XXXIX Workshop on Geometric Methods in Physics | 27.06-3.07.2021 |

IX School on Geometry and Physics | 21-25.06.2021 |

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## Janusz Grabowski## $\mathbb{Z}_2^n$-supermanifoldsThe concept of a $\mathbb{Z}_2^n$-supermanifold will be presented with its local model made of formal power series with $\mathbb{Z}_2^n$-gradation and $\mathbb{Z}_2^n$-commutation rules. The classical Batchelor-Gawedzki theorem says that any smooth supermanifold is (non-canonocally) diffeomorphic to the `superization' $\Pi E$ of a vector bundle $E$. It is also known that this result fails in the complex analytic category. We show that any smooth $\mathbb{Z}_2^n$-supermanifold is (non-canonocally) diffeomorphic to the `superization' $\Pi E$ of an $n$-fold vector bundle $E$. The latter can be chosen split. |

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University of Bialystok |

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