Change of fiber

If β is a path in B that starts at, say, b, then we have the homotopy $h:\; p^\{-1\}(b)\; \backslash times\; I\; \backslash to\; I\; \backslash overset\{\backslash beta\}\backslash to\; B$ where the first map is a projection. Since p is a fibration, by the homotopy lifting property, h lifts to a homotopy $g:\; p^\{-1\}(b)\; \backslash times\; I\; \backslash to\; E$ with $g\_0:\; p^\{-1\}(b)\; \backslash hookrightarrow\; E$. We have:

:$g\_1:\; p^\{-1\}(b)\; \backslash to\; p^\{-1\}(\backslash beta(1))$.

(There might be an ambiguity and so $\backslash beta\; \backslash mapsto\; g\_1$ need not be well-defined.)

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

:$\backslash tau:\; \backslash operatorname\{Pc\}(B)\; \backslash to$ 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

:$K\; =\; I\; \backslash times\; \backslash \{0,\; 1\backslash \}\; \backslash cup\; \backslash \{0\backslash \}\; \backslash times\; I\; \backslash subset\; I^2$.

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

Then, by the homotopy lifting property, we can lift the homotopy $p^\{-1\}(b)\; \backslash times\; I^2\; \backslash to\; I^2\; \backslash overset\{h\}\backslash to\; B$ to w such that w restricts to $f$. In particular, we have $g\_1\; \backslash sim\; g\_1\text{'}$, establishing the claim.

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

:$\backslash tau([c\_b])\; =\; \backslash operatorname\{id\},\; \backslash ,\; \backslash tau([\backslash beta]\; \backslash cdot\; [\backslash gamma])\; =\; \backslash tau([\backslash beta])\; \backslash circ\; \backslash tau([\backslash gamma])$

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

:$\backslash tau:\; \backslash operatorname\{Pc\}(B)\; \backslash to$ the set of homotopy classes of homotopy equivalences.

Also, we have: for each b in B,

:$\backslash tau:\; \backslash pi\_1(B,\; b)\; \backslash to$ { [ƒ] | homotopy equivalence $f\; :\; p^\{-1\}(b)\; \backslash to\; p^\{-1\}(b)$ }

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.

One consequence of the construction is the below:

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

• 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]

Category:Algebraic topology