Fibration

Homotopy lifting property A mapping p \colon E \to B satisfies the homotopy lifting property for a space X if: • for every homotopy h \colon X \times [0, 1] \to B and • for every mapping (also called lift) \tilde h_0 \colon X \to E lifting h|_{X \times 0} = h_0 (i.e. h_0 = p \circ \tilde h_0) there exists a (not necessarily unique) homotopy \tilde h \colon X \times [0, 1] \to E lifting h (i.e. h = p \circ \tilde h) with \tilde h_0 = \tilde h|_{X \times 0}. The following commutative diagram shows the situation: Fibration A fibration (also called Hurewicz fibration) is a mapping p \colon E \to B satisfying the homotopy lifting property for all spaces X. The space B is called the base space and the space E is called the total space. The fiber over b \in B is the subspace F_b = p^{-1}(b) \subseteq E. Serre fibration A Serre fibration (also called weak fibration) is a mapping p \colon E \to B satisfying the homotopy lifting property for all CW-complexes. Every Hurewicz fibration is a Serre fibration. Quasifibration A mapping p \colon E \to B is called quasifibration, if for every b \in B, e \in p^{-1}(b) and i \geq 0 holds that the induced mapping p_* \colon \pi_i(E, p^{-1}(b), e) \to \pi_i(B, b) is an isomorphism. Every Serre fibration is a quasifibration. == Examples ==

• The projection onto the first factor p \colon B \times F \to B is a fibration. That is, trivial bundles are fibrations. • Every covering p \colon E \to B is a fibration. Specifically, for every homotopy h \colon X \times [0, 1] \to B and every lift \tilde h_0 \colon X \to E there exists a uniquely defined lift \tilde h \colon X \times [0,1] \to E with p \circ \tilde h = h. • Every fiber bundle p \colon E \to B satisfies the homotopy lifting property for every CW-complex. • A fiber bundle with a paracompact and Hausdorff base space satisfies the homotopy lifting property for all spaces. • An example of a fibration which is not a fiber bundle is given by the mapping i^* \colon X^{I^k} \to X^{\partial I^k} induced by the inclusion i \colon \partial I^k \to I^k where k \in \N, X a topological space and X^{A} = \{f \colon A \to X\} is the space of all continuous mappings with the compact-open topology. • The Hopf fibration S^1 \to S^3 \to S^2 is a non-trivial fiber bundle and, specifically, a Serre fibration. == Basic concepts ==

Fiber homotopy equivalence A mapping f \colon E_1 \to E_2 between total spaces of two fibrations p_1 \colon E_1 \to B and p_2 \colon E_2 \to B with the same base space is a fibration homomorphism if the following diagram commutes: The mapping f is a fiber homotopy equivalence if in addition a fibration homomorphism g \colon E_2 \to E_1 exists, such that the mappings f \circ g and g \circ f are homotopic, by fibration homomorphisms, to the identities \operatorname{Id}_{E_2} and \operatorname{Id}_{E_1}. Pullback fibration Given a fibration p \colon E \to B and a mapping f \colon A \to B, the mapping p_f \colon f^*(E) \to A is a fibration, where f^*(E) = \{(a, e) \in A \times E\ |\ f(a) = p(e)\} is the pullback and the projections of f^*(E) onto A and E yield the following commutative diagram: The fibration p_f is called the pullback fibration or induced fibration. Pathspace fibration With the pathspace construction, any continuous mapping can be extended to a fibration by enlarging its domain to a homotopy equivalent space. This fibration is called pathspace fibration. The total space E_f of the pathspace fibration for a continuous mapping f \colon A \to B between topological spaces consists of pairs (a, \gamma) with a \in A and paths \gamma \colon I \to B with starting point \gamma (0) = f(a), where I = [0, 1] is the unit interval. The space E_f = \{ (a, \gamma) \in A \times B^I | \gamma (0) = f(a) \} carries the subspace topology of A \times B^I, where B^I describes the space of all mappings I \to B and carries the compact-open topology. The pathspace fibration is given by the mapping p \colon E_f \to B with p(a, \gamma) = \gamma (1). The fiber F_f is also called the homotopy fiber of f and consists of the pairs (a, \gamma) with a \in A and paths \gamma \colon [0, 1] \to B, where \gamma(0) = f(a) and \gamma(1) = b_0 \in B holds. For the special case of the inclusion of the base point i \colon b_0 \to B, an important example of the pathspace fibration emerges. The total space E_i consists of all paths in B which starts at b_0. This space is denoted by PB and is called path space. The pathspace fibration p \colon PB \to B maps each path to its endpoint, hence the fiber p^{-1}(b_0) consists of all closed paths. The fiber is denoted by \Omega B and is called loop space. == Properties ==

