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35
The homotopy fixed point spectra of profinite Galois extensions
"... Let E be a klocal profinite GGalois extension of an E∞ring spectrum A (in the sense of Rognes). We show that E may be regarded as producing a discrete Gspectrum. Also, we prove that if E is a profaithful klocal profinite extension which satisfies certain extra conditions, then the forward dir ..."
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Let E be a klocal profinite GGalois extension of an E∞ring spectrum A (in the sense of Rognes). We show that E may be regarded as producing a discrete Gspectrum. Also, we prove that if E is a profaithful klocal profinite extension which satisfies certain extra conditions, then the forward direction of Rognes’s Galois correspondence extends to the profinite setting. We show the function spectrum FA((EhH)k, (EhK)k) is equivalent to the homotopy fixed point spectrum ((E[[G/H]]) hK)k where H and K are closed subgroups of G. Applications to Morava Etheory are given, including showing that the homotopy fixed points defined by Devinatz and Hopkins for closed subgroups of the extended Morava stabilizer group agree with those defined with respect to a continuous action and in terms of the derived functor of fixed points.
CONGRUENCES BETWEEN MODULAR FORMS GIVEN BY THE DIVIDED β FAMILY IN HOMOTOPY THEORY
"... Abstract. We characterize the 2line of the plocal AdamsNovikov spectral sequence in terms of modular forms satisfying a certain explicit congruence condition for primes p ≥ 5. We give a similar characterization of the 1line, reinterpreting a computation of A. Baker. These results are then used t ..."
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Abstract. We characterize the 2line of the plocal AdamsNovikov spectral sequence in terms of modular forms satisfying a certain explicit congruence condition for primes p ≥ 5. We give a similar characterization of the 1line, reinterpreting a computation of A. Baker. These results are then used to deduce that, for ℓ a prime which generates Z × p, the spectrum Q(ℓ) detects the α and β families in the stable stems. Contents
The homotopy of the K(2)local Moore spectrum at the prime 3 revisited
"... Abstract. In this paper we use the approach introduced in [5] in order to analyze the homotopy groups of LK(2)V (0), the mod3 Moore spectrum V (0) localized with respect to Morava Ktheory K(2). These homotopy groups have already been calculated by Shimomura [12]. The results are very complicated s ..."
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Abstract. In this paper we use the approach introduced in [5] in order to analyze the homotopy groups of LK(2)V (0), the mod3 Moore spectrum V (0) localized with respect to Morava Ktheory K(2). These homotopy groups have already been calculated by Shimomura [12]. The results are very complicated so that an independent verification via an alternative approach is of interest. In fact, we end up with a result which is more precise and also differs in some of its details from that of [12]. An additional bonus of our approach is that it breaks up the result into smaller and more digestible chunks which are related to the K(2)localization of the spectrum TMF of topological modular forms and related spectra. Even more, the AdamsNovikov differentials for LK(2)V (0) can be read off from those for TMF. 1.
HIGHER REAL KTHEORIES AND TOPOLOGICAL AUTOMORPHIC FORMS
"... Abstract. Given a maximal finite subgroup G of the nth Morava stabilizer group at a prime p, we address the question: is the associated higher real Ktheory EOn a summand of the K(n)localization of a TAFspectrum associated to a unitary similitude group of type U(1, n − 1)? We answer this question ..."
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Abstract. Given a maximal finite subgroup G of the nth Morava stabilizer group at a prime p, we address the question: is the associated higher real Ktheory EOn a summand of the K(n)localization of a TAFspectrum associated to a unitary similitude group of type U(1, n − 1)? We answer this question in the affirmative for p ∈ {2, 3, 5, 7} and n = (p − 1)p r−1 for a maximal finite subgroup containing an element of order p r. We answer the question in the negative for all other odd primary cases. In all odd primary cases, we to give an explicit presentation of a global division algebra with involution in which the group G embeds unitarily. Contents
The 3local tmfhomology of BΣ3
 Proc. Amer. Math. Soc
"... (Communicated by Paul Goerss) Abstract. In this paper, we introduce a Hopf algebra, developed by the author and André Henriques, which is usable in the computation of the tmfhomology of a space. As an application, we compute the tmfhomology of BΣ3 in a manner analogous to Mahowald and Milgram’s co ..."
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(Communicated by Paul Goerss) Abstract. In this paper, we introduce a Hopf algebra, developed by the author and André Henriques, which is usable in the computation of the tmfhomology of a space. As an application, we compute the tmfhomology of BΣ3 in a manner analogous to Mahowald and Milgram’s computation of the kohomology RP ∞ in [7]. 1.
POWER OPERATIONS IN MORAVA ETHEORY: STRUCTURE AND CALCULATIONS
, 2013
"... We review what is known about power operations for height 2 Morava Etheory, and carry out some sample calculations. ..."
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We review what is known about power operations for height 2 Morava Etheory, and carry out some sample calculations.
String bordism and chromatic characteristics
, 2013
"... We introduce characteristics into chromatic homotopy theory. This parallels the prime characteristics in number theory as well as in our earlier work on structured ring spectra and unoriented bordism theory. Here, the K(n)local HopkinsMiller classes ζn take the places of the prime numbers, and thi ..."
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We introduce characteristics into chromatic homotopy theory. This parallels the prime characteristics in number theory as well as in our earlier work on structured ring spectra and unoriented bordism theory. Here, the K(n)local HopkinsMiller classes ζn take the places of the prime numbers, and this allows us to discuss higher bordism theories. We prove that the K(2)localizations of the spectrum of topological modular forms as well as the string bordism spectrum have characteristic ζ2. 2010 MSC: primary 55N22, 55P43; secondary 19L41, 57R90, 58J26. 1
TOPOLOGICAL MODULAR FORMS [after Hopkins, Miller, and Lurie]
"... In the early 1970s, Quillen [Qui] noticed a strong connection between 1parameter formal Lie groups and cohomology theories with a natural theory of Chern classes. The algebraic geometry of these formal Lie groups allowed Morava, Ravenel, and others to make predictions about large scale phenomena in ..."
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In the early 1970s, Quillen [Qui] noticed a strong connection between 1parameter formal Lie groups and cohomology theories with a natural theory of Chern classes. The algebraic geometry of these formal Lie groups allowed Morava, Ravenel, and others to make predictions about large scale phenomena in stable homotopy theory, and the
THE BROWNCOMENETZ DUAL OF THE K(2)LOCAL SPHERE AT THE PRIME 3
"... Abstract. We calculate the homotopy type of the BrownComenetz dual I2 of the K(2)local sphere at the prime 3 and show that there is an equivalence in the K(2)local ..."
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Abstract. We calculate the homotopy type of the BrownComenetz dual I2 of the K(2)local sphere at the prime 3 and show that there is an equivalence in the K(2)local