Stable homotopy
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Joel M. Cohen University of Pennsylvania, Philadelphia, PA/USA
Stable Homotopy
Springer-Verlag Berlin' Heidelberg' NewYork 1970
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Title No.
Offsetdmck: Julius Beltz, WeiDheimlBerptr.
Introduction These notes are essentially the lecture notes of a course I gave at the University of Chicago in the summer of 1968.* Most aspects of stable homotopy are touched on and some are studied in very great detail.
It should, however, be emphasized that we are
only concerned with finite
CW
complexes.
Thus one never has to
worry about the problems which may arise for infinite
CW
complexes:
i.e. certain long exact sequences which are easy to get for finite dimensional
CW
complexes become very difficult in general unless
one takes great care in defining the morphisms (as J. M. Boardman has done in his Warwick lecture notes: or see Tierney). It is assumed that the reader has had a year of algebraic topology (a course which covers the equivalent of most of Spanier. sa0. I quote without proof some theorems from first year topology (e.g. the Hurewicz theorem) and prove others.
In addition I assume the reader
has some understanding of spectral sequences and what they can do. Specifically, I assume existence of the Serre Spectral Sequence in homology.
Spanier covers quite adequately the necessary material. For the computations of the stable homotopy groups of
spheres in Chapter V,I quote a lot of results on the Steenrod Algebra-all of which can be found in Steenrod-Epstein or Mosher-Tangora.
Lack
of prior knowledge of cohomology operations will not interfere with the understanding of this section, although the reader may have to accept some results on faith (or study the above-mentioned books). This set of notes has a quite different point of view on the whole from Frank Adams' lecture notes on stable homolopy.
I feel
*The author was partially supported by the National Science Foundation during the preparation of these notes.
IV
that to some degree, these complement the other.
Although I do con-
struct the Adams spectral sequence for completeness, not very much is said about it here and the reader is encouraged to pursue the sUbject either in Adams' notes or in MosberTangora.
The present method of
computing the stable homotopy groups of spheres is somewhat simpler than the Adams spectral sequence in the dimensions where it is done. (Higher up this method seems to break down and the Adams method is much neater.) Chap
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