Let $A$ be two distinct points in $R^3$. How would I go about showing that $R^3\backslash A$ is homotopy equivalent to the one-point union $S^2\vee S^2$?
Any help appreciated
Let $A$ be two distinct points in $R^3$. How would I go about showing that $R^3\backslash A$ is homotopy equivalent to the one-point union $S^2\vee S^2$?
Any help appreciated
You probably want an explicit formula. Suppose the points are $(-1,0,0)$ and $(1,0,0)$; take the two spheres with center at these points, with radius $1$. First consider a deformation retraction of the interiors of the balls onto the spheres: just move along the radii (by multiplying the distance from the sphere by $1-t$, $t\in[0,1]$). Then you can deformation retract the exterior onto the two spheres and the axis $x$: move along lines orthogonal to the $x$ axis (again by multiplying by $1-t$ the distance along the line). Finally, deformation retract the two pieces of the $x$ axis. From this you can get a formula for a deformation retraction.