In
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
, a fiber-homotopy equivalence is a map over a space ''B'' that has homotopy inverse over ''B'' (that is we require a homotopy be a map over ''B'' for each time ''t''.) It is a relative analog of a
homotopy equivalence
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a defor ...
between spaces.
Given maps ''p'': ''D'' → ''B'', ''q'': ''E'' → ''B'', if ƒ: ''D'' → ''E'' is a fiber-homotopy equivalence, then for any ''b'' in ''B'' the restriction
:
is a homotopy equivalence. If ''p'', ''q'' are fibrations, this is always the case for homotopy equivalences by the next proposition.
Proof of the proposition
The following proof is based on the proof of Proposition in Ch. 6, § 5 of . We write
for a homotopy over ''B''.
We first note that it is enough to show that ƒ admits a left homotopy inverse over ''B''. Indeed, if
with ''g'' over ''B'', then ''g'' is in particular a homotopy equivalence. Thus, ''g'' also admits a left homotopy inverse ''h'' over ''B'' and then formally we have
; that is,
.
Now, since ƒ is a homotopy equivalence, it has a homotopy inverse ''g''. Since
, we have:
. Since ''p'' is a fibration, the homotopy
lifts to a homotopy from ''g'' to, say, ''g
''' that satisfies
. Thus, we can assume ''g'' is over ''B''. Then it suffices to show ''g''ƒ, which is now over ''B'', has a left homotopy inverse over ''B'' since that would imply that ƒ has such a left inverse.
Therefore, the proof reduces to the situation where ƒ: ''D'' → ''D'' is over ''B'' via ''p'' and
. Let
be a homotopy from ƒ to
. Then, since
and since ''p'' is a fibration, the homotopy
lifts to a homotopy
; explicitly, we have
. Note also
is over ''B''.
We show
is a left homotopy inverse of ƒ over ''B''. Let
be the homotopy given as the composition of homotopies
. Then we can find a homotopy ''K'' from the homotopy ''pJ'' to the constant homotopy
. Since ''p'' is a fibration, we can lift ''K'' to, say, ''L''. We can finish by going around the edge corresponding to ''J'':
:
References
* {{cite book , last=May , first=J. Peter , title=A concise course in algebraic topology , publisher=University of Chicago Press , series=Chicago Lectures in Mathematics , publication-place=Chicago , date=1999 , isbn=0-226-51182-0 , oclc=41266205 , url=http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf , postscript=. (See chapter 6.)
Algebraic topology
Homotopy theory