First, deformation retract $\mathbb{R}^3$ minus $L$ to $S^2$ minus $2n$ points (you can do this since you've removed the origin). Stereographically project from one of the punctures, and you've got $\mathbb{R}^2$ minus $2n-1$ points. Choose a point away from one of the punctures and draw disjoint based loops around each of the remaining holes. Now deformation retract to those loops and you've got a wedge of 2n-1 $2n-1$ circles.
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First deformation retract $\mathbb{R}^3$ minus $L$ to $S^2$ minus $2n$ points (since you've removed the origin). Stereographically project from one of the punctures, and you've got $\mathbb{R}^2$ minus $2n-1$ points. Choose a point away from one of the punctures and draw disjoint based loops around each of the remaining holes. Now deformation retract to those loops and you've got a wedge of 2n-1 circles. |
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