Boundary parallel

<h2 id="an-example">An example</h2>
<p>Consider the <a href="/facts/Annulus_(mathematics)/rhxKuoem">annulus</a> 
  
    
      
        I
        ×
        
          S
          
            1
          
        
      
    
    {\displaystyle I\times S^{1}}
  
. Let π denote the projection map
</p>

π
        :
        I
        ×
        
          S
          
            1
          
        
        →
        
          S
          
            1
          
        
        ,
        
        (
        x
        ,
        z
        )
        ↦
        z
        .
      
    
    {\displaystyle \pi \colon I\times S^{1}\rightarrow S^{1},\quad (x,z)\mapsto z.}

<p>If a circle <i>S</i> is embedded into the annulus so that π <a href="/facts/Restriction_(mathematics)/JV59DhKy">restricted</a> to <i>S</i> is a <a href="/facts/Bijection/ayC7YE3z">bijection</a>, then <i>S</i> is boundary parallel. (The <a href="/facts/Converse_(logic)/VhlUrxxx">converse</a> is not true.)
</p><p>If, on the other hand, a circle <i>S</i> is embedded into the annulus so that π restricted to <i>S</i> is not <a href="/facts/Surjection/KezhLpCb">surjective</a>, then <i>S</i> is not boundary parallel. (Again, the converse is not true.)
</p>

<h2 id="context-and-applications">Context and applications</h2>

<h2 id="further-reading">Further reading</h2>
<ul><li><a href="/facts/Marc_Culler/ISsVlsa7">Culler, Marc</a>, and <a href="/facts/Peter_Shalen/zHvUVWXm">Peter B. Shalen</a>. "Bounded, separating, incompressible surfaces in knot manifolds." <i><a href="/facts/Inventiones_mathematicae/bARcL95Y">Inventiones mathematicae</a></i> 75 (1984): 537–545.</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Atoroidal/srUhgk57">Atoroidal</a></li>
<li><a href="/facts/Satellite_knot/Apee9e4L">Satellite knot</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Definition 3.4.7 in Schultens, Jennifer (2014). Introduction to 3-manifolds. Graduate studies in mathematics. Vol. 151. AMS. ISBN 978-1-4704-1020-9. <a href="978-1-4704-1020-9" target="_blank">978-1-4704-1020-9</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Boundary parallel open-in-new

Boundary parallel