• The fibers p^{-1}(b) over b \in B are homotopy equivalent for each path component of B. • For a homotopy f \colon [0, 1] \times A \to B the pullback fibrations f^*_0(E) \to A and f^*_1(E) \to A are fiber homotopy equivalent. • If the base space B is contractible, then the fibration p \colon E \to B is fiber homotopy equivalent to the product fibration B \times F \to B. • The pathspace fibration of a fibration p \colon E \to B is very similar to itself. More precisely, the inclusion E \hookrightarrow E_p is a fiber homotopy equivalence. • For a fibration p \colon E \to B with fiber F and contractible total space, there is a weak homotopy equivalence F \to \Omega B. == Puppe sequence ==

For a fibration p \colon E \to B with fiber F and base point b_0 \in B the inclusion F \hookrightarrow F_p of the fiber into the homotopy fiber is a homotopy equivalence. The mapping i \colon F_p \to E with i (e, \gamma) = e, where e \in E and \gamma \colon I \to B is a path from p(e) to b_0 in the base space, is a fibration. Specifically it is the pullback fibration of the pathspace fibration PB \to B along p. This procedure can now be applied again to the fibration i and so on. This leads to a long sequence: \cdots \to F_j \to F_i \xrightarrow {j} F_p \xrightarrow i E \xrightarrow p B. The fiber of i over a point e_0 \in p^{-1}(b_0) consists of the pairs (e_0, \gamma) where \gamma is a path from p(e_0) = b_0 to b_0, i.e. the loop space \Omega B. The inclusion \Omega B \hookrightarrow F_i of the fiber of i into the homotopy fiber of i is again a homotopy equivalence and iteration yields the sequence:\cdots \Omega^2B \to \Omega F \to \Omega E \to \Omega B \to F \to E \to B.Due to the duality of fibration and cofibration, there also exists a sequence of cofibrations. These two sequences are known as the Puppe sequences or the sequences of fibrations and cofibrations. == Principal fibration ==

A fibration p \colon E \to B with fiber F is called principal, if there exists a commutative diagram: The bottom row is a sequence of fibrations and the vertical mappings are weak homotopy equivalences. Principal fibrations play an important role in Postnikov towers. == Long exact sequence of homotopy groups ==

For a Serre fibration p \colon E \to B there exists a long exact sequence of homotopy groups. For base points b_0 \in B and x_0 \in F = p^{-1}(b_0) this is given by:\cdots \rightarrow \pi_n(F,x_0) \rightarrow \pi_n(E, x_0) \rightarrow \pi_n(B, b_0) \rightarrow \pi_{n - 1}(F, x_0) \rightarrow \cdots \rightarrow \pi_0(F, x_0) \rightarrow \pi_0(E, x_0).The homomorphisms \pi_n(F, x_0) \rightarrow \pi_n(E, x_0) and \pi_n(E, x_0) \rightarrow \pi_n(B, b_0) are the induced homomorphisms of the inclusion i \colon F \hookrightarrow E and the projection p \colon E \rightarrow B. Hopf fibration Hopf fibrations are a family of fiber bundles whose fiber, total space and base space are spheres:S^0 \hookrightarrow S^1 \rightarrow S^1, S^1 \hookrightarrow S^3 \rightarrow S^2, S^3 \hookrightarrow S^7 \rightarrow S^4, S^7 \hookrightarrow S^{15} \rightarrow S^8.The long exact sequence of homotopy groups of the hopf fibration S^1 \hookrightarrow S^3 \rightarrow S^2 yields:\cdots \rightarrow \pi_n(S^1,x_0) \rightarrow \pi_n(S^3, x_0) \rightarrow \pi_n(S^2, b_0) \rightarrow \pi_{n - 1}(S^1, x_0) \rightarrow \cdots \rightarrow \pi_1(S^1, x_0) \rightarrow \pi_1(S^3, x_0) \rightarrow \pi_1(S^2, b_0). This sequence splits into short exact sequences, as the fiber S^1 in S^3 is contractible to a point:0 \rightarrow \pi_i(S^3) \rightarrow \pi_i(S^2) \rightarrow \pi_{i-1}(S^1) \rightarrow 0.This short exact sequence splits because of the suspension homomorphism \phi \colon \pi_{i - 1}(S^1) \to \pi_i(S^2) and there are isomorphisms:\pi_i(S^2) \cong \pi_i(S^3) \oplus \pi_{i - 1}(S^1).The homotopy groups \pi_{i - 1}(S^1) are trivial for i \geq 3, so there exist isomorphisms between \pi_i(S^2) and \pi_i(S^3) for i \geq 3. Analog the fibers S^3 in S^7 and S^7 in S^{15} are contractible to a point. Further the short exact sequences split and there are families of isomorphisms: \pi_i(S^4) \cong \pi_i(S^7) \oplus \pi_{i - 1}(S^3) and \pi_i(S^8) \cong \pi_i(S^{15}) \oplus \pi_{i - 1}(S^7). == Spectral sequence ==

