Math 283 - Topics in algebraic topology - Spring 2011

Homepage http://math.stanford.edu/~galatius/283S11/

Classes TTh 8:35 - 9:50 in 200-201

Instructor Soren Galatius

Office 383-DD

e-mail galatius@stanford.edu

Office hours By appointment

Prerequisites Math 282B or similar.

Topic The goal of the class is to cover the recent paper by Hill-Hopkins-Ravenel about the Kervaire invariant one problem. We will not say much about the history of this problem, but start right away with one of the main tools in the proof, namely equivariant stable homotopy theory.

Literature Can all be downloaded from Ravenel's page.

Hill, Hopkins, Ravenel: On the non-existence of elements of Kervaire invariant one. arXiv:0908.3724

Greenlees, May: Equivariant stable homotopy theory.

Mandell, May: Equivariant orthogonal spectra.

See also May's "Alaska notes" for a lot of material about equivariant stable homotopy theory. See also Mandell-May-Schwede-Shipley for foundational material about (non-equivariant) orthogonal spectra and how they relate to other notions of spectra.

Lecture plan. Will be updated during the quarter.

Date Topic

3/29 Nothing mathematical happened, due to qualifying exams. There was a fire alarm.

3/31 Unstable equivariant homotopy theory. Main reference: Greenlees-May. G-spaces, equivariant maps, equivariant homotopy equivalences, equivariant weak equivalences, G-CW complexes, equivariant Whitehead theorem.

4/5 Basepoints. The orbit category (objects the G/H, morphisms equivariant maps). Homotopy groups (as contravariant functors from the orbit category). Coefficient systems (contravariant functors from the orbit category to abelian groups). Cohomology of a G-CW complex with coeffiencents in a coefficient system. Eilenberg-MacLane spaces. Examples from HHR (prop 3.5).

4/7 More examples of Bredon cohomology: Both H^*(X) and H^*(X^G), the cohomology of the underlying space and of the fixed points, can be written as Bredon cohomology H^*(X;M) for suitable coefficient systems M (see e.g. May's "Alaska notes", the chapter about Smith theory). Families F of subgroups: X is an F-space if all isotropy groups of points in X are in F. If X is a G-CW complex, this is equivalent to it being built from (G/H)xD^n with H in F. If F is closed under conjugation and taking subgroups, there is a space EF such that (EF)^H is contractible for all H in F. This has the universal property that any G-CW F-space X admits a unique equivariant map X -> EF. The isotropy separation sequence EF_+ -> S^0 -> \tilde EF. Explicit examples: For F=P the family of proper subgroups, a model for EP is colim_n S(n(\rho_G - 1)), the direct limit of the sphere in the n-fold direct sum of the reduced regular representation. A model for \tilde EP is the direct limit of S^{n(\rho_G-1)}, the limit of the one-point compactifications. Finally, I talked about the definition of equivariant orthogonal spectra.

4/12 Equivariant orthogonal spectra: These are functors X: I_G -> T_G of categories enriched over T^G (here T^G is the category of based G-spaces and equivariant maps. T_G has the same objects, but T_G(A,B) is the based G-space of non-equivariant maps (with compact-open topology and conjugation action)). For two such spectra X and Y, there is a based G-space of (non-equivariant) maps X -> Y. Thus, the category of orthogonal G-spectra is enriched over T_G (the category of based G-spaces). The underlying "naive" spectrum is obtained by evaluating X at the objects R^n in I_G. Examples: The sphere spectrum S and the suspension spectrum of a based G-space. Change of universe. Unbelievable miracle: This gives an equivalence of categories between S_G (orthogonal equivariant spectra) and spectra with a G-action (i.e. pairs consisting of a non-equivariant spectrum X and a homomorphism G -> Aut(X)). Various constructions and adjunctions in S_G: Smash product of A in T_G and X in S_G, the shift suspension Sigma^V(X) of a spectrum X, the function spectrum F(X,Y), the smash product of two spectra, and finally the negative sphere S^{-W} associated to an orthogonal G-representation W. Important note: I haven't yet talked about homotopy theory; all isomorphisms today were actual (categorical) isomorphisms, not just "weak equivalences" (which I'll defined next time). A good reference for constructions (smash product etc) of (non-equivariant) orthogonal spectra is Mandell-May-Schwede-Shipley.

