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\parbox[b]{3.5in}{
  {\LARGE\bf Cheat Sheet 3D Topology}
  \newline \footnotesize Material from Hatcher's notes and from Hempel's book.
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  \null\hfill\sheeturl
  \newline\null\hfill initiated 31/8/13; modified \today, \ampmtime; continued \href{http://drorbn.net/AcademicPensieve/2013-09/}{2013-09}
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{\bf Theorem} (Alexander, 1920s). An embedded 2-sphere in $\bbR^3$ bounds a 3-ball.

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{\bf Dehn's Lemma} (Dehn 1910 (wrong), Papakyriakopoulos 1950s). $M$ a 3-manifold, $f\colon B^2\to M$ s.t.\ for some neighborhood $A$ of $\partial B^2$ in $B^2$ the restriction $F\mid_A$ is an embedding and $f^{-1}(f(A))=A$. Then $f\mid_{\partial B^2}$ extends to an embedding $g\colon B^2\to M$.

{\bf The Loop Theorem} (Stallings 1960, implies Dehn's lemma). $M$ a 3-manifold, $F$ a connected 2-manifold in $\partial M$, $\ker(\pi_1(F)\to\pi_1(M)\not\subset N\triangleleft\pi_1(F)$. Then there is a proper embedding $g\colon(B^2,\partial B^2)\to(M,F)$ s.t.\ $[g\mid_{\partial B^2}]\not\in N$.

{\bf The Sphere Theorem.} $M$ orientable 3-manifold, $N$ a $\pi_1(M)$-invariant proper subgroup of $\pi_2(M)$. Then there is an embedding $g\colon S^2\to M$ s.t.\ $[g]\not\in N$.

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{\bf To do.}
$\bullet$ Schoenlies, JSJ.

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