http://www.math.uio.no/∼rognes/kurs/ma422h03/

Morava K-teori

1. (Ko-)homologi-teorier.

Singulær homologi. CW-komplekser. Cellulær homologi.

Homologi med koeffisienter. Singulær kohomologi. Tor og Ext. De universelle koeffisient-teoremene.

Eilenberg–Steenrod aksiomene. Kofibreringer og Puppe-sekvensen. Grenser og kogrenser. Generaliserte homologi-teorier og kohomologi-teorier.

Høyere homotopi-grupper. Hurewicz-homomorfien. Hurewicz-teoremet. Fibreringer. Freudenthals suspensjonsteorem. Stabil homotopi.

Topologisk K-teori. Bordisme.

Kategorien av homologiteorier. Entydighet for ordinær homologi.

Atiyah–Hirzebruch spektralsekvensen. Konstruksjon og konvergens av spektralsekvenser.

Jevne (ko-)homologi-teorier. CP^{∞}.

2. Spektra.

Brown representabilitet. Eilenberg–Mac Lane rom K(π, n). Spektra. Representerte (ko-)homologiteorier på rom og spektra.

Den stabile homotopi-kategorien. Triangulert kategori.

3. Produkter.

Eilenberg–Zilber teoremet. Paringer i homologi og kohomologi. Kunneth teoremene. Cup-produkt i kohomologi. Koprodukt i homologi. Algebraer og koalgebraer.

Multiplikative (ko-)homologi-teorier. Ring-spektra.

4. Operasjoner og kooperasjoner.

E^{∗}E og E_{∗}E. Steenrod algebraen. Adams spektralsekvensen.

5. Kompleks kobordisme.

(Ko-)homologi av CP^{∞}, BU, MU.

Den komplekse kobordisme-ringen MU_{∗}.

6. Formelle gruppelover.

7. Brown–Peterson spekteret.

Morava–Johnson–Wilson spektrene.

Morava K-teori av Eilenberg–Mac Lane rom.

S-moduler og S-algebraer.

Produkter på MU-moduler.

Hopkins–Kuhn–Ravenel karakter-teori.

A_{∞}-produkter på MU-moduler.

S-algebraisk geometri.

Referanser

[Ad74] Adams, J. F., Stable homotopy and generalised homology, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, Ill.-London, 1974.

[BJ02] Baker, Andrew; Jeanneret, Alain, Brave new Hopf algebroids and extensions of MUalgebras, Homology Homotopy Appl. 4 (2002), 163–173.

[Bo99] Boardman, J. Michael, Conditionally convergent spectral sequences, Homotopy invariant algebraic structures (Baltimore, MD, 1998), Contemp. Math., vol. 239, Amer. Math. Soc., Providence, RI, 1999, pp. 49–84.

[Ha02] Hatcher, Allen, Algebraic topology, Cambridge University Press, Cambridge, 2002.

[HKR00] Hopkins, Michael J.; Kuhn, Nicholas J.; Ravenel, Douglas C., Generalized group characters and complex oriented cohomology theories, J. Amer. Math. Soc. 13 (2000), 553–594.

[HPS97] Hovey, Mark; Palmieri, John H.; Strickland, Neil P., Axiomatic stable homotopy theory, Mem. Amer. Math. Soc., vol. 610, 1997.

[HS99] Hovey, Mark; Strickland, Neil P., Morava K-theories and localisation, Mem. Amer. Math. Soc., vol. 666, 1999.

[JW75] Johnson, David Copeland; Wilson, W. Stephen, BP operations and Morava’s extraordinary K-theories, Math. Z. 144 (1975), 55–75.

[La01] Lazarev, A., Homotopy theory of A^{∞} ring spectra and applications to MU-modules, K-Theory 24 (2001), 243–281.

[MM79] Madsen, Ib; Milgram, R. James, The classifying spaces for surgery and cobordism of manifolds, Annals of Mathematics Studies, vol. 92, Princeton University Press, Princeton, N.J., 1979.

[Mi58] Milnor, John, The Steenrod algebra and its dual, Ann. of Math. (2) 67 (1958), 150–171.

[Mi60] Milnor, J., On the cobordism ring Ω^{∗} and a complex analogue. I, Amer. J. Math. 82 (1960), 505–521.

[MM65] Milnor, John W.; Moore, John C., On the structure of Hopf algebras., Ann. of Math. (2) 81 (1965), 211–264.

[MS74] Milnor, John W.; Stasheff, James D., Characteristic classes, Annals of Mathematics Studies, No. 76, Princeton University Press, Princeton, N. J., 1974.

[RW80] Ravenel, Douglas C.; Wilson, W. Stephen, The Morava K-theories of Eilenberg–Mac Lane spaces and the Conner–Floyd conjecture, Amer. J. Math. 102 (1980), 691–748.

[St99] Strickland, N. P., Products on MU-modules, Trans. Amer. Math. Soc. 351 (1999), 2569–2606.

[Wh62] Whitehead, George W., Generalized homology theories, Trans. Amer. Math. Soc. 102 (1962), 227–283.