4/14 Homotopy groups of a spectrum: The slogan is that pi_*(X) is an RO(G)-graded Mackey functor. RO(G)-graded meant that there is an abelian group pi_W^H(X) for each virtual representation W and each subgroup H of G, and Mackey means that it is functorial in H in a certain way. Then we defined weak equivalences as maps of spectra that induce isomorphisms in pi_n^H for all H and all integers n. Important example, which I did in some detail: The natural map S^V smash S^{-V} to S^0 is a weak equivalence. Another example (requires an equivariant version of Freudenthal's suspension theorem): The adjoint S^{-V} -> F(S^V,S^0) is a weak equivalence.

4/19 Homotopy fibers and cofibers of maps of spectra: These are defined "levelwise" (i.e. just take the (co)fiber of the maps of V'th spaces in the spectra). Induce long exact sequences of RO(G)-graded Mackey functors. Elements in pi_n^H(X) are not necessarily represented by spectrum maps G/H_+ smash S^n to X, but they can (by definition, pretty much) be represented by maps G/H_+ smash S^{n+V} smash S^{-V} to X for some large enough representation V. Killing homotopy groups. Eilenberg-MacLane spectra HM for any Mackey functor M. G-Omega spectra and G-cellular (alias CW) spectra. Any spectrum can be approximated from the left by a CW spectrum and from the right by an Omega spectrum. For spectra X and Y, [X,Y] should be defined as spectrum maps from a CW approximation of X to an Omega approximation of Y.

4/21 More on Mackey functor cohomology: We calculated part of the RO(G) graded homology groups H^*(S;Z), where S is the sphere spectrum and Z is the "constant" Mackey functor. This translates to homotopy groups of the Eilenberg-MacLane spectrum HZ. We calculated part of the RO(G) graded homotopy groups (which is non-trivial when the element of RO(G) is not a trivial virtual representation). Then we talked about categorical fixed points, its derived functor (homotopy invariant fixed points), and introduced geometric fixed points.

4/26 More on geometric fixed points, which we defined as the derived fixed points of E-tilde-P smash X. The homotopy groups of the geometric fixed points of X are obtained from the RO(G)-graded homotopy groups of X by a taking direct limit. More explicitly, these form a module over the RO(G)-graded homotopy groups of the sphere spectrum, and one inverts the class represented by the inclusion of 0 into the reduced regular representation. Finally, we talked about a model for the geometric fixed points which has better categorical properties.

