Lifting of a continuous map through a covering map

From Maths
Jump to: navigation, search
Stub grade: A**
This page is a stub
This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
Very important page, make sure that it is done with the utmost generality!


[ilmath]\xymatrix{ & & E \ar[d]_p \\ Y \ar[rr]^f \ar@{-->}[rru]^\varphi & & X }[/ilmath]
If [ilmath]\varphi[/ilmath] exists such that all maps are continuous and the diagram commutes then [ilmath]\varphi[/ilmath] is a lifting of [ilmath]f[/ilmath] through [ilmath]p[/ilmath]

Let [ilmath](X,\mathcal{ J })[/ilmath] be a topological space, let [ilmath](E,\mathcal{ H })[/ilmath] be a covering space of [ilmath]X[/ilmath], with covering map [ilmath]p:E\rightarrow X[/ilmath]. Then[1][2]:

  • if we're given a continuous map [ilmath]f:Y\rightarrow X[/ilmath] for an arbitrary topological space [ilmath](Y,\mathcal{ K })[/ilmath] such that:
    • there exists a continuous map, [ilmath]\varphi:Y\rightarrow E[/ilmath], such that [ilmath]p\circ\varphi\eq f[/ilmath] (the diagram on the right commutes)
      • then [ilmath]\varphi[/ilmath] is called a lifting of [ilmath]f[/ilmath] (through [ilmath]p[/ilmath])

Caveat:I am not sure if we require [ilmath]Y[/ilmath] be a connected topological space or not[Note 1] - however if we do then the unique lifting property applies.

See next


  1. Author notes for future use:


  1. Introduction to Topological Manifolds - John M. Lee
  2. Introduction to Topology - Theodore W. Gamelin & Robert Everist Greene