Problems in homotopy theory
Problems from Hovey's list

Major Problems in Algebraic Topology

The Generating Hypothesis

The generating hypothesis may be stated as follows:

Generating Hypothesis: Let Ω and Γ be finite spectra with a stable map F:ΩΓ. If π *F=0, then F is stably null-homotopic. (Cf. Stable and Unstable Homotopy edited by William G. Dwyer).

Alternative Statement of the Generating Hypothesis: The stable homotopy functor is faithful on the category of finite spectra. (Cf. Hovey’s list.)

One consequence of this hypothesis is that the stable homotopy functor is full.

A Geometric Meaning for Elliptic Cohomology

The following is the excerpt from Hovey’s list:

“Find some geometric meaning for elliptic cohomology. I believe this problem may be solvable–we keep learning new things about it. One thing I will say here; if I am called to referee a paper on elliptic cohomology that does not deal with the Hopkins viewpoint on elliptic spectra, I will almost surely reject it. The time is gone when one could write papers about the Landweber-Ravenel-Stong elliptic cohomology based on the Jacobi quartic–we now understand that that is only one of many different elliptic cohomology theories, and all papers on elliptic cohomology should now accept that and deal with it. The fundamental reference here is M. J. Hopkins, Topological modular forms, the Witten genus, and the theorem of the cube, in {\it Proceedings of the International Congress of Mathematicians, Vol.\ 1, 2 (Z"urich, 1994)}, 554–565, Birkh"auser, Basel, 1995; MR 97i:11043. But one should also see Grojnowski’s approach to equivariant elliptic cohomology–unfortunately, this does not seem to be published, but there is a preprint. Matthew Ando has also thought about this, see M. Ando, Power operations in elliptic cohomology and representations of loop groups, Trans. Amer. Math. Soc. ; CNO CMP 1 637 129. I am sure I have left something out here as well.”

Again quoting Hovey’s list:

”… find some way of doing index theory related to elliptic cohomology…”

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