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Cappell-Shaneson knots are a special class of knots in homotopy 4-spheres. In a sense they were designed as an example of knots with a well-known but prescribed 2-type.

The definition is like this. Look at bundles over $S^1$ whose fiber is a once-punctured $(S^1)^3$. The monodromy is an element of $A \in GL_3(\mathbb Z)$. A non-trivial (but fun) exercise is to check this manifold $M = \left((S^1)^3\setminus{*}\right) \rtimes_A S^1$ is the complement of a smoothly embedded $S^2$ in a homotopy $4$-sphere if and only if $det(A) = \pm 1$.

$\pi_2 M$ has a single generator as a module over $\pi_1 M$, but it's far from a free module over $\pi_1 M$. Perhaps the best way to describe $\pi_2 M$ is that it's a Laurent polynomial ring $\mathbb Z[x^\pm,y^\pm,z^\pm]$ where the $x,y,z$ correspond to the generators of $\pi_1 ((S^1)^3 \setminus{*})$ i.e. this is $H_2$ of the universal cover, which has a natural identification with $(\mathbb R^3 \setminus \mathbb Z^3) \times \mathbb R$. The action of $\pi_1 M$ is just the action on this covering space, so you get not just multiplication by units $x^ay^bz^c$ but also the automorphisms of $\mathbb Z^3$ (coming from $A \in GL_3(\mathbb Z)) Z))$ acting on the exponents of the polynomials.

The homotopy 4-spheres that contain Cappell-Shaneson knots were once considered possible counter-examples to the smooth 4-dimensional Poincare conjecture. Recent work of Akbulut and Gompf seem to have largely removed this possibility.

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Cappell-Shaneson knots are a special class of knots in homotopy 4-spheres. In a sense they were designed as an example of knots with a well-known but prescribed 2-type.

The definition is like this. Look at bundles over $S^1$ whose fiber is a once-punctured $(S^1)^3$. The monodromy is an element of $A \in GL_3(\mathbb Z)$. A non-trivial (but fun) exercise is to check this manifold $M = \left((S^1)^3\setminus{*}\right) \rtimes_A S^1$ is the complement of a smoothly embedded $S^2$ in a homotopy $4$-sphere if and only if $det(A) = \pm 1$.

$\pi_2 M$ has a single generator as a module over $\pi_1 M$, but it's far from a free module over $\pi_1 M$. Perhaps the best way to describe $\pi_2 M$ is that it's a Laurent polynomial ring $\mathbb Z[x^\pm,y^\pm,z^\pm]$ where the $x,y,z$ correspond to the generators of $\pi_1 ((S^1)^3 \setminus{*})$ i.e. this is $H_2$ of the universal cover, which has a natural identification with $(\mathbb R^3 \setminus \mathbb Z^3) \times \mathbb R$. The action of $\pi_1 M$ is just the action on this covering space, so you get not just multiplication by units $x^ay^bz^c$ but also the automorphisms of $\mathbb Z^3$ (coming from \$A \in GL_3(\mathbb Z)) acting on the exponents of the polynomials.

The homotopy 4-spheres that contain Cappell-Shaneson knots were once considered possible counter-examples to the smooth 4-dimensional Poincare conjecture. Recent work of Akbulut and Gompf seem to have largely removed this possibility.