= Homology of the complement of a smooth conic
{title2=$H_0=\mathbb Z,\quad H_1=\mathbb Z/2,\quad H_{k\geq2}=0$}
The complement of a closed <tubular neighborhood> of a <smooth plane conic> has the displayed integral <homology>. The <Excision theorem> and <Thom isomorphism theorem> reduce the calculation to the relative long exact sequence: the ambient degree-four <fundamental class> restricts with coefficient one, while the degree-two map is intersection with the conic and has coefficient two. A <collar neighborhood> identifies the open and compact exterior <homotopy equivalence> types. The boundary instead has first <homology> $\mathbb Z/4$, from the <Gysin sequence> and the normal <Euler class> four.
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