On the Wu classes of the topological blow up

TL;DR

This paper investigates Wu classes and Steenrod operations in the context of topological blow ups, providing formulas and applications to geometric obstructions.

Contribution

It derives explicit formulas for Wu classes and Steenrod operations of topological blow ups, advancing understanding of their algebraic and geometric properties.

Findings

Formulas for Wu classes of blow ups

Formulas for Steenrod operations of blow ups

Application to geometric obstructions

Abstract

We mainly discuss the Wu classes $v(M)$ and the Steenrod operation $Sq$ of the topological blow up $\tilde{M}$. The formula of the Wu class $v(\tilde{M})$ will be given as well as the formula of the Steenrod operation $Sq$. As an application, we will use our results to describe a geometric obstruction.