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Research Statement Research Statement Michael Andrews Department of Mathematics MIT November 1, 2014 1 Introduction Algebraic topologists are interested in the class of spaces which can be built from spheres. For this reason, when one tries to understand the continuous maps between two spaces up to homotopy, it is natural to restrict attention to the maps between spheres first. In this case, the sets of interest have S a group structure and these groups stabilize to give the stable homotopy groups of spheres fπn gn≥0. Calculating all of these groups is an impossible task but one can ask for partial information. In particular, one can try to understand the global structure of these groups by proving the existence of recurring patterns. One can also make complete calculations of related groups. This represents a touchstone problem in homotopy theory and has motivated all of my research so far. S Serre observed that π∗ is locally finitely generated and so one can understand the stable homo- S topy groups better by studying π∗ ⊗ Z(p) for each prime p separately. This gave the first hint that localization would be a useful tool in algebraic topology. Faced with the impossibility of calculating all the stable homotopy groups, topologists have resorted to calculating the homotopy of certain localizations of the sphere spectrum. A theme in my work has been to understand the homotopy of such localizations through spectral sequence calculations where one can start by addressing the localization at the algebraic level of an E2-page. The Adams spectral sequence (ASS) and the Adams-Novikov spectral sequence (ANSS) are useful tools for homotopy theorists. Theoretically, they enable a calculation of the stable homotopy groups but they have broader utility than this. Much of contemporary homotopy theory has been inspired by analyzing the structure of these spectral sequences. In particular, many periodic patterns are visible on the Adams E2-page and these arise because of the way that chromatic homotopy theory manifests itself in the spectral sequence. Although it is impossible to give a complete description of the Adams spectral sequence, at odd primes, I have succeeded [3] in giving a description above a line. As the prime tends to infinity, the fraction of the ASS described tends to 1. As a consequence of this work, I have been able to find, for the first time, differentials of arbitrarily long length in the ASS. The stable motivic homotopy category contains the spaces of topology and the schemes of alge- braic geometry. It is an enrichment of the stable homotopy category because one can \tensor down" to recover the classical story. Even though the global structure of the homotopy of the motivic S sphere spectrum is more complicated than that of π∗ , the motivic perspective provides insight into the classical problem. Moreover, the ASS and ANSS inherit extra gradings, so that some difficult classical computations become formal. 1 When one works motivically over C, the nilpotence theorem [11] of Devinatz, Hopkins and Smith breaks down and I conjecture, with strong evidence, that there is a new world of non-nilpotent self maps waiting to be discovered. In my recent work, I have started to explore these possibilities. In [5], by answering a longstanding question [34] about the Adams-Novikov E2-page, Haynes Miller and I were able to give a simple description of the homotopy of the η-local motivic sphere over C, proving a conjecture of Guillou and Isaksen appearing in [14]. In [4], I constructed a non-nilpotent self map of S/η and used this to construct infinitely many new nontrivial homotopy classes in the homotopy of the motivic sphere spectrum. One might see this as the motivic analog of Adams' work on the mod 2 Moore spectrum in [1], although the techniques used are somewhat different. I am fascinated by this new world of chromatic motivic homotopy theory. I believe there is an interesting story to be investigated and I envision much of my future work to be dedicated to this exciting field. I would like to discover whether there is a homology theory for which a nilpotence theorem does hold, to make analogous calculations to those of Mahowald in [17{19], and to set up a chromatic spectral sequence in this new setting. More broadly, I would like to understand to what extent this new story exists over arbitrary ground fields. In keeping with my original motivation, I would like to determine the implications of this new world for classical homotopy theory. 