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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 .
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. Although algebraic topology primarily uses algebra to study topological ...
Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces Subcategories. This category has the following ...
Let = be a -graded algebra, with product , equipped with a map : of degree (homologically graded) or degree + (cohomologically graded). We say that (,,) is a differential graded algebra if is a differential, giving the structure of a chain complex or cochain complex (depending on the degree), and satisfies a graded Leibniz rule.
In other words, a characteristic class associates to each principal G-bundle in () an element c(P) in H*(X) such that, if f : Y → X is a continuous map, then c(f*P) = f*c(P). On the left is the class of the pullback of P to Y ; on the right is the image of the class of P under the induced map in cohomology.
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology.
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Edwin Henry Spanier (August 8, 1921 – October 11, 1996) was an American mathematician at the University of California at Berkeley, working in algebraic topology.He co-invented Spanier–Whitehead duality and Alexander–Spanier cohomology, and wrote what was for a long time the standard textbook on algebraic topology (Spanier 1981).