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Higher chromatic Thom spectra via unstable homotopy theory | | Sanath K Devalapurkar
; | | Date: |
19 Apr 2020 | | Abstract: | We investigate implications of an old conjecture in unstable homotopy theory
related to the Cohen-Moore-Neisendorfer theorem and a conjecture about the
$mathbf{E}_{2}$-topological Hochschild cohomology of certain Thom spectra
(denoted $A$, $B$, and $T(n)$) related to Ravenel’s $X(p^n)$. We show that
these conjectures imply that the orientations $mathrm{MSpin} o mathrm{ko}$
and $mathrm{MString} o mathrm{tmf}$ admit spectrum-level splittings. This is
shown by generalizing a theorem of Hopkins and Mahowald, which constructs
$mathrm{H}mathbf{F}_p$ as a Thom spectrum, to construct
$mathrm{BP}langle{n-1}
angle$, $mathrm{ko}$, and $mathrm{tmf}$ as Thom
spectra (albeit over $T(n)$, $A$, and $B$ respectively, and not over the
sphere). This interpretation of $mathrm{BP}langle{n-1}
angle$,
$mathrm{ko}$, and $mathrm{tmf}$ offers a new perspective on Wood equivalences
of the form $mathrm{bo} wedge Ceta simeq mathrm{bu}$: they are related to
the existence of certain EHP sequences in unstable homotopy theory. This
construction of $mathrm{BP}langle{n-1}
angle$ also provides a different lens
on the nilpotence theorem. Finally, we prove a $C_2$-equivariant analogue of
our construction, describing $underline{mathrm{H}mathbf{Z}}$ as a Thom
spectrum. | | Source: | arXiv, 2004.8951 | | Services: | Forum | Review | PDF | Favorites | |
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