2 Chromatic motivic homotopy theory over C at p = 2 All of my research has been an effort to understand the global structure of the stable homotopy groups of spheres. The objects of the stable homotopy category are called spectra and with this terminology, it is the homotopy groups of the sphere spectrum S0 that are of interest to me. There is a theory of localization for spectra due to Bousfield [9], and we work one prime at a time by studying the homotopy of the p-local sphere spectrum. In this section, we take p = 2 and write S0 0 for S(2), the 2-local sphere spectrum. 0 Chromatic homotopy theory provides a systematic procedure for trying to understand π∗(S ). One has a recursive algorithm for calculating the homotopy of any finite 2-local spectrum and we can start this algorithm with X = S0. d −1 1. Find a non-nilpotent self map f :Σ X −! X and calculate f π∗(X). 2. Attack the problem of calculating the f-torsion elements in π∗(X) using a Bockstein spectral L 1 sequence v<0 π∗(X=f) =) π∗(X=f ). For the input we need to calculate π∗(X=f) and we do this by replacing X with X=f and going back to step 1. To say that f is non-nilpotent, we mean that each composite f e :ΣdeX −! X is not nullhomotopic. By X=f we mean the cofiber of f, a construction one can make in any triangulated category. We 1 leave X=f as a black box; its homotopy stores, in some sense, the f-torsion elements of π∗(X). Before the work of Devinatz, Hopkins and Smith, in [11] and [15] it was not known that one could always construct the non-nilpotent self maps which appear in the algorithm. However, in [1], 4 8 Adams constructed a non-nilpotent map v1 :Σ S=2 −! S=2 and this gave the first hint that the above procedure is, in fact, implementable. One might say that Adams' work gave birth to chromatic homotopy theory. An informal statement of the nilpotence theorem of [11] is that any non-nilpotent self map is a \vn-self map" for some n 2 f0; 1; 2;:::g. Working motivically over C, one can find a non-nilpotent 0;0 self map on the motivic sphere spectrum S , which is not a vn-self map and it is this observation that has inspired my recent work. 2 To describe such a non-nilpotent self map on S0;0 we first recall the construction of the classical Hopf map. It is given by the following composite: 3 2 1 2 η : S ⊂ C − 0 −! P (C) = S : 0 Stably, this defines an element of π1(S ). Motivically, spheres are bigraded: we have the topological 1;0 1 1;1 1 1-sphere S = S , the algebraic 1-sphere S = A − 0, and other spheres are smash products of 3;2 2;1 2 1 these spheres. One can describe the spheres S and S by the schemes A −0 and P , respectively, and so we can make a construction analogous to the one above: 3;2 2 1 2;1 η : S = A − 0 −! P = S : 0;0 Stably, this defines an element η 2 π1;1(S ) and it was shown in [16] that this homotopy class is non-nilpotent. Moreover, η is not a vn-self map. Influenced by the algorithm above and the conjecture of Guillou and Isaksen appearing in [14], Haynes Miller and I proved the following theorem in [5]. 0;0 0;0 Theorem 2.1 (Andrews, Miller, [5]). Let η 2 π1;1(S ) and σ 2 π7;4(S ) be the motivic Hopf 0;0 invariant one elements described in [13] and let µ9 2 π9;5(S ) be the element defined by the Toda bracket h8σ; 2; ηi. Then −1 0;0 ±1 2 η π∗;∗(S ) = F2[η ; σ; µ9]=(ησ ): We have written ησ2 instead of σ2, since the relation ησ2 = 0 holds before localization. −1 0 The analogous classical calculation is that of 2 π∗(S ). Because we are working at 2, this is the rational homotopy of the sphere spectrum, which is Q concentrated in degree 0. Thus, although theorem 2.1 is simple to state, it is already far more complicated than the analogous classical result. To prove theorem 2.1 we made use of the motivic Adams-Novikov spectral sequence. It is noted in [16] and [12] that the ANSS and its motivic analog are closely related and so the theorem follows relatively quickly from the following proposition, which concerns the classical Adams-Novikov E2- page. This proposition resolves a question raised by Zahler in [34]. Proposition 2.2 (Andrews, Miller, [5]). See figure 1. 0 Write E2(S ; BP ) for the E2-page of the ANSS and write αn for the generator of the cyclic 1;2n 0 group E2 (S ; BP ). The localization map 0 −1 0 E2(S ; BP ) −! α1 E2(S ; BP ) is an isomorphism above a line of slope 1=5 and intercept 2 in the standard (t − s; s) coordinates.
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