J. Mukai
Self-homotopy of a suspension of the real 4-projective space

Abstract: We denote by $\Sigma X$ the reduced suspension of a space X. We denote by ${\bf P}^n$ the real n-dimensional projective space and by ${\bf P}^n_m = {\bf P}^n/{\bf P}^{m-1}$ for $n\geq m$ the stunted real projective space. We set ${\bf Z}_n = {\bf Z}/n{\bf Z}$ and denote by $({\bf Z}_m)^r\oplus({\bf Z}_n)^s$

$$\underbrace{{\bf Z}_m\oplus\cdots\oplus{\bf Z}_m}_r \oplus\underbrace{{\bf Z}_n\oplus\cdots\oplus{\bf Z}_n}_s.$$

The first purpose of this note is to determine the homotopy groups $\pi_i(\Sigma{\bf P}^3)$ for $i\leq 7$.

Theorem 1  

(i)

$\pi_3(\Sigma{\bf P}^3)\cong{\bf Z}_2, \pi_4(\Sigma{\bf P}^3) \cong{\bf Z}\oplus{\bf Z}_4$ and $\pi_5(\Sigma{\bf P}^3)\cong({\bf Z}_2)^5$.

(ii)

$\pi_6(\Sigma{\bf P}^3)\cong({\bf Z}_2)^7$ and $\pi_7(\Sigma{\bf P}^3) \cong{\bf Z}\oplus{\bf Z}_8\oplus ({\bf Z}_4)^5\oplus ({\bf Z}_2)^2\oplus{\bf Z}_3$.

The second purpose of this paper is to determine the group of the homotopy set $[\Sigma{\bf P}^4, \Sigma{\bf P}^n]$ for $n\geq 1$.

Theorem 2   $[\Sigma{\bf P}^4, \Sigma{\bf P}^n]$ for $n\geq 1$ is abelian and its group structure is given by the following.

(i)

$[\Sigma{\bf P}^4, S^2]\cong{\bf Z}_2, [\Sigma{\bf P}^4, \Sigma{\bf P}^2] \cong{\bf Z}_4\oplus{\bf Z}_2$ and $[\Sigma{\bf P}^4, \Sigma{\bf P}^3] \cong{\bf Z}_4\oplus ({\bf Z}_2)^2$.

(ii)

$[\Sigma{\bf P}^4, \Sigma{\bf P}^n]\cong{\bf Z}_8\oplus ({\bf Z}_2)^2$ for $n\geq 4$.

Our method is to use the quasi-fibration $h: \Sigma{\bf P}^3\wedge{\bf P}^3\rightarrow\Sigma{\bf P}^3$ induced from the Hopf construction of the multiplication of the Hopf space ${\bf P}^3$. In fact we use the following direct sum decomposition

$$\pi_n(\Sigma{\bf P}^3) = h_*\pi_n(\Sigma{\bf P}^3\wedge{\bf P}^3) \oplus\Sigma\pi_{n-1}({\bf P}^3).$$

We denote by $M^n = \Sigma^{n-2}{\bf P}^2$ the Moore space of type $({\bf Z}_2, n-1)$. Let $\nu'$ be a generator of the 2-primary component of $\pi_6(S^3)\cong{\bf Z}_{12}$. To show the first result, we need a cellular decomposition of $\Sigma{\bf P}^3\wedge{\bf P}^3$.

Lemma 0   $\Sigma{\bf P}^3\wedge{\bf P}^3 = ((\Sigma M^2\wedge M^2)\cup_{i\nu'}e^7) \vee M^6\vee M^6,$ where $i: S^3\hookrightarrow\Sigma M^2\wedge M^2$ is the inclusion.

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