Commit a4ca0524 authored by Jannes Bantje's avatar Jannes Bantje

fix thom spectrum command

\M is already def’ed
parent 1071085d
Pipeline #59709 passed with stages
in 42 seconds
......@@ -209,11 +209,11 @@ The $\hat{A}$-genus can be generalised to the spin bordism ring, where a morphis
emerges.
In some degrees the group on the right is $\mathbb{Z}/2$ and in particular has torsion.
One could say, this $\alpha$-invariant remedies the need to tensor with $\mathbb{Q}$.
This map can be represented by a map of spectra $\M \Spin \to \KO$, which factors through the connective cover $\kO$ since $\M\Spin$ is connective.
This map can be represented by a map of spectra $\thoM \Spin \to \KO$, which factors through the connective cover $\kO$ since $\thoM\Spin$ is connective.
This generalisation process is illustrated in the following diagram
\[
\begin{tikzcd}
\pi_* \M\Spin \dar \rar & \pi_* \kO \dar \\
\pi_* \thoM\Spin \dar \rar & \pi_* \kO \dar \\
\Omega^{\Spin}_* \rar["\alpha"] \dar & \KO_*(\pt) \dar \\
\Omega^{\Spin}_* \otimes \mathbb{Q} \rar["\Ahat"] \dar & \mathbb{Z} \dar \\
\Omega^{\SO}_* \otimes \mathbb{Q} \rar["\Ahat"] & \mathbb{Q}
......@@ -223,7 +223,7 @@ In the end we are interested in the so-called Witten genus, which will be a genu
One now asks for a generalisation of the Witten genus in the same fashion as the generalisation of the \Ahat-genus.
The main observation is here, that the Witten genus maps string manifolds to \emph{topological modular forms}, \tmf.
By means of homotopy theory there already is a map of spectra $\sigma \colon \M\Strng \to \tmf$.
By means of homotopy theory there already is a map of spectra $\sigma \colon \thoM\Strng \to \tmf$.
The million dollar question is: Is there a geometrical description as for the \Ahat-genus?
Some of the \emph{conjectures} in this direction are:
......
......@@ -230,7 +230,7 @@
\newcommand{\CP}{\mathbb{C}\mathbb{P}}
\newcommand{\HP}{\mathbb{H}\mathbb{P}}
% \newcommand{\M}{\mathrm{M}\mkern-3mu}
\newcommand{\M}{M\mkern-3mu}
\newcommand{\thoM}{M\mkern-3mu}
\newcommand{\tmf}{\ensuremath{\mathrm{tmf}}}
\DeclareMathOperator{\Map}{Map}
\newcommand{\sa}{\mathrm{sa}}
......
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