Thom space ($\mathrm{Th}(\xi )=D(\xi )/S(\xi )$)

For a real vector bundle over a compact base, its Thom space is the quotient of its disk bundle by its sphere bundle, with the collapsed sphere bundle as basepoint. Equivalently it is the one-point compactification of the total space. An oriented bundle's Thom class and Thom isomorphism identify the reduced cohomology of this space with shifted cohomology of the base.

Decompose the ambient vector space as $V\oplus W$, of dimensions $n+1,m+1$. A projective line outside $\mathbb{P}(V)$ is the graph of a unique linear map from a line in $W$ to $V$. Thus the complement is the total space of ${\gamma }^{\ast }\otimes V$, which is complex-isomorphic to ${\bar{\gamma }}^{\oplus (n+1)}$ after choosing a Hermitian metric. Collapsing $\mathbb{P}(V)$ gives its one-point compactification, hence the Thom space because the base is compact.

// Target: geometry-and-topology.bigb