Seeking 'simple' space with specified homotopy

Question

I am looking for a 'named' space $S$ such that $\pi_1(S) = \mathbb{Z}_2$ and $\pi_n(S) = \star$ (the one-point group) for all $n\geq 2$.

Commentary: I know that the projective plane fits the first requirement but not the second.

If a 'named' space is lacking, then a construction of such a space would be a second-best solution.

Answer 1

The space you're looking for is the $K(\mathbb Z_2,1)$, up to homotopy an example of such space is give by the $\mathbb P^\infty(\mathbb R)$ the infinite projective space.

This can be easily described as the CW-complex having one cell in every dimension. You can find about such space in Hatcher's Algebraic Topology.

The $\pi_1(\mathbb P^\infty(\mathbb R)) = \mathbb Z_2$ since it's obtained by the infinite sphere $S^\infty$ (which is contractible) by quotienting for the (properly discontinuous action of the antipodal map, so is a quotient of $S^\infty$ by the action of $\mathbb Z_2$.

Being $S^\infty$ a covering space of $\mathbb P^\infty(\mathbb R)$ it follows that all the $\pi_n(\mathbb P^\infty(\mathbb R))=\pi_n(S^\infty)=0$ for $n \geq 2$.

Answer 2

You are looking for an Eilenberg-Maclane space, $K(\mathbb{Z}_2,1)$, which is unique up to homotopy equivalence. A representative of this equivalence class is the infinite real projective space:

$$\mathbb{RP}^{\infty}=\bigcup_{n\ge0}\mathbb{RP}^n$$