Spectral sequence - Wikipedia
In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations.
Associated to a chain complex with a ltration is an algebraic gadget generalizing the long exact sequence, which is called a spectral sequence, and which can help compute the homology of the chain complex.
spectral sequence in nLab
2023年1月23日 · The spectral sequence is the sequence of higher-order corrections that make this naive idea actually work. Being, therefore, an iterative perturbative approximation scheme of bigraded differential objects, fully-fledged spectral sequences can look a bit intricate.
Spectral sequences are a useful tool for computing the homology and cohomology of topological spaces. A spectral sequence is a bit like a sequence of two-dimensional chain complexes, with each successive element of the sequence arising by taking the homology of the previous object.
There is an algorithm for computing H*(T) up to extension, called a spec-tral sequence, and we have just performed the first two steps of this algorithm. The next two steps are illustrated in Figure 5.1. 1. A family {Er pq} of objects of A defined for all integers p, q, and r > a 2.
Spectral Sequence -- from Wolfram MathWorld
5 天之前 · A spectral sequence is a tool of homological algebra that has many applications in algebra, algebraic geometry, and algebraic topology. Roughly speaking, a spectral sequence is a system for keeping track of collections of exact sequences that have maps between them.
These notes are intended to serve as a practical and painless introduction to the Serre spectral sequence, based on a lecture by Andre Henriques. Through 10 step-by-step examples, we'll see that a lot can be deduced from just the co-homology of spheres and some well-known brations.
The first spectral sequence that appeared in algebraic topology, and still the most important one, is the Serre spectral sequence which relates the homology or cohomol-ogy groups of the fiber, base, and total space of a fibration. The homotopy groups of these three spaces fit into a long exact sequence, but for homology or cohomology the
While spectral sequences arise with different patterns of gradings, the most com-monly encountered homologically and cohomologically graded spectral sequences fit into the patterns given in the following pair of definitions. Definition 1.1.
This notes aims to give a brief introduction to spectral sequences from a purely homological algebra point of view. Speci cally, we only describe the set-up of spectral sequences, without going into their many exciting applications in topology and geometry. For this notes, we closely follow the treatment found in [1].