as this example already shows, algebraic topology tends to be less about topological spaces themselves as rather about the homotopy types which they present. Therefore the topological invariants in question are typically homotopy invariants of spaces with some exceptions, like the shape invariants for spaces with bad local behaviour.
A general and powerful such method is the assignment of homology and cohomology groups to topological spaces, such that these abelian groups depend only on the homotopy type. The simplest such are ordinary homology and ordinary cohomology groups, given by singular simplicial complexes. This way algebraic topology makes use of tools of homological algebra.
The axiomatization of the properties of such cohomology group assignments is what led to the formulation of the trinity of concepts of category, functor and natural transformations, and algebraic topology has come to make intensive use of category theory.
In particular this leads to the formulation of generalized (Eilenberg-Steenrod) cohomology theories which detect more information about classes of homotopy types. By the Brown representability theorem such are represented by spectra (generalizing chain complexes), hence stable homotopy types, and this way algebraic topology comes to use and be about stable homotopy theory.