Computation of the homotopy of the spectrum tmf

By Tilman Bauer. 2003

116 citations recorded by the source corpus.

This paper contains a complete computation of the homotopy ring of the spectrum of topological modular forms constructed by Hopkins and Miller. The computation is done away from 6, and at the (interesting) primes 2 and 3 separately, and in each of the latter two cases, a sequence of algebraic Bockstein spectral sequences is used to compute the E2 term of the elliptic Adams‐Novikov spectral sequence from the elliptic curve Hopf algebroid. In a further step, all the differentials in the latter spectral sequence are determined. The result of this computation is originally due to Hopkins and Mahowald (unpublished). 55N34; 55T15

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b737b0286d14377a78caf962027f0a82f75df91e

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In Geometry and Topology

1. Lennart Meier
2. A. Mathew
3. M. Hill
4. Vesna Stojanoska
5. Mark Behrens
6. Paul G. Goerss
7. I. Bobkova
8. Zhouli Xu

In Mathematical Physics

1. Lennart Meier
2. A. Mathew
3. M. Hill
4. Vesna Stojanoska
5. I. Bobkova
6. Mark Behrens
7. Paul G. Goerss
8. Zhouli Xu

Citations in the corpus

• The homology of $\tmf$ (2013, methodology): “We refer to [7, Lectures 11–12] for a treatment of the construction of the sheaf O and to [4] and [7, Lecture 13] for some of the computations of the homotopy groups.”
• The Picard group of topological modular forms via descent theory (2014, background): “1; 5/; we will denote the generating class by the generic name x (in [6] it bears the name a(2) 1h1 ).”
• The C2–spectrum Tmf1(3) and its invertible modules (2015, methodology): “As computed in the beginning of Section 7 of [6], the map p is surjective and we have Spec A=Gm Mcub Spec A=Gm' .”
• A modular description of the K(2)-local sphere at the prime 3 (2005, methodology): “MF∧V(1) is the cohomology of the quotient Hopf algebroid of (B,ΓB) (see Section 1.4) given by (B/I2,ΓB/I2) where I2 is the invariant regular ideal (3,q2) ⊂B. The ANSS E2 term for TMF∧V(1) is computed [4], [41] to be E2(TMF∧V(1)) = H∗(Γ B/I2) = F3[q ±1 4 ,β]⊗E[α] 31 where the generators have bidegree”
• Topological resolutions in K(2) ‐local homotopy theory at the prime 2 (2016, methodology): “What we will need can be read off of Figure 4, which is adapted from the charts created by Tilman Bauer [2, Section 8].”
• The triviality of the 61-stem in the stable homotopy groups of spheres (2016, background): “s shows that the attaching map factors through P2 1 . S4 // ❅˜˜ ❅ ❅ ❅ P 4 1 /S 3 P2 1 << The group π4(P2 1 ) is generated by η2[2] and ν[1]. However, the element η2[2] is killed by η[4] in the Atiyah-Hirzebruch spectral sequence of P4 1 /S 3. The element ν[1] does not detect the attaching m”
• Stable Stems (2014, methodology): “For this, we use that β10/2 is the only element of the Adams-Novikov spectral sequence in the 54-stem with filtration 2 that is not divisible by α1, and that this element maps to ∆ (2)h(2)2 in the AdamsNovikov spectral sequence for tmf [5] [37].”