Talk:Kernel formation (Ex)

The lens space $L(m,n)$$\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}{^}L(m,n)$ is a union of two solid tori glued along the boundary. The gluing diffeomorphism $\psi \colon T^2 \to T^2$$\psi \colon T^2 \to T^2$ is such that $\psi_{\ast}\mu = m \lambda + n \mu$$\psi_{\ast}\mu = m \lambda + n \mu$, where $\lambda$$\lambda$ and $\mu$$\mu$ denote the longtitude and meridian of the solid torus. From this Heegard decomposition we obtain the kernel formation

$\displaystyle (H_{(-1)}(\mathbb{Z}); F,G).$ $$\displaystyle (H_{(-1)}(\mathbb{Z}); F,G).$$

The first lagrangian $F$$F$ is spanned by the generator of $H_2(S^1 \times D^2, S^1 \times S^1)$$H_2(S^1 \times D^2, S^1 \times S^1)$. This generator is exactly the meridian $\mu$$\mu$. The same argument shows, that the second lagrangian is generated by the image of the meridian of complementary torus $\psi_{\ast} \mu = m \lambda + n \mu$$\psi_{\ast} \mu = m \lambda + n \mu$. Certainly we have $F \cap G = \{0\}$$F \cap G = \{0\}$. Furthermore it is easy to see that the index of $F+G$$F+G$ in $\mathbb{Z} \times \mathbb{Z}$$\mathbb{Z} \times \mathbb{Z}$ is equal to $m$$m$. Thus the kernel formation is trivial if and only if $m=1$$m=1$. We have $L(1,n) = S^3$$L(1,n) = S^3$, thus we obtain trivial formations for $S^3$$S^3$:

$\displaystyle (H_{(-1)}(\mathbb{Z}),\mu \mathbb{Z}, (\lambda + n \mu) \mathbb{Z}).$ $$\displaystyle (H_{(-1)}(\mathbb{Z}),\mu \mathbb{Z}, (\lambda + n \mu) \mathbb{Z}).$$

For $\mathbb{R}P^3$$\mathbb{R}P^3$ it is sufficient to notice that $L(2,1) = \mathbb{R}P^3$$L(2,1) = \mathbb{R}P^3$. Thus the associated formation is:

$\displaystyle (H_{(-1)}(\mathbb{Z}); \mu \mathbb{Z}, (2 \lambda + \mu) \mathbb{Z}).$ $$\displaystyle (H_{(-1)}(\mathbb{Z}); \mu \mathbb{Z}, (2 \lambda + \mu) \mathbb{Z}).$$