Funding Proposal for Cecil King Travel Scholarship 2007

David Barnes

February 20, 2007

Research Interests

• Algebraic Topology

• Stable Homotopy Theory

• Equivariant Homotopy Theory

• Model Categories

• Enriched Categories

• Related areas of Algebra and Representation Theory

I intend to use this grant to fund a three month period of study at the the University of Chicago with J. Peter May.

Background

Cohomology theories provide information about the abstract nature of topological spaces and allow us to distinguish between and work with these spaces. To each space X, a cohomology theory E^{∗} gives a group E^{∗}(X). For a map of spaces f: X → Y we have a map of groups E^{∗}(f): E^{∗}(Y) → E^{∗}(X) and this structure satisfies many useful axioms. Since spaces with a group action abound in all areas of mathematics, an understanding of how these spaces differ and behave is of particular importance. A G-equivariant cohomology theory is a special kind of cohomology theory that is designed to be used on spaces with a G-action.

Particular examples include: equivariant K-theory [Seg68] (which is constructed in the same way as K-theory but using G-vector bundles) and equivariant cobordism [tD72]. The Borel construction takes any cohomology theory E^{∗} and makes a Gequivariant cohomology theory from it: E_{G}^{∗}(X) = E^{∗}(EG_{+} ∧_{G} X) (EG_{+} is the universal free G-space with an additional G-fixed point adjoined). A rational Gequivariant cohomology theory satisfies the additional condition that each group E^{∗}(X) is a rational vector space and that the induced maps E^{∗}(f): E^{∗}(Y) → E^{∗}(X) are rational maps. This extra assumption simplifies the area so that it becomes amenable to study.

I study rational O(2)-equivariant cohomology theories (O(2) is the group of twoby-two orthogonal matrices) the collection of such theories is defined in [MM02]. These theories are classified in [Gre98b] and I am attempting to improve on this work using the techniques of highly-structured model categories. I intend to show that this classification can be given the structure of a monoidal Quillen equivalence: so that the classification is compatible with taking smash products of cohomology theories and with the homotopy information.

This project extends and relies upon a similar classification of rational SO(2)- equivariant cohomology theories [Gre99, Shi02, GS06]. In [Gre99] the author provides an algebraic model A for rational SO(2)-equivariant cohomology theories and proves that the derived category of this model DA is equivalent to the derived category of rational SO(2)-equivariant cohomology theories. The later papers [Shi02, GS06] extend this result to the case of a torus (a product of copies of SO(2)) and also improves the result, showing that the equivalence has the structure of a monoidal Quillen equivalence. The paper [Shi04] is an important part of this result.

Completed Work

In this project so far, [Bar07], I have defined several different model categories to represent rational O(2)-equivariant cohomology theories and I have shown that these are all equivalent. I have proven the following result, which splits such a category into two other model categories, it improves upon the result of [Gre98b] which gives a version of this splitting with less of the structure preserved.

Theorem 1 The adjoint pair below is a (monoidal) Quillen equivalence and hence an equivalence of derived categories.

O(2)-cohomology/ℚ ⇄ (cyclic O(2)-cohomology/ℚ × dihedral O(2)-cohomology/ℚ)

So for an O(2)-space X, we have an isomorphism of rational vector spaces

E^{∗}(X) ≅ E_{c}^{∗}(X) ⊕ E_{d}^{∗}(X)

that is compatible with maps f: X → Y, for E_{c}^{∗} a cyclic O(2)-cohomology theory and E_{d}^{∗} a dihedral O(2)-cohomology theory.

The use of this theorem is to separate these cohomology theories into those which behave like rational SO(2)-cohomology theories (the cyclic part) and those which behave like products of rational D_{2n}-cohomology theories (D_{2n} the dihedral group of order 2n). The cyclic part is much more interesting, to understand this part I have constructed a theory of categories (and model categories) with involution.

Definition 1 A category with involution (𝒞 , τ) is a category 𝒞 with a functor τ: 𝒞 → 𝒞 such that τ^{2} is naturally isomorphic to 1_{𝒞} by the natural transformation λ: 1_{𝒞} → τ^{2}.

Definition 2 Given a category with involution (𝒞 , τ), we define the associated halftwisted category τ ⋉ 𝒞 to be the category with objects (A, w) for A ∈ 𝒞 and w: A → τA, such that τw ∘ w = λ_{A}. A morphism of such objects is a commutative square as below.

\xymatrix{A \ar[d]^{w} \ar[r]^{f} & B \ar[d]^{w′} \\ τA \ar[r]^{τf} & τB}

The half-twisted category of a category with involution is a way of altering the following result to the case of a semi-direct product. Given two groups A and B, there is an isomorphism of categories

(A × B) − spaces ≅ A − (B − spaces).

