\documentstyle[12pt]{amsart} \textheight=574pt \textwidth=432pt \oddsidemargin=18.88pt \evensidemargin=18.88pt \topmargin=14.21pt \makeatletter %restore definition changed by amsart \def\cases#1{\left\{\,\vcenter{\normalbaselines\m@th \ialign{$##\hfil$&\quad##\hfil\crcr#1\crcr}}\right.} \makeatother \let\<\langle \let\>\rangle \author{Jie Wu} \address{MSRI, 1000 Centennial Drive, Berkeley, CA 94720, USA} \email{jwu@@msri.org} \title{A Product Decomposition of $\Omega^3_0\Sigma\bold RP^2$} \thanks{Research at MSRI is supported in part by NSF grant DMS-9022140.} \newtheorem{theorem}{Theorem}[section] \newtheorem{definition}[theorem]{Definition} \newtheorem{definitions}[theorem]{Definitions} \newtheorem{notation}[theorem]{Notation} \newtheorem{notations}[theorem]{Notations} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{example}[theorem]{Example} \newtheorem{examples}[theorem]{Examples} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{remarks}[theorem]{Remarks} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{sublemma}[theorem]{Sublemma} \newtheorem{otherlemma}[theorem]{Lemma} \begin{document} \maketitle \begin{abstract} We give a specific product decomposition of the base-point path connected component of the triple loop space of the suspension of the projective plane. \end{abstract} \section{Introduction} In this paper, we give a specific product decompositions of the base-point path connected component of the triple loop space of the suspension of the projective plane, which is denoted by $\Omega^3_0\Sigma\bold RP^2$. Let $\bold RP_a^b=\bold RP^b/\bold RP^{a-1}$, let $X$ be the $n$-connected covering of the space $X$ and let $P^n(2)=\Sigma^{n-2}\bold RP^2$, the $n$-dimensional mod $2$ Moore space. In particular, $P^3(2)=\Sigma\bold RP^2$. Our theorem is as follows. \begin{theorem} There is a homotopy equivalence $$ \Omega^3_0P^3(2)\simeq\Omega^2(S^3\<3\>)\times\Omega^3_0(P^6(2)\vee\Sigma\bold RP^4_2) $$ localized at $2$. \end{theorem} This theorem was first given in my thesis where we only gave a product decomposition of $\Omega^4_0(\Sigma\bold RP^2)$. The author would like to thank his thesis advisor Fred Cohen for pointing out that our original decomposition for $\Omega^4_0(\Sigma\bold RP^2)$ can be delooping. He showed that the power map $$ 4:\Omega^2(S^3\<3\>)\rightarrow\Omega^2(S^3\<3\>) $$ is null localized at $2$, his proof will be used in this paper. The application of this theorem to calculate the homotopy groups $\pi_r(P^3(2))$ for $r\leq 11$ was given in my thesis [Wu]. An independent calculation for $\pi_*(P^3(2))$ can be found out in [Mu]. The homotopy groups of $\Sigma\bold