Sandbox

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1 Images

$\mathscr{A}$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}\newcommand{\ZZZ}{\mathbb{Z}}\mathscr{A}$ $\mathscr{B}$$\mathscr{B}$

File:Foliation.png

3-dimensional Reeb foliation

2 Fonts

;

$a > b < c/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_ns4Eac$$a > b < c$ CM: The connected sum of $\RP^2/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_mgc3AA$$\RP^2$ with $T^2ZZZ/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_53jmnZ$$T^2ZZZ$ is homeomorphic to $\RP^2\mathbin{\sharp}\RP^2\mathbin{\sharp}\RP^2X/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_MDlswo$$\RP^2\mathbin{\sharp}\RP^2\mathbin{\sharp}\RP^2X$.

HV: The connected sum of $\RP^2/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_1166cO$$\RP^2$ with $T^2XXX/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_ydXTge$$T^2XXX$ is homeomorphic to $\RP^2\mathbin{\sharp}\RP^2\mathbin{\sharp}\RP^2Y/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_5xaNIE$$\RP^2\mathbin{\sharp}\RP^2\mathbin{\sharp}\RP^2Y$.

LM: The connected sum of $\RP^2/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_SRLzz5$$\RP^2$ with $T^2YYYY/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_9aZUNw$$T^2YYYY$ is homeomorphic to $\RP^2\mathbin{\sharp}\RP^2\mathbin{\sharp}\RP^2Z/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_WW6epY$$\RP^2\mathbin{\sharp}\RP^2\mathbin{\sharp}\RP^2Z$.

MF: The connected sum of $\RP^2/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_HckaUq$$\RP^2$ with $T^2VVVV/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_8e71MT$$T^2VVVV$ is homeomorphic to $\RP^2\mathbin{\sharp}\RP^2\mathbin{\sharp}\RP^2ZZ/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_LkU63m$$\RP^2\mathbin{\sharp}\RP^2\mathbin{\sharp}\RP^2ZZ$.

The connected sum of $\RP^2/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_Iu55JQ$$\RP^2$ with $T^2/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_TUlxNk$$T^2$ is homeomorphic to $\RP^2\mathbin{\sharp}\RP^2\mathbin{\sharp}\RP^2ZZZ/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_OpDWdP$$\RP^2\mathbin{\sharp}\RP^2\mathbin{\sharp}\RP^2ZZZ$.

3 Tests

[Ranicki1981]

Proof.

$\square$$\square$

frog

4 Section

4.1 Subsection

Refert to subsection 4.1

Theorem 4.1. test

Refer to theorem 4.1

5 Section

$\stackrel{CW^2}{\xrightarrow{\hspace*{0.75cm}}}$$\stackrel{CW^2}{\xrightarrow{\hspace*{0.75cm}}}$

An inter-Wiki link.

Another; inter-Wiki link.

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Template:Ref Poincare complexes are great:

$\displaystyle H_i(X) \cong H^{n-i}(X)$ $$\displaystyle H_i(X) \cong H^{n-i}(X)$$

$\displaystyle \oplus, \bigoplus$ $$\displaystyle \oplus, \bigoplus$$