The group O(2) is a semi-direct product, O(2) = W ⋉ SO(2) where W is the group of order two. We can describe the category of SO(2)-spaces as a category with involution (given by inverting the SO(2)-action) and then we have an isomorphism of categories

W ⋉ (SO(2)-spaces) ≅ O(2)-spaces.

Theorem 2 There is a (monoidal) Quillen equivalence

W ⋉ (SO(2)-cohomology/ℚ)\overrightarrow{⟵}cyclic O(2)-cohomology/ℚ.

Planned Work

The investigation of the dihedral side still remains and this should be easy to understand using the results of [GM95], which proves that for finite G, a rational G-equivariant cohomology theory splits as a product of ordinary (non-equivariant) cohomology theories. The category of dihedral O(2)-cohomology theories has a description as a category of Mackey functors, or equally, as sheaves over a space [Gre98a, Gre98b]. I will improve on this by constructing a symmetric monoidal Quillen equivalence between these categories.

The next step in cyclic cohomology part of the project is to show that the equivalence between rational SO(2)-cohomology theories and the category A passes to an equivalence of the half-twisted categories W ⋉ (SO(2)-cohomology/ℚ) and W ⋉ A. Then I will investigate the structure of the category W ⋉A and see which of the nice properties of A (such as the ease of constructing injective or projective resolutions) pass to this new category.

In particular this project will create a framework to prove equivalent classification theorems for many more Lie groups by performing similar steps. First splitting out a part containing a maximal torus and understanding this via [GS06], then dealing with the remainder, which should behave somewhat like the finite group case.

Peripherally this project will improve the application of model categories to highly structured stable homotopy theory. Model structures are an axiomatic way of encoding homotopy information: data about how shapes and spaces can be deformed into each other. By ‘highly structured’ I mean that these categories will have a symmetric monoidal product with compatible internal function object, along with enrichments, tensoring and cotensoring over various other model categories, such that these structures pass to the homotopy categories. This extra information will be preserved in the classification and is important in proving such results like a universal coefficient theorem (relating cohomology and homology) and in potential applications such as equivariant elliptic cohomology [AB02]. It is also highly desirable when considering cohomology theories with a ring structure and modules over such a cohomology theory.

Benefits of the Visit

As well as assisting with my studies, this visit will also improve international links between mathematicians working in related areas of mathematics (or, equally, areas of mathematics that use similar ideas and concepts). Within a short distance of Chicago are many experts in related fields including: Matt Ando and Charles Rezk at the University of Illinois at Urbana-Champaign, Paul Goerss at Northwestern University and Brooke Shipley at the University of Illinois at Chicago. There are also mathematicians of all ranges of experience who may provide me with new and interesting ideas to investigate or incorporate into the project or who may benefit from sharing the methods and results of my project. In particular the proposed supervisor is an internationally renowned mathematician with many significant results in similar areas of mathematics.

References

[AB02] Matthew Ando and Maria Basterra. The Witten genus and equivariant elliptic cohomology. Math. Z., 240(4):787–822, 2002.

[Bar07] David Barnes. Phd thesis: Rational O(2)-equivariant spectra. In preperation, 2007.

[GM95] J. P. C. Greenlees and J. P. May. Generalized Tate cohomology. Mem. Amer. Math. Soc., 113(543):viii+178, 1995.

[Gre98a] J. P. C. Greenlees. Rational Mackey functors for compact Lie groups. I. Proc. London Math. Soc. (3), 76(3):549–578, 1998.

[Gre98b] J. P. C. Greenlees. Rational O(2)-equivariant cohomology theories. In Stable and unstable homotopy (Toronto, ON, 1996), volume 19 of Fields Inst. Commun., pages 103–110. Amer. Math. Soc., Providence, RI, 1998.

[Gre99] J. P. C. Greenlees. Rational S^{1}-equivariant stable homotopy theory. Mem. Amer. Math. Soc., 138(661):xii+289, 1999.

[GS06] J. P. C. Greenlees and B. Shipley. An algebraic model for rational torusequivariant stable homotopy theory. Preprint, 2006.

[MM02] M. A. Mandell and J. P. May. Equivariant orthogonal spectra and Smodules. Mem. Amer. Math. Soc., 159(755):x+108, 2002.

[Seg68] Graeme Segal. Equivariant K-theory. Inst. Hautes Études Sci. Publ. Math., (34):129–151, 1968.

[Shi02] Brooke Shipley. An algebraic model for rational S^{1}-equivariant stable homotopy theory. Q. J. Math., 53(1):87–110, 2002.

[Shi04] Brooke Shipley. Hℤ-algebra spectra are differential graded algebras. Preprint, 2004.

[tD72] Tammo tom Dieck. Kobordismentheorie klassifizierender Räume und Transformationsgruppen. Math. Z., 126:31–39, 1972.