Ehresmann's lemma

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, or specifically, in <a href="/facts/Differential_topology/Y21eGQe0">differential topology</a>, Ehresmann's lemma or Ehresmann's fibration theorem states that if a <a href="/facts/Smooth_mapping/mffL4ch2">smooth mapping</a> 
 
 
 
 f
 :
 M
 →
 N
 
 
 {\displaystyle f\colon M\rightarrow N}
 
, where 
 
 
 
 M
 
 
 {\displaystyle M}
 
 and 
 
 
 
 N
 
 
 {\displaystyle N}
 
 are <a href="/facts/Smooth_manifold/aSnbXKcV">smooth manifolds</a>, is

<ol><li>a surjective <a href="/facts/Submersion_(mathematics)/ARWB4sfr">submersion</a>, and</li>
<li>a <a href="/facts/Proper_map/jJS4YK7o">proper map</a> (in particular, this condition is always satisfied if M is <a href="/facts/Compact_space/d0cgXJH7">compact</a>),</li></ol>
then it is a <a href="/facts/Locally_trivial/SJ7Ek6cI">locally trivial</a> <a href="/facts/Fibration/h51Cx7xU">fibration</a>. This is a foundational result in <a href="/facts/Differential_topology/Y21eGQe0">differential topology</a> due to <a href="/facts/Charles_Ehresmann/KNtePSlk">Charles Ehresmann</a>, and has many variants.

Ehresmann's lemma open-in-new

Ehresmann's lemma