Spectral sequences are important tools in algebraic topology for computing (co-)homology groups. The Leray-Serre spectral sequence connects the (co-)homology of the total space and the fiber with the (co-)homology of the base space of a fibration. For a fibration p \colon E \to B with fiber F, where the base space is a path connected CW-complex, and an additive homology theory G_* there exists a spectral sequence: :H_k (B; G_q(F)) \cong E^2_{k, q} \implies G_{k + q}(E). Fibrations do not yield long exact sequences in homology, as they do in homotopy. But under certain conditions, fibrations provide exact sequences in homology. For a fibration p \colon E \to B with fiber F, where base space and fiber are path connected, the fundamental group \pi_1(B) acts trivially on H_*(F) and in addition the conditions H_p(B) = 0 for 0 and H_q(F) = 0 for 0 hold, an exact sequence exists (also known under the name Serre exact sequence):H_{m+n-1}(F) \xrightarrow {i_*} H_{m+n-1}(E) \xrightarrow {f_*} H_{m+n-1} (B) \xrightarrow \tau H_{m+n-2} (F) \xrightarrow {i^*} \cdots \xrightarrow {f_*} H_1 (B) \to 0.This sequence can be used, for example, to prove Hurewicz's theorem or to compute the homology of loopspaces of the form \Omega S^n: H_k (\Omega S^n) = \begin{cases} \Z & \exist q \in \Z \colon k = q (n-1)\\ 0 & \text{otherwise} \end{cases}.For the special case of a fibration p \colon E \to S^n where the base space is a n-sphere with fiber F, there exist exact sequences (also called Wang sequences) for homology and cohomology: \cdots \to H_q(F) \xrightarrow{i_*} H_q(E) \to H_{q-n}(F) \to H_{q-1}(F) \to \cdots \cdots \to H^q(E) \xrightarrow{i^*} H^q(F) \to H^{q-n+1}(F) \to H^{q+1}(E) \to \cdots == Orientability ==

For a fibration p \colon E \to B with fiber F and a fixed commutative ring R with a unit, there exists a contravariant functor from the fundamental groupoid of B to the category of graded R-modules, which assigns to b \in B the module H_*(F_b, R) and to the path class [\omega] the homomorphism h[\omega]_* \colon H_*(F_{\omega (0)}, R) \to H_*(F_{\omega (1)}, R), where h[\omega] is a homotopy class in [F_{\omega(0)}, F_{\omega (1)}]. A fibration is called orientable over R if for any closed path \omega in B the following holds: h[\omega]_* = 1. == Euler characteristic ==

For an orientable fibration p \colon E \to B over the field \mathbb{K} with fiber F and path connected base space, the Euler characteristic of the total space is given by:\chi(E) = \chi(B)\chi(F).Here the Euler characteristics of the base space and the fiber are defined over the field \mathbb{K}. == See also ==