Supertransversality and $\Pi$-symmetric supermanifolds

The main objective of this article is to extend the concept of transversality to supergeometry. Transversality has two important properties in the classical case, namely " stability" and " genericity", which we show in the following that in the category of smooth supermanifolds, supertransversality has stable property. By extending Sard's theorem to supergeometry, genericity property is proved. In the final section, we examine transversality in the category of $\Pi$-symmetric supermanifolds. The theory presented here is a step towards an extension of the concept of Euler-Poincar\'e characteristic to supermanifolds.