The topological modular forms of $\mathbb{R}{P}^2$ and $\mathbb{R}{P}^2\wedge \mathbb{C}{P}^2$

Abstract

In this paper, we study the elliptic spectral sequence computing $tm{f}_{\ast }(\mathbb{R}{P}^2)$ and $tm{f}_{\ast }(\mathbb{R}{P}^2\wedge \mathbb{C}{P}^2)$. Specifically, we compute all differentials and resolve exotic extensions by 2, $\eta$, and $\nu$. For $tm{f}_{\ast }(\mathbb{R}{P}^2\wedge \mathbb{C}{P}^2)$, we also compute the effect of the $v_1$-self maps of $\mathbb{R}{P}^2\wedge \mathbb{C}{P}^2$ on $tmf$-homology.

References in corpus (11)

• On the Adams Spectral Sequence for R-modules
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• Computation of the homotopy of the spectrum tmf
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• Chromatic splitting for the $K(2)$-local sphere at $p=2$
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• On the periodic $v_2$-self-map of $A_1$
• The telescope conjecture at height 2 and the tmf resolution
• Some extensions in the Adams spectral sequence and the 51-stem
• Towards the $K(2)$-local homotopy groups of $Z$

Cited by in corpus (2)

• Bordism for the 2-group symmetries of the heterotic and CHL strings
• The localization of orthogonal calculus with respect to homology