RP^2$ may be useful in low dimensional topology. \section{Proof of the Theorem 1.1} In this section, all spaces are localized at $2$ and $H_*(X)$ means the homology with coefficients in field $\bold F_2$, the field with two elements. Let $j:P^3(2)\rightarrow BSO(3)$ be the inclusion of the $3$-skeleton of $BSO(3)$ and let $X_2$ be the pull back in the diagram $$ \hspace{1.5in} \begin{array}{cccccc} & BSO(2) &\rightarrow & BSO(3) \\ & \uparrow & & \uparrow \\ & X_2 &\rightarrow &P^3(2), \\ \end{array} $$ where the map $BSO(2)\rightarrow BSO(3)$ is induced by the canonical injection $SO(2)\rightarrow SO(3)$. Thus there is a fibration $$ S^2\rightarrow X_2 \rightarrow P^3(2). $$ Let $i:S^2\rightarrow X_2$ be the inclusion of the fibre. \begin{lemma} The map $\Omega^2(i):\Omega^2(S^2\<3\>)\rightarrow\Omega^2(X_2\<3\>)$ is null. \end{lemma} \noindent{\em Proof:} Consider the commutative diagram $$ \hspace{0.5in} \begin{array}{cccccccccccc} &\rightarrow&\pi_3(BSO(3)) &\stackrel{0}{\rightarrow} &\pi_2(S^2) &\hookrightarrow &\pi_2(BSO(2))&\rightarrow&\pi_2(BSO(3)) &\rightarrow0\\ & & \uparrow & & \Vert & &\uparrow & & \uparrow & \\ &\rightarrow&\pi_3(P^3(2))&\stackrel{0}{\rightarrow} &\pi_2(S^2) &\stackrel{i_*}{\hookrightarrow}&\pi_2(X_2) &\rightarrow &\pi_2(P^3(2))&\rightarrow0\\ \end{array} $$ where the top and bottom rows are long exact sequence of homotopy groups. Thus $\pi_1(X_2)=0$, $\pi_2(X_2)\cong\pi_2(BSO(2))=\pi_2(\bold CP^{\infty})\cong\bold Z$ and $i_*:\pi_2(S^2)\rightarrow\pi_2(X_2)$ is of degree $2$. Let $\phi:S^2\rightarrow X_2$ be a representive of the generator in $\pi_2(X_2)$. Then there is a homotopy commutative diagram $$ \hspace{1.5in} \begin{array}{cccccc} & S^2 & \stackrel{i}{\rightarrow} & X_2 \\ & \Vert & & \uparrow \phi \\ & S^2&\stackrel{[2]}{\rightarrow} & S^2, \\ \end{array} $$ where $[2]:S^2\rightarrow S^2$ is of degree $2$. Now consider the homotopy commutative diagram $$ \hspace{1.5in} \begin{array}{cccccc} & \Omega S^2 & \stackrel{H_2}{\rightarrow} & \Omega S^3 \\ &\Omega([2])\downarrow & & \downarrow \Omega([4]) \\ &\Omega S^2&\stackrel{H_2}{\rightarrow} & \Omega S^3, \\ \end{array} $$ where $H_2$ is the second James-Hopf map. Notice that $S^3$ is an H-space. We have $$ \Omega([4])\simeq 4:\Omega S^3\rightarrow \Omega S^3, $$ where $4:\Omega S^3\rightarrow \Omega S^3$ is the power map of degree $4$. Consider the EHP sequence $S^3\rightarrow\Omega(S^3\<3\>)\stackrel{H_2}{\rightarrow}\Omega S^5$, the map $S^3\rightarrow\Omega(S^3\<3\>)$ is a representative of the nontrivial element $\eta$ in $\pi_4(S^3)\cong\bold Z/2$. Notice that the composite $$ S^3\stackrel{\eta}{\rightarrow}\Omega(S^3\<3\>)\stackrel{2}{\rightarrow}\Omega(S^3\<3\>) $$ is null. Thus the map $2\simeq\Omega(2):\Omega^2(S^3\<3\>)\rightarrow\Omega^2(S^3\<3\>)$ factors through $\Omega(H_2):\Omega^2(S^3\<3\>)\rightarrow\Omega^2S^5$ up to homotopy. By a result of Cohen [C1, or C2], the map $$ 2\cdot\Omega(H_2):\Omega^2(S^3\<3\>)\rightarrow\Omega^2S^5 $$ is null. Thus the power map $$ 4:\Omega^2(S^3\<3\>)\rightarrow\Omega^2(S^3\<3\>) $$ is null. Notice that the map $\Omega (H_2):\Omega^2(S^2\<3\>)\rightarrow \Omega^2(S^3\<3\>)$ is a homotopy equivalence. Thus $\Omega^2([2]):\Omega^2(S^2\<3\>)\rightarrow \Omega^2(S^2\<3\>)$ is null and so the boundary map $\Omega^2(i):\Omega^2(S^2\<3\>)\rightarrow \Omega^2(X_2\<3\>)$ is null, which is the assertion. \begin{corollary} There is a homotopy equivalence $$ \Omega^3_0P^3(2)\simeq\Omega^2(S^3\<3\>)\times\Omega^3_0(X_2). $$ \end{corollary} \begin{lemma} The (mod $2$) homology of $X_2\<2\>$ is as follows: $$ {\bar H}_r(X_2\<2\>)=\cases{\bold Z/2&r=3,4,6\cr \bold Z/2\oplus\bold Z/2&r=5\cr 0&otherwise.} $$ \end{lemma} \noindent{\em Proof:} Consider the homotopy commutative of fibre sequences $$ \hspace{0.5in} \begin{array}{cccccccccc} & S^2 & \rightarrow&BSO(2)=\bold CP^{\infty}&\rightarrow &BSO(3) \\ &\Vert& &\uparrow & & \uparrow \\ &S^2 &\rightarrow &X_2 &\rightarrow & P^3(2) \\ &\uparrow& &\uparrow & &\uparrow\\ &* &\rightarrow &X_2\<2\> &\stackrel{=}{\rightarrow}&X_2\<2\>\\ \end{array} $$ Thus the homotopy fibre of $X_2\<2\>\rightarrow P^3(2)$ is $SO(3)$. The assertion follows by inspecting the Serre spectral sequence for the principle fibre sequence $$ SO(3)\rightarrow X_2\<2\>\rightarrow P^3(2). $$ \par \par Recall that$ H_*(\Omega P^3(2))\cong T(u,v)$ as algebras with $dim(v)=2$ and $Sq^1_*v=u$, where $T(u,v)$ is the tensor algebra generated by $u$ and $v$. \begin{lemma} $H_*(\Omega (X_2\<2\>))\cong T(V)$ as subalgebras of $H_*(\Omega P^3(2))$, where $V$ has a basis $$ \{u^2,[u,v],v^2+u[u,v],[u^2,v],[v^2,u]+u[u^2,v]\} $$ in $T(u,v)\cong H_*(\Omega P^3(2))$ and $[x,y]$ is the commutator of $x$ and $y$. \end{lemma} \noindent{\em Proof:} Notice that $\Omega j_*:H_*(\Omega P^3(2))\rightarrow H_*(SO(3))$ is an epimorphism and $$\pi_1(\Omega P^3(2))\cong\pi_1(SO(3)).$$ Thus the Serre spectral sequence for the fibre sequence $$ \hspace{0.5in} \begin{array}{cccccc} &\Omega (X_2\<2\>) & \stackrel{\Omega i}{\rightarrow} &\Omega P^3(2)&\stackrel{\Omega j}{\rightarrow} & SO(3) \\ \end{array} $$ collapses and therefore there is a short exact sequence of Hopf algebras $$ 1\rightarrow H_*(\Omega (X_2\<2\>))\rightarrow H_*(\Omega P^3(2))\rightarrow H_*(SO(3))\rightarrow1. $$ Thus $H_*(\Omega (X_2\<2\>))$ is the cotensor product of $\bold F_2$ and $H_*(\Omega P^3(2))$ over $H_*(SO(3))$ ( see [MM] ). \par We first show that $V\subseteq H_*(\Omega (X_2\<2\>))$. Let $\psi$ be the comultiplication in the Hopf algebra $H_*(\Omega P^3(2))\cong T(u,v)$. Then $\psi$ is determined by the generators. $$ \psi(u)=u\otimes1+1\otimes u $$ and $$ \psi(v)=v\otimes1+1\otimes v+u\otimes u. $$ The comultiplication $\psi$ on the basis for $V$ is as follows: $$ \psi(u^2)=u^2\otimes 1+1\otimes u^2, $$ $$ \psi([u,v])=[u,v]\otimes1+1\otimes[u,v], $$ $$ \psi(v^2+u[u,v])=(v^2+u[u,v])\otimes1+1\otimes(v^2+u[u,v])+u^2\otimes u^2, $$ $$ \psi([u^2,v])=[u^2,v]\otimes1+1\otimes[u^2,v], $$ and $$ \psi([u,v^2]+u[u^2,v])=([u,v^2]+u[u^2,v])\otimes1+1\otimes([u,v^2]+u[u^2,v]). $$ \par Notice that $H_*(SO(3))\cong E(u,v)$, the exterior algebra generated by $u$ and $v$. Thus $V\subseteq H_*(\Omega (X_2\<2\>))$. Let $\phi:T(V)\rightarrow H_*(\Omega (X_2\<2\>))$ be the algebra map induced by the inclusion $V\subseteq H_*(\Omega (X_2\<2\>))$ and let $A=\Omega i_*\circ\phi(T(V))$. Then $A$ is the subalgebra of $H_*(\Omega P^3(2))$ generated by $V$, which is also a tensor algebra. Notice that $V\rightarrow Q(A)$ is an isomorphism. Thus $A\cong T(V)$ ( see [CMN, Proposition 3.9]) and so $\phi:T(V)\rightarrow H_*(\Omega (X_2\<2\>))$ is a monomorphism. Notice that Poincare series $$ \chi(H_*(\Omega (X_2\<2\>)))=\chi(H_*(\Omega P^3(2)))/\chi(H_*(SO(3))) $$ $$ =(1-t-t^2)^{-1}(1+t+t^2+t^3)^{-1}=(1-t^2-t^3-2t^4-t^5)^{-1}=\chi(T(V)). $$ Thus $\phi:T(V)\rightarrow H_*(\Omega (X_2\<2\>))$ is an isomorphism, which is the assertion. \begin{lemma} The homology suspension $\sigma_*:Q{\bar H}_*(\Omega (X_2\<2\>))\rightarrow {\bar H}_{*+1}(X_2\<2\>)$ is an isomorphism. \end{lemma} \noindent{\em Proof:} Let $x_2,x_3,x_4,y_4$ and $y_5$ be in $H_*(\Omega(X_2\<2\>))$ corresponding to $u^2,[u,v],v^2+u[u,v],[u^2,v]$ and $[v^2,u]+u[u^2,v]$ in $T(u,v)$, respectively. Consider mod $2$ Serre spectral sequence $E^r_{*,*}$ for the principle fibre sequence $$ \Omega (X_2\<2\>)\rightarrow *\rightarrow X_2\<2\>. $$ Notice that, for $*\leq5$, ${\bar H}_*(\Omega (X_2\<2\>))$ has a basis $$ x_2,x_3,x_2^2,y_4,x_4,[x_2,x_3],x_2x_3,y_5. $$ Let $a_{j+1}=\sigma_*(x_j)$ and $b_{j+1}=\sigma_*(y_j)$. Then $$ d_3(a_3)=x_2, $$ $$ d_3(a_3x_2)=x_2^2, $$ $$ d_3(a_3x_3)=x_2x_3, $$ $$ d_4(a_4)=x_3, $$ $$ d_4(a_4x_2)=x_3x_2, $$ and $$ E^r_{s,t}=0 $$ for $r\geq 6$,$s>0$,$t>0$ and $s+t\leq 6$. Thus there exists $a_5,b_5,b_6\in E^2_{*,0}$ such that $d_5(a_5)=x_4$,$d_5(b_5)=y_4$ and $d_6(b_6)=y_5$. The assertion follows. \begin{lemma} The element $$x_3=[u,v]$$ in $H_3(\Omega (X_2\<2\>))$ is spherical. \end{lemma} \noindent{\em Proof:} Consider the exact sequence of homotopy groups $$ \pi_4(P^3(2))\rightarrow\pi_3(SO(3))\rightarrow\pi_3(X_2\<2\>)\rightarrow\pi_3(P^3(2))\rightarrow\pi_2(SO(3))=0. $$ Notice that $\pi_3(SO(3))=\bold Z$ and $\pi_4(P^3(2))$ is torsion. Thus $\pi_3(X_2\<2\>)=\bold Z$ or $\bold Z\oplus G$ for some $2$-torsion abelian group $G$. Now $X_2\<2\>$ is 2-connected and $\bold F_2=H_3(X_2\<2\>;\bold F_2)\cong \pi_3(X_2\<2\>)\otimes\bold F_2$. Thus $\pi_3(X_2\<2\>)=\bold Z$ and so $\pi_2(\Omega (X_2\<2\>))=\bold Z$. Now the $3$-skeleton $(\Omega (X_2\<2\>))_{(3)}$ of $\Omega (X_2\<2\>)$ is the homotopy cofibre of an attaching map $\phi:S^2\rightarrow S^2$. Notice that $$ H_2((\Omega (X_2\<2\>))_{(3)};\bold Z)\cong H_2(\Omega (X_2\<2\>);\bold Z)=\bold Z. $$ Thus $\phi$ is of degree $0$ and so the $3$-skeleton $(\Omega (X_2\<2\>))_{(3)}\simeq S^2\vee S^3$. The assertion follows.\\ \\ \par \noindent{\em Proof of Theorem 1.1:} By Corollary 2.2, it suffices to show that $$ X_2\<2\>\simeq P^6(2)\vee\Sigma\bold RP^4_2. $$ We use the notation in Lemmas 2.4 and 2.5. By Lemma 2.4, Steenrod operations on ${\bar H}_*(X_2\<2\>)$ are given by $$ Sq^2_*a_5=a_3,Sq^1_*a_5=a_4,Sq^1_*b_6=b_5. $$ Thus $$ {\bar H}_*(X_2\<2\>)\cong M\oplus N $$ as modules over Steenrod algebra, where $M$ is generated by $a_3,a_4,a_5$ and $N$ is generated by $b_5,b_6$. By Lemma 2.6, the element $x_3$ is spherical in $H_3(\Omega (X_2\<2\>))$. Let $f:S^3\rightarrow \Omega (X_2\<2\>)$ so that $f_*(\iota_3)=x_3$, where $\iota_3$ is a generator for $H_3(S^3)$. Consider the relative Samelson product for the multiplicative fibre sequence $$ \Omega (X_2\<2\>)\rightarrow\Omega P^3(2)\rightarrow SO(3) $$ ( see [N1] ). There is a relative Samelson product $$ [f,E]:S^3\wedge\bold RP^2=P^5(2)\rightarrow\Omega (X_2\<2\>), $$ where $E:\bold RP^2\rightarrow \Omega P^3(2)$ is the suspension. Notice that both $\iota_3$ and $u$ are primitive. Then $$ [f,E]_*(\iota_3\otimes u)=[f_*(\iota_3),u]=[[u,v],u]=[u^2,v] $$ in $T(u,v)$. Let $\theta=\sigma\circ\Sigma[f,E]$ be the composite $P^6(2)\rightarrow\Sigma\Omega (X_2\<2\>)\stackrel{\sigma}{\rightarrow} X_2\<2\>$ and let $u_5$ and $v_6$ be the bases for $H_5(P^6(2))$ and $H_6(P^6(2))$, repectively. Then $\theta_*(u_5)=\sigma_*([u^2,v])=b_5$ and $Sq^1_*\theta_*(v_6)=b_5$. Thus $\theta_*(v_6)=b_6$ which is the only nonzero element in $H_6(X_2\<2\>)$ and so $$ \theta_*:H_*(P^6(2))\rightarrow H_*( X_2\<2\>) $$ is one to one. Let $B^5$ be the homotopy cofibre of $\theta:P^6(2)\rightarrow X_2\<2\>$. Then ${\bar H}_*(B^5)$ has a basis $\{a_3,a_4,a_5\}$. Notice