1
$\begingroup$

Let $I=[-1,1],$ and $J=\partial I=\{-1,1\},$ and consider $A=(I\times I\times J)\cup(I\times J\times I)$ ($A$ is a cube without two parallel faces). The fundamental group $\pi_1(C)$ is $\mathbb{Z}$ since $A$ deformation retracts to the square $\{0\}\times I\times I.$ I want to calculate $\pi_1(\mathbb{R}^3-A).$ Since I usually have problem with these kind of problems, I chose this one to make some questions. Are there some usual techniques to solve these problems: in particular based on van Kampen or deformation retracts? Are there some links between $\pi_1(X)$ and $\pi_1(Y-X)?$

$\endgroup$
0
$\begingroup$

Let $S^2$ be a sphere of radius $2$ centred at the origin, and let $L$ be the straight line segment connecting the points $(2, 0, 0)$ and $(-2, 0, 0)$. The space $R^3 - A$ deformation retracts onto $S^2 \cup L$, whose fundamental group is $\mathbb Z$.

Hatcher, page 46, example 1.23 contains pictures of these kinds of deformation retractions for a number of examples, including $\mathbb R^3 - S^1$, which is very similar to the present example.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.