Non-reducible Spivak Normal Fibrations (Ex)

Let $[\iota_2, \iota_3] \in \pi_4(S^2 \vee S^3)$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}[\iota_2, \iota_3] \in \pi_4(S^2 \vee S^3)$ be the Whitehead product of the inclusion of the two factors: $[\iota_2, \iota_3]$$[\iota_2, \iota_3]$ is the attaching map for the top cell of $S^2 \times S^3$$S^2 \times S^3$.

Let

$\displaystyle X \simeq S^2 \vee S^3 \cup_\phi E^5$ $$\displaystyle X \simeq S^2 \vee S^3 \cup_\phi E^5$$

be the space obtained by attaching a $5$$5$-cell as indicated where the map

$\displaystyle \phi \colon S^4 \to S^2 \vee S^3$ $$\displaystyle \phi \colon S^4 \to S^2 \vee S^3$$

is given by $\phi = [\iota_2, \iota_3] + \iota_2 \circ \eta^2_2$$\phi = [\iota_2, \iota_3] + \iota_2 \circ \eta^2_2$ and here $\iota_i \colon S^i \to S^2 \vee S^3$$\iota_i \colon S^i \to S^2 \vee S^3$ is the obvious inclusion, $[\iota_2, \iota_3]$$[\iota_2, \iota_3]$ is the Whitehead product and $\eta^2_2 \colon S^4 \to S^2$$\eta^2_2 \colon S^4 \to S^2$ is essential.

Exercise 0.1 c.f. [Madsen&Milgram1979, 2.5].

1. Show that $X$$X$ is a Poincaré complex.
2. Find a self-homotopy equivalence $f \colon S^2 \times S^2 \simeq S^2 \times S^2$$f \colon S^2 \times S^2 \simeq S^2 \times S^2$ such that there is a homotopy equivalence
   $\displaystyle X \simeq (S^2 \times D^3) \cup_f (S^2 \times D^3).$ $$\displaystyle X \simeq (S^2 \times D^3) \cup_f (S^2 \times D^3).$$
3. Show that the Spivak normal fibration of $X$$X$ has no vector bundle reduction.

References

• [Madsen&Milgram1979] I. Madsen and R. J. Milgram, The classifying spaces for surgery and cobordism of manifolds, Princeton University Press, Princeton, N.J., 1979. MR548575 (81b:57014) Zbl 0446.57002