How to check whether a lifting between given mappings is possible?Suppose mappings M → P 1 , M → P 2 ,where M is some manifold, while P 1 , 2 are in general different spaces. I want to clarify if there exists a lifting between these two mappings.Suppose M = S n , n &gt; 1. Then have I to compare the homotopy groups π i ( P 1 ) with π i ( P 2 ) , i = 1 , . . . , n ,or only π n ( P 1 ) with π n ( P 2 ) ?

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How to check whether a lifting between given mappings is possible?Suppose mappings  M  →      P          1        ,    M  →      P          2        ,where M is some manifold, while       P          1      ,      2       are in general different spaces. I want to clarify if there exists a lifting between these two mappings.Suppose   M  =      S          n        ,  n  &amp;gt;  1. Then have I to compare the homotopy groups      π          i        (      P          1        )        with            π          i        (      P          2        )  ,    i  =  1  ,  .  .  .  ,  n  ,or only      π          n        (      P          1        )        with            π          n        (      P          2        )  ?

London Maldonado · Accepted Answer

Step 1I assume you are looking for a morphism   h  :      P    1    →      P    2   such that   h  ∘  f  =  g, where   f  :  M  →      P    1   denote your morphisms. Since every object is fibrant with respect to the Quillen model structure on the category Top of topological spaces and continuous maps, it is enough to check that f is an acyclic cofibration.To answer your question, it means that in particular (but it is not sufficient) you have to check that f is a weak homotopy equivalence, meaning that for all   x  ∈  M and for all   n  ∈      N  ,      f          ∗        :      π    n    (  M  ,  x  )  →      π    n    (      P    1    ,  f  (  x  )  )is an isomorphism.Step 2Note that if you assume your manifold M to be       S    n   for some n, then (at least for   n  ≥  2) the homotopy groups       π    i    (      S    n    ,  x  ) are not zero in general, hence you have to compare possibly non-trivial homotopy groups       π    i    (      S    n    ,  x  )  ,        π    i    (      P    1    ,  f  (  x  )  ) for all   x  ∈      S    n   and for all   i  ∈      N  .Fortunately, there is a shortcut. Indeed, as I mentioned it is enough to check that f is an acyclic cofibration with respect to the Quillen model structure, meaning that f is a retract of a transfinite composition of pushouts of morphisms among the set of inclusions   {      D    i    →      D    i    ×  I        |      i  ≥  0  } (the inclusions that map x to (x,0)). In particular, this is the case if your space       P    1   is obtained from M by glueing (possibly infinitely many) cylinders along disks, and f is the inclusion from M to       P    1  .

How to check whether a lifting between given mappings is possible? Suppose mappings M rightarrow P_1, M rightarrow P_2 where M is some manifold, while P_{1,2} are in general different spaces. I want to clarify if there exists a lifting between these two mappings.

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