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It is a simple exercise in topology to see that every three elements of a Puppe sequence are, up to a homotopy, of the form: X → Y → C ( f ) {\displaystyle X\to Y\to C(f)} . By "up to a homotopy", we mean here that every 3 elements in a Puppe sequence are of the above form if regarded as objects and morphisms in the homotopy category .
In homotopy theory and algebraic topology, the word "space" denotes a topological space.In order to avoid pathologies, one rarely works with arbitrary spaces; instead, one requires spaces to meet extra constraints, such as being compactly generated weak Hausdorff or a CW complex.
12 Foundations of algebraic topology. 13 Topology and algebra. 14 See also. Toggle the table of contents. ... Download as PDF; Printable version; In other projects ...
In mathematics, Lehrbuch der Topologie (German for "textbook of topology") is a book by Herbert Seifert and William Threlfall, first published in 1934 and published in an English translation in 1980. It was one of the earliest textbooks on algebraic topology , and was the standard reference on this topic for many years.
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 ...
Allen Hatcher, Algebraic Topology, Cambridge University Press, Cambridge, 2002. ISBN 0-521-79540-0. A modern, geometrically flavored introduction to algebraic topology. The book is available free in PDF and PostScript formats on the author's homepage. Kainen, P. C. (1971). "Weak Adjoint Functors". Mathematische Zeitschrift. 122: 1– 9.
In algebraic topology, a fiber-homotopy equivalence is a map over a space B that has homotopy inverse over B (that is if is a homotopy between the two maps, is a map over B for t.) It is a relative analog of a homotopy equivalence between spaces.
Singular cohomology is a powerful invariant in topology, associating a graded-commutative ring with any topological space. Every continuous map: determines a homomorphism from the cohomology ring of to that of ; this puts strong restrictions on the possible maps from to .