Infinite Loop Spaces: Hermann Weyl Lectures, The Institute for Advanced Study

John Frank Adams “Infinite Loop Spaces: Hermann Weyl Lectures, The Institute for Advanced Study”
Princeton University Press | 1978-09-01 | ISBN: 0691082065 | 230 pages | Djvu | 2,7 Mb

Reader’s review:
Although published in 1978, this book could be used as an introduction to the theory of operads and other recent work on homotopy theory and vertex operators. Vertex operators are not discussed in this book, but the theory elucidated herein is good background material for their study.
The author does a great job in motivating the subject in chapter 1. Loop spaces are function spaces of maps from the unit interval to a space with a chosen basepoint, with the property that each map sends 0 and 1 to the base point. The mathematician Jean Pierre Serre introduced the path space in order to study loop spaces, resulting in the famous Serre fibering. The nth homotopy group of the loop space can be shown to be equivalent to the (n+1)-th homotopy group of the original space. The homology of loop spaces can be calculated for some types of spaces, such as wedges of spheres. Infinite loop spaces are essentially sequences of spaces such that the nth element of this sequence is equivalent to the loop space of the (n+1)-th element. This sequence is also known as an “Omega-spectrum” and has the infinite loop space as its zeroth term. The name “spectrum” comes from general considerations involving sequences of spaces where the nth term is equivalent to the loop space of the (n+1)-th term; equivalently, where the suspension of the nth term is equal to the (n+1)-th term. The author reviews how a generalized cohomology theory yields an Omega-spectrum, giving two examples involving Eilenberg-Maclane spaces and complex and real K-theory. One can also start with a spectrum and construct a generalized homology and cohomology theory. Spectra and cohomology theory are thus essentially equivalent.
Chapter 2 is an overview of techniques needed to construct a category of spaces with enough structure so that the infinite loop space functor yields an equivalence from the category of spectra to the category of certain spaces. An example of the latter is given by the Stasheff A-infinity space, and its now ubiquitous property of having a product which is strictly associative. This property allows one to prove that a space is equivalent to a loop space if and only if the space is a Stasheff A-infinity space and that the zeroth homotopy of the space is a group. The Stasheff A-infinity spaces are also used to motivate the construction of ‘operads’.
The next chapter the author is concerned with the concept of a space being like another one without being equivalent to it. He discusses the use of ‘localization’ in homotopy theory, an idea that is analogous to the one in algebra. The use of localization in homotopy theory is due to D. Sullivan, and involves use of the notion of a space being ‘A-local’, where A is a subring of the rationals. Remembering that a Z-module is A-local if it has the structure of an A-module, a space is A-local if its homotopy groups are A-local. Examples of the use of localization in constructing certain spaces are given. The author also discusses the use of the ‘plus construction’ that allows the alteration of fundamental groups without affecting the cohomology groups. Then after the construction of the Quillen higher algebraic K-theory groups in this regard, the author describes the relation between a topological monoid and the loop space of the classifying space of this monoid. This involves the notion of ‘group completion’, which is essentially an isomorphism between the homology of the path components of the monoid and the homology of the loop space of the classifying space of the monoid, but in the (infinite) direct limit.
Chapter 4 introduces the concept of a transfer map. A very elusive idea at first glance, the transfer map is motivated via the n-sheeted covering map of a space on another. The (singular) simplices of each then get matched up by the covering, and the transfer map between the spaces is then defined so that it is equal to the sum of the singular simplices of the covering space. It is in fact a chain map as shown by the author. The transfer maps are related to homotopy classes of the ’structure’ maps of chapter 2, and the author gives a few examples of how they are used.
Chapter 5 is a quick overview of the Adams conjecture, which is essentially an assertion that the image of KO(X) in KF(X) can be characterized explicitly. Detailed proofs are omitted but references are given for the interested reader.
In chapter 6, the author restricts his attention to the K-theory of spectra. The treatment is concerned in large degree with the question of the existence of infinite loop map between infinite loop structures, and finding such a map, checking whether it is unique. This question is answered for particular types of spectra, via the Madsen, Snaith, and Tornehave theorem. Also, the Adams-Priddy theorem is proved, showing that one can construct on a space a unique infinite loop space structure. The reader gets more examples of the use of localization, in that some spaces can become equivalent as infinite loop spaces upon localization. The origin of K-theory in this chapter comes from the replacing of spectra that are not known by ones that are (namely the ones in classical K-theory). The author shows how the Madsen-Snaith-Tornehave theorem works in the context of both complex and real (periodic) K-theory. Detailed proofs are given for all of these results.

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