The first part, in which i attempted to motivate the notion of spectral sequence, and constructed the basic example that well be using, is at the ncategory cafe here. Notes on cup product and intersections updated 20110315 spectral sequences. Some indication of the power of spectral sequences will be apparent from the results obtained by its use here. Spectral sequences, exact couples and persistent homology of.
The third chapter on spectral sequences is the most difficult one, but also the richest one by the various applications and digressions into other topics of algebraic topology. Such spaces exhibit a hidden symmetry, which is the culminationof18. By algebraic ktheory i understand the study of the following process. Fibrations and the lerayserre spectral sequence 4 3. Introductory books on algebraic topology usually focus on the di. Spectral sequences in algebraic topology download book. I have now returned to an earlier plan of having this material be an extra chapter of the algebraic topology book, rather than a separate book. In this paper we study the relationship between a very classical algebraic object associated to a filtration of spaces, namely a spectral sequence introduced by leray in the 1940s, and a more recently invented object that has found many applications namely, its persistent homology groups. Homological algebra and the eilenbergmoore spectral sequence by larry smitho in 6 eilenberg and moore have developed a spectral sequence of great use in algebraic topology. The first third of the book covers the fundamental group, its definition and its application in the study of covering spaces. Spectral sequences in string topology researchgate. This algebraic concept is introduced for the study of the homology and cohomology properties of arbitrary fibrations, but it has other important applications in algebraic topology, and the number of these is constantly increasing.
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. The main topics include the serre spectral sequence, constructing spectral sequences via. This chapter contains those results about spectral sequences that we used earlier in the book, incorporated into a brief background compendium of the very minimum that anybody interested in algebraic topology needs to know about spectral sequences. First quadrant spectral sequences in algebraic ktheory by r. Additional resources michael hopkins notes on steenrod operations chapter 2 of cohomology operations in homotopy theory by mosher and tangora. In the end, the overriding pedagogical goal has been the introduction of basic ideas and methods of thought. Introduction to spectral sequences ucb mathematics. Its main focus is the serre spectral sequence and its applications, but there is also some coverage of the adams spectral sequence and.
There he took up algebraic topology, and the result was a spectacular owering of highly original ideas, ideas which. Introduction to spectral sequences michael hutchings april 28, 2011 abstract the words \spectral sequence strike fear into the hearts of many hardened mathematicians. By associating to a space x an algebraic invariant the cohomology groups h. We show the existence of a long exact sequence of groups linking these two objects and using it derive. Curtis the unstable adamsnovikov spectral sequence, topology 17 1978 229248.
Spectral sequence are a tool to organize complex computations. Suitable as a text for an intermediate course in algebraic topology, this book provides a direct exposition of the basic concepts of bordism, characteristic classes, adams spectral sequences, brownpeterson spectra and the computation of stable stems. It should qualify as a thorough first introduction to the topic, but i dont think all of the important examples have made it into the text. They are arguably the most important such tool in modern algebraic topology. A typical application which is common in discrete geometry, as well as in quantitative real algebraic geometry, is to use the initial terms of a certain spectral sequence to give upper bounds on the topological complexity for example, the sum of betti numbers of. An introduction to the serre spectral sequence, with a number of applications, mostly fairly standard. Spectral sequences are highpowered algebraic machines used in a variety of settings. This chapter begins with the abstract properties of spectral sequences and their relation to the.
We will use this to compute the cohomology of eilenbergmaclane spaces, which is the starting point for the adams spectral sequence. Math 885 computational methods in algebraic topology at uva this course is a selfcontained introduction to spectral sequences with an emphasis on the spectral sequences in algebraic topology. Also james davis, paul kirk, lecture notes in algebraic topology gives many nice applications of the serre spectral sequence. A spectral sequence for the homology of an in nite delooping, pac. Spectral sequences in algebraic topology jane tan amsi vacation. Our understanding of the foundations of algebraic topology has undergone subtle but serious changes since i began teaching this course.
Spectral sequences are a useful technique in algebraic topology traditionally applied to calculate homology and homotopy groups of spaces see 7 or 8. Spectral sequences in topology university of michigan. In algebraic geometry i think all of the long exact sequences youll run into ultimately come from deriving some functor between abelian categories. It can be nicely supplemented by homotopic topology by a. Im not comfortable enough with spectral sequences to answer this question, but let me answer an easier version of this question with spectral sequences replaced by long exact sequences. Pdf lecture notes in algebraic topology semantic scholar. Introduction to the serre spectral sequence, with a number of applications, mostly fairly standard, the adams spectral sequence, eilenbergmoore spectral sequences. Spectral sequences are among the most elegant and powerful methods of computation in mathematics. Asidefromrnitself,theprecedingexamples are also compact. The first two chapters cover the material of the fall semester. Spectral sequences in algebraic topology jane tan supervised by dr vigleik angeltveit australian national university vacation research scholarships are funded jointly by the department of education and training and the australian mathematical sciences institute. In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences daniel murfet october 5, 2006 in this note we give a minimal presentation of spectral sequences following ega. We will start with the serre spectral sequence, which computes the cohomology of the total space of a fibration from.
