Stable Homotopy Theory

Content

Please log in to view module content:

log in

Description

Wednesday 11.00-13.00

We will study the stable homotopy category and the Adams spectral sequence.  The central goal is to better understand computations of stable homotopy groups of spheres, but understanding the stable homotopy category and spectra turns out to be extremely useful in much wider contexts. 

1. Unstable homotopy theory and the Hopf invariant. 
2.  Stable cohomology operations, Steenrod squares,  properties, and immediate applications. 
3.  Adem relations, the Hopf invariant, and the ring of stable homotopy groups of spheres. 
4.  Generalised cohomology theories and  Brown representability. 
5.  Spectra and the stable homotopy category, properties of spectra and basic constructions. 
6. The long exact sequence of the mapping cone and additivity of the stable homotopy category. 
7.  An interpretation of the Steenrod algebra.  Construction of the Adams spectral sequence I. 
8.  Construction of the Adams spectral sequence II and basic results. 
9. Computations in the Adams spectral sequence.  
10. Computing with the Adams spectral sequence and  student talks.

ASSESSMENT

Five written homework assignments, due in at the end of weeks 2, 4, 6, 8, and 10. (short, 2-3 questions). 
One 10-20 minute talk (depending on numbers) on a topic in homotopy theory, during the last lecture. 
Best 5 out of 6 to count, 20% each. 
Each lecturer will set and grade one homework. 

REFERENCES (for Homotopy Theory (Semester 1) & Stable Homotopy Theory (Semester 2)

References

Frank Adams, Stable homotopy theory and generalised homology, Chicago Lectures in Mathematics, 1974. 

James F. Davis and Paul Kirk. Lecture notes in algebraic topology. 
Grad. Stud. Math., 35, American Mathematical Society, Providence, RI, 2001. xvi+367 pp.

Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. xii+544 pp.

Allen Hatcher, Spectral sequences. Unfinished book project, available at pi.math.cornell.edu/~hatcher/AT/SSpage.html. 

Peter May, A concise course in algebraic topology, Chicago Lectures in Math.
University of Chicago Press, Chicago, IL, 1999. x+243 pp.

Robert E. Mosher and Martin C. Tangora.  Cohomology operations and applications in homotopy theory. Harper \& Row, Publishers, New York-London, 1968. x+214 pp.

Yuli Rudyak, On Thom spectra, orientability, and cobordism. Springer Monogr. Math.
Springer-Verlag, Berlin, 1998. xii+587 pp.
 
Stefan Schwede, Symmetric spectra. Unpublished lecture notes, available at https://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf. 

Edwin H. Spanier, Algebraic topology.
McGraw-Hill Book Co., New York-Toronto-London, 1966. xiv+528 pp. 

Robert M. Switzer, Algebraic topology -- homology and homotopy. Die Grundlehren der mathematischen Wissenschaften, Band 212 Springer-Verlag, New York-Heidelberg, 1975. xii+526 pp.
 
George Whitehead, Elements of homotopy theory.
Grad. Texts in Math., 61 Springer-Verlag, New York-Berlin, 1978. xxi+744 pp.