that $a_4=\sigma_*(x_3)$ is spherical. Thus $B^5$ is the homotopy cofibre of an attaching map $\phi:S^4\rightarrow S^3\vee S^4=(X_2\<2\>)_{(4)}$. Notice that the homotopy fibre of the pinch map $X_2\<2\>\rightarrow B^5$ is $4$-connected and $\pi_4(X_2\<2\>)\cong\pi_4(B^5)$. Thus the composite $$ S^4\stackrel{\phi}{\rightarrow}S^4\vee S^3\rightarrow X_2\<2\> $$ is null, or the injection $S^4\vee S^3\rightarrow X_2\<2\>$ extends to a map $s:B^5\rightarrow X_2\<2\>$. Now the composite $$ P^6(2)\vee B^5\stackrel{\theta\vee s}{\rightarrow}X_2\<2\>\vee X_2\<2\>\stackrel{\bigtriangledown}{\rightarrow} X_2\<2\> $$ is a homotopy equivalence, where $\bigtriangledown$ is the fold map. Consider homotopy commutative diagram of cofibre sequences $$ \hspace{0.5in} \begin{array}{cccccccccc} & S^4 & \stackrel{\phi}{\rightarrow}&S^3\vee S^4 &\rightarrow& B^5 & \rightarrow &S^5 \\ &\Vert& &p\downarrow & & \downarrow & &\Vert \\ &S^4 &\stackrel{[k]}{\rightarrow} & S^4 &\rightarrow&P^5(k) &\rightarrow &S^5, \\ \end{array} $$ where $p$ is the projection. Notice that $Sq^1_*a_5=a_4$. Thus $P^5(k)\simeq P^5(2)$ localized at $2$ and $k=2k'$ with $k'\equiv 1(2)$. Notice that $Sq^2_*a_5=a_3$. Thus $B^5$ is not homotopy equivalent to $P^5(2)\vee S^3$ and therefore the composite $$ \hspace{0.5in} \begin{array}{cccccccccc} &S^4 & \stackrel{\phi}{\rightarrow}&S^3\vee S^4&\stackrel{p}{\rightarrow} &S^3 \\ \end{array} $$ is essential which is equal to $\eta$ in $\pi_4(S^3)=\bold Z/2$, where $p$ is the projection. Thus the homotopy class $[\phi]=\eta+2k'[\iota]$ in $\pi_4(S^3\vee S^4)\cong\bold Z/2\oplus\bold Z$ with generators $\eta$ and $[\iota]$, where $k'\equiv 1(2)$. Notice that $\Sigma\bold RP^4_2$ is the homotopy cofibre of $(\eta,[2]):S^4\rightarrow S^3\vee S^4$. Thus $B^5\simeq\Sigma\bold RP^4_2$ and the assertion follows. \begin{thebibliography}{Lellmannaa} \bibitem[C1]{C1} F. R. Cohen, {\em Two-Primary analogues of Selick's theorem and the Kahn-Priddy theorem for the $3$-sphere, } Topology, 23(1984),401-421. \bibitem[C2]{C2} F. R. Cohen, {\em A course in some aspects of classical homotopy theory, } S.L.M 1286(1986),1-92. \bibitem[CMN]{CMN} F. R. Cohen, J. Moore and J. Neisendorfer, {\em Torsion in homotopy groups,} Ann. of Math. 109(1979),121-168. \bibitem[Mu]{Mu} J. Mukai, {\em Some homotopy groups of $M^3$,} preprint. \bibitem[MM]{MM} J. Milnor and J. Moore, {\em On the structure of Hopf Algebras, } Ann. of Math. 81 (1965), 211-264. \bibitem[N1]{N1} J. Neisendorfer, {\em Primary homotopy theory}, Memoirs. AMS, 25 (1980). \bibitem[Wu]{Wu} J. Wu, Thesis, University of Rochester (1995). \end{thebibliography} \end{document}