Double of a manifold
= Double of a manifold
{title2=$D(W)=W\cup_{\partial W}W$}
Glue two copies of a compact <manifold> along their boundaries to form its double. A collar gives the double a smooth structure when $W$ is smooth. Folding the copies onto $W$ gives a <retraction>; in particular, inclusion of either copy is injective on <homology>. For an oriented <manifold>, use the opposite <orientation> on the second copy.