Date	Topic
3/29	Nothing mathematical happened, due to qualifying exams. There was a fire alarm.
3/31	Unstable equivariant homotopy theory. Main reference: Greenlees-May. G-spaces, equivariant maps, equivariant homotopy equivalences, equivariant weak equivalences, G-CW complexes, equivariant Whitehead theorem.
4/5	Basepoints. The orbit category (objects the G/H, morphisms equivariant maps). Homotopy groups (as contravariant functors from the orbit category). Coefficient systems (contravariant functors from the orbit category to abelian groups). Cohomology of a G-CW complex with coeffiencents in a coefficient system. Eilenberg-MacLane spaces. Examples from HHR (prop 3.5).
4/7	More examples of Bredon cohomology: Both H^(X) and H^(X^G), the cohomology of the underlying space and of the fixed points, can be written as Bredon cohomology H^*(X;M) for suitable coefficient systems M (see e.g. May's "Alaska notes", the chapter about Smith theory). Families F of subgroups: X is an F-space if all isotropy groups of points in X are in F. If X is a G-CW complex, this is equivalent to it being built from (G/H)xD^n with H in F. If F is closed under conjugation and taking subgroups, there is a space EF such that (EF)^H is contractible for all H in F. This has the universal property that any G-CW F-space X admits a unique equivariant map X -> EF. The isotropy separation sequence EF_+ -> S^0 -> \tilde EF. Explicit examples: For F=P the family of proper subgroups, a model for EP is colim_n S(n(\rho_G - 1)), the direct limit of the sphere in the n-fold direct sum of the reduced regular representation. A model for \tilde EP is the direct limit of S^{n(\rho_G-1)}, the limit of the one-point compactifications. Finally, I talked about the definition of equivariant orthogonal spectra.
4/12	Equivariant orthogonal spectra: These are functors X: I_G -> T_G of categories enriched over T^G (here T^G is the category of based G-spaces and equivariant maps. T_G has the same objects, but T_G(A,B) is the based G-space of non-equivariant maps (with compact-open topology and conjugation action)). For two such spectra X and Y, there is a based G-space of (non-equivariant) maps X -> Y. Thus, the category of orthogonal G-spectra is enriched over T_G (the category of based G-spaces). The underlying "naive" spectrum is obtained by evaluating X at the objects R^n in I_G. Examples: The sphere spectrum S and the suspension spectrum of a based G-space. Change of universe. Unbelievable miracle: This gives an equivalence of categories between S_G (orthogonal equivariant spectra) and spectra with a G-action (i.e. pairs consisting of a non-equivariant spectrum X and a homomorphism G -> Aut(X)). Various constructions and adjunctions in S_G: Smash product of A in T_G and X in S_G, the shift suspension Sigma^V(X) of a spectrum X, the function spectrum F(X,Y), the smash product of two spectra, and finally the negative sphere S^{-W} associated to an orthogonal G-representation W. Important note: I haven't yet talked about homotopy theory; all isomorphisms today were actual (categorical) isomorphisms, not just "weak equivalences" (which I'll defined next time). A good reference for constructions (smash product etc) of (non-equivariant) orthogonal spectra is Mandell-May-Schwede-Shipley.
4/14	Homotopy groups of a spectrum: The slogan is that pi_*(X) is an RO(G)-graded Mackey functor. RO(G)-graded meant that there is an abelian group pi_W^H(X) for each virtual representation W and each subgroup H of G, and Mackey means that it is functorial in H in a certain way. Then we defined weak equivalences as maps of spectra that induce isomorphisms in pi_n^H for all H and all integers n. Important example, which I did in some detail: The natural map S^V smash S^{-V} to S^0 is a weak equivalence. Another example (requires an equivariant version of Freudenthal's suspension theorem): The adjoint S^{-V} -> F(S^V,S^0) is a weak equivalence.
4/19	Homotopy fibers and cofibers of maps of spectra: These are defined "levelwise" (i.e. just take the (co)fiber of the maps of V'th spaces in the spectra). Induce long exact sequences of RO(G)-graded Mackey functors. Elements in pi_n^H(X) are not necessarily represented by spectrum maps G/H_+ smash S^n to X, but they can (by definition, pretty much) be represented by maps G/H_+ smash S^{n+V} smash S^{-V} to X for some large enough representation V. Killing homotopy groups. Eilenberg-MacLane spectra HM for any Mackey functor M. G-Omega spectra and G-cellular (alias CW) spectra. Any spectrum can be approximated from the left by a CW spectrum and from the right by an Omega spectrum. For spectra X and Y, [X,Y] should be defined as spectrum maps from a CW approximation of X to an Omega approximation of Y.
4/21	More on Mackey functor cohomology: We calculated part of the RO(G) graded homology groups H^*(S;Z), where S is the sphere spectrum and Z is the "constant" Mackey functor. This translates to homotopy groups of the Eilenberg-MacLane spectrum HZ. We calculated part of the RO(G) graded homotopy groups (which is non-trivial when the element of RO(G) is not a trivial virtual representation). Then we talked about categorical fixed points, its derived functor (homotopy invariant fixed points), and introduced geometric fixed points.
4/26	More on geometric fixed points, which we defined as the derived fixed points of E-tilde-P smash X. The homotopy groups of the geometric fixed points of X are obtained from the RO(G)-graded homotopy groups of X by a taking direct limit. More explicitly, these form a module over the RO(G)-graded homotopy groups of the sphere spectrum, and one inverts the class represented by the inclusion of 0 into the reduced regular representation. Finally, we talked about a model for the geometric fixed points which has better categorical properties.