The goal is to give some examples of spectral sequences, and some example computations, in the context of algebra and algebraic geometry. We cover essentially only that part of the theory needed in algebraic geometry. Boris botvinnik lecture notes on algebraic topology. Chapter 11 simplehomotopy theory introduces the ideas which lead to the subject of algebraic ktheory and. Topological library part 3, spectral sequences in topology. On relations between adams spectral sequences, with an application to the. The final volume of the threevolume edition, this book features classical papers on algebraic and differential topology published in 195060s.
The key ideas are presented in complete detail without becoming encyclopedic. But my advisor, peter may, is an algebraic topologist with categorical leanings, so i sat in on a lot of algebric topology lectures as a grad student. Bordism, stable homotopy and adams spectral sequences. Alan hatcher, spectral sequences in algebraic topology web. Spectral sequences in algebraic topology anu mathematical. The real theme of this book is to get the reader to some powerful and compelling applications of algebraic topology and comfort with spectral sequences. In chapter 10 further applications of spectral sequences many of the fruits of the hard labor that preceded this chapter are harvested.
Aug 08, 20 page 5 shows the two spectral sequences using the notation for postnikov towers of spectra on p. These notes will attempt to demonstrate that spectral sequences are not so scary, and also very powerful. Introduction to spectral sequences michael hutchings april 28, 2011 abstract the words \ spectral sequence strike fear into the hearts of many hardened mathematicians. A typical application which is common in discrete geometry, as well as in quantitative real algebraic geometry, is to use the initial terms of a certain spectral sequence to give upper bounds on the topological complexity for example, the sum of. Differential forms in algebraic topology pp 154265 cite as. A primer on spectral sequences university of chicago. Allen hatcher has some textbooks available for free download on his website, including the first 100 pages of an unfinished text on spectral sequences in algebraic topology.
Spectral sequences in algebraic topology we will start with the serre spectral sequence, which computes the cohomology of the total space of a fibration from the cohomology of the fiber and base. The completely decomposed topology on schemes and associated. The course will focus on applications to algebraic topology. Spectral sequences are a generalization of exact sequences, and since their introduction by jean leray, they have become important computational tools, particularly in algebraic topology, algebraic geometry and homological algebra. Intended for use both as a text and a reference, this book is an exposition of the fundamental ideas of algebraic topology. The use of differential forms avoids the painful and for the beginner unmotivated homological algebra in algebraic topology. This course will serve as an introduction to spectral sequences as a computational tool. Lecture notes updated 20110427, but still very incomplete. Spectral sequences the modus operandi of algebraic topology is that \algebra is easy. It expresses, in the language of homological algebra, the singular cohomology of the total space x of a serre fibration in terms of the cohomology of the base space b and the fiber f. It is impossible to describe everything about spectral sequences in the duration of a single course, so we will focus on a special and important example. The course website includes notes, homework sets, spectral sequence pictures, and. In mathematics, the serre spectral sequence sometimes lerayserre spectral sequence to acknowledge earlier work of jean leray in the leray spectral sequence is an important tool in algebraic topology. Spectral sequences in algebraic topology, by alan hatcher.
Other introductions can be found in many texts in algebraic topology and homological algebra 1, 4, 5. See also hatcher, algebraic topology, chapter 4, which has some overlap with the topics to be covered. Thomason department of mathematics massachusetts institute of technology cambridge, massachusetts 029 u. Spectral sequences in topology this volume belongs to a threevolume edition devoted to reprint classical papers on algebraic and differential topology published in the 1950s1960s. Some algebraic aspects of the adamsnovikov spectral. The cohomology spectral sequence additional topics that it would be nice to include. Some algebraic aspects of the adamsnovikov spectral sequence. Spectral sequences, exact couples and persistent homology. Some of these papers may be difficult to find, and several of them have been translated by v. This book describes some of the most important examples of spectral sequences and some of their most spectacular applications. Alan hatcher, spectral sequences in algebraic topology stanley kochmann, bordism, stable homotopy and adams spectral sequences, ams 1996 the general discussion in the context of stable. By the help of these sequences, one can compute topological graded invariants of a topological space such as homotopy, homology or cohomology groups by taking approximations. The serre spectral sequence is covered in most textbooks on algebraic topology, e.
Spectral sequences in algebraic topology cornell university. Chain complexes, homology, and cohomology homological algebra products fiber bundles homology with local coefficients fibrations, cofibrations and homotopy groups obstruction theory and eilenbergmaclane spaces bordism, spectra, and generalized homology spectral sequences further applications of spectral sequences simplehomotopy theory bibliography index. As a consequence, we will derive some homotopytheoretic applications. One is the standard ahss, the other one should be mikes version from above i havent checked in detail, but it seems clear enough. Notably there is a spectral sequence for computing the homology of the total. Last time we defined cohomology in homotopy type theory. The spectral sequence of a filtered complex and the serre. Spectral sequences in topology emily clader student geometry and topology seminar, university of michigan outline.
Spectral sequences are now ubiquitous in mathematics. In algebraic topology, one of the most important tools for computing the cohomology of the space of free loops on a space is the cohomological eilenbergmoore spectral sequence. In this paper, we investigate the behaviour of the serre spectral sequence with respect to the algebraic structures of string topology in generalized homology theories, specificially with the chassullivan product and the corresponding coproduct and module structures. And always, one of the most mystifying things in algebraic topology to me was the notion of a spectral sequence.