---
date: 2022-04-05 23:42
modification date: Tuesday 5th April 2022 23:42:25
title: "A hat genus"
aliases: [A hat genus]
---

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# A hat genus

-  Notation: $\Ahat(M)$. Defined as the multiplicative sequence genus of the power series
$$
\frac{\sqrt{z} / 2}{\sinh (\sqrt{z} / 2)}=1-\frac{z}{24}+\frac{7 z^{2}}{5760}-\cdots
$$
- Genus: like a ring homomorphism $g: \K _0(X, \coprod, \times) \to (\ZZ, +, \cdot)$ for $X$ a manifold with boundary up to [cobordism](cobordism.md), where $g(X) = 0 \iff X = \bd X'$.
-  In $\ZZ$ for [spin](spin.md) manifolds, and is *even* if additionally $\dim_\RR M = 4 \mod 8$
-  The [Aatiyah-Singer Index Theorem](Aatiyah-Singer%20Index%20Theorem.md) implies that $\Ahat(M) = \Ind(\dirac)$ for spin manifolds