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In algebraic topology, given a fibration p:E→B, the change of fiber is a map between the fibers induced by paths in B.

Since a covering is a fibration, the construction generalizes the corresponding facts in the theory of covering spaces.

Definition

If β is a path in B that starts at, say, b, then we have the homotopy $h:p^{-1}(b)\times I\to I{\overset {\beta }{\to }}B$ where the first map is a projection. Since p is a fibration, by the homotopy lifting property, h lifts to a homotopy $g:p^{-1}(b)\times I\to E$ with $g_{0}:p^{-1}(b)\hookrightarrow E$ . We have:

g_{1}:p^{-1}(b)\to p^{-1}(\beta (1))

.

(There might be an ambiguity and so $\beta \mapsto g_{1}$ need not be well-defined.)

Let $\operatorname {Pc} (B)$ denote the set of path classes in B. We claim that the construction determines the map:

\tau :\operatorname {Pc} (B)\to

the set of homotopy classes of maps.

Suppose β, β' are in the same path class; thus, there is a homotopy h from β to β'. Let

K=I\times \{0,1\}\cup \{0\}\times I\subset I^{2}

.

Drawing a picture, there is a homeomorphism $I^{2}\to I^{2}$ that restricts to a homeomorphism $K\to I\times \{0\}$ . Let $f:p^{-1}(b)\times K\to E$ be such that $f(x,s,0)=g(x,s)$ , $f(x,s,1)=g'(x,s)$ and $f(x,0,t)=x$ .

Then, by the homotopy lifting property, we can lift the homotopy $p^{-1}(b)\times I^{2}\to I^{2}{\overset {h}{\to }}B$ to w such that w restricts to $f$ . In particular, we have $g_{1}\sim g_{1}'$ , establishing the claim.

It is clear from the construction that the map is a homomorphism: if $\gamma (1)=\beta (0)$ ,

\tau ([c_{b}])=\operatorname {id} ,\,\tau ([\beta ]\cdot [\gamma ])=\tau ([\beta ])\circ \tau ([\gamma ])

where $c_{b}$ is the constant path at b. It follows that $\tau ([\beta ])$ has inverse. Hence, we can actually say:

\tau :\operatorname {Pc} (B)\to

the set of homotopy classes of homotopy equivalences.

Also, we have: for each b in B,

\tau :\pi _{1}(B,b)\to

{ [ƒ] | homotopy equivalence

f:p^{-1}(b)\to p^{-1}(b)

}

which is a group homomorphism (the right-hand side is clearly a group.) In other words, the fundamental group of B at b acts on the fiber over b, up to homotopy. This fact is a useful substitute for the absence of the structure group.

Consequence

One consequence of the construction is the below:

The fibers of p over a path-component is homotopy equivalent to each other.

References

James F. Davis, Paul Kirk, Lecture Notes in Algebraic Topology
May, J. A Concise Course in Algebraic Topology

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-Given a [[fibration]] ''p'':''E''→''B'', the '''change of fiber''' is a map between the fibers induced by paths in ''B''.
+In algebraic topology, given a [[fibration]] ''p'':''E''→''B'', the '''change of fiber''' is a map between the fibers induced by paths in ''B''.
 Since a covering is a fibration, the construction generalizes the corresponding facts in the theory of [[covering space]]s.
 == Definition ==
-If ''β'' is a path in ''B'', then we have the homotopy <math>h: p^{-1}(b) \times I \to I \overset{\beta}\to B</math> where the first map is a projection. Since ''p'' is a fibration, by the [[homotopy lifting property]], ''h'' lifts to a homotopy <math>g: E \times I \to E</math> with <math>g_0</math> = the identity. is the inclusion. We have:
+If ''β'' is a path in ''B'' that starts at, say, ''b'', then we have the homotopy <math>h: p^{-1}(b) \times I \to I \overset{\beta}\to B</math> where the first map is a projection. Since ''p'' is a fibration, by the [[homotopy lifting property]], ''h'' lifts to a homotopy <math>g: p^{-1}(b) \times I \to E</math> with <math>g_0: p^{-1}(b) \hookrightarrow E</math>. We have:
-:<math>g_t: p^{-1}(b) \to p^{-1}(\beta(t))</math>.
+:<math>g_1: p^{-1}(b) \to p^{-1}(\beta(1))</math>.
-We let <math>\tau(\beta) = g_1</math>. We claim
+(There might be an ambiguity and so <math>\beta \mapsto g_1</math> need not be well-defined.)
-Let <math>\operatorname{Pc}(B)</math> denotes the set of path classes in ''B''. Then we thus get a map
+Let <math>\operatorname{Pc}(B)</math> denote the set of [[path class]]es in ''B''. We claim that the construction determines the map:
-:<math>\tau: Pa(B) \to </math>
+:<math>\tau: \operatorname{Pc}(B) \to </math> the set of homotopy classes of maps.
+Suppose β, β' are in the same path class; thus, there is a homotopy ''h'' from β to β'. Let
-It is immediate from the construction that the map is a homomorphism:
-:<math>\tau([\beta] \cdot [\gamma]) = \tau([\beta]) \tau([\gamma])</math>.
+:<math>K = I \times \{0, 1\} \cup \{0\} \times I \subset I^2</math>.
+Drawing a picture, there is a homeomorphism <math>I^2 \to I^2</math> that restricts to a homeomorphism <math>K \to I \times \{0\}</math>. Let <math>f: p^{-1}(b) \times K \to E</math> be such that <math>f(x, s, 0) = g(x, s)</math>, <math>f(x, s, 1) = g'(x, s)</math> and <math>f(x, 0, t) = x</math>.
+Then, by the homotopy lifting property, we can lift the homotopy <math>p^{-1}(b) \times I^2 \to I^2 \overset{h}\to B</math> to ''w'' such that ''w'' restricts to <math>f</math>. In particular, we have <math>g_1 \sim g_1'</math>, establishing the claim.
+It is clear from the construction that the map is a homomorphism: if <math>\gamma(1) =\beta(0)</math>,
+:<math>\tau([c_b]) = \operatorname{id}, \, \tau([\beta] \cdot [\gamma]) = \tau([\beta]) \circ \tau([\gamma])</math>
+where <math>c_b</math> is the constant path at ''b''. It follows that <math>\tau([\beta])</math> has inverse. Hence, we can actually say:
+:<math>\tau: \operatorname{Pc}(B) \to </math> the set of homotopy classes of homotopy equivalences.
+Also, we have: for each ''b'' in ''B'',
+:<math>\tau: \pi_1(B, b) \to</math> { [ƒ] | homotopy equivalence <math>f : p^{-1}(b) \to p^{-1}(b)</math> }
+which is a group homomorphism (the right-hand side is clearly a group.) In other words, the fundamental group of ''B'' at ''b'' acts on the fiber over ''b'', up to homotopy. This fact is a useful substitute for the absence of the [[structure group]].
 == Consequence ==
+One consequence of the construction is the below:
-One can get a substitute for a [[structure group]].
+*The fibers of ''p'' over a path-component is homotopy equivalent to each other.
+== References ==
+*James F. Davis, Paul Kirk, [http://www.maths.ed.ac.uk/~aar/papers/davkir.pdf Lecture Notes in Algebraic Topology]
+*May, J. [http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf A Concise Course in Algebraic Topology]
-{{topology-stub}}
+[[Category:Algebraic topology]]