Grain

In algebraic topology , a branch of mathematics, a grain (also Hurewicz grain , after the Polish mathematician Witold Hurewicz ) is a continuous mapping of topological spaces, which satisfies the homotopy elevation property with regard to every topological space. Fibers play a major role in homotopy theory, a sub-area of ​​algebraic topology. Roughly speaking, fibers are pairs of spaces with a mapping one below the other, which allow any homotopias to be withdrawn into the image space along the given image onto the original image space.

definition

Homotopy elevation property

Denote the unit interval . ${\displaystyle I}$ ${\displaystyle [0,1]\subset \mathbb {R} }$

A continuous mapping of topological spaces fulfills the homotopy elevation property for the topological space if it is for all continuous maps ${\displaystyle p\colon E\longrightarrow B}$ ${\displaystyle X}$

${\displaystyle f\colon X\times I\longrightarrow B}$

such as

${\displaystyle {\bar {f}}\colon X\times \{0\}\to E}$,

so the diagram

${\displaystyle {\begin{matrix}\qquad {\bar {f}}\colon X\times \{0\}&\longrightarrow &E\\\operatorname {incl} {\bigg \downarrow }&&{\bigg \downarrow }p\\\qquad f\colon X\times I\ &\longrightarrow &B\end{matrix}}}$

commutates, a mapping

${\displaystyle F\colon X\times I\longrightarrow E}$

there so that and is. ${\displaystyle f=p\circ F}$ ${\displaystyle F|_{X\times \{0\}}={\bar {f}}}$

Hurewicz fibers

A fiber (also Hurewicz fiber ) is a continuous mapping that fulfills the homotopy elevation property for all topological spaces . ${\displaystyle p\colon E\longrightarrow B}$ ${\displaystyle X}$

${\displaystyle E}$is called total space , the basis of the grain. The archetype of a point is called fiber over . ${\displaystyle B}$ ${\displaystyle p^{-1}(b)}$ ${\displaystyle b\in B}$ ${\displaystyle b}$

If the base is contiguous, the fibers are homotopy equivalent across different points from . ${\displaystyle B}$ ${\displaystyle B}$

Serre fibers

A Serre fibration is a continuous map , the homotopy high elevation property for all CW complexes met. ${\displaystyle p\colon E\longrightarrow B}$ ${\displaystyle X}$

This is sufficient (and therefore equivalently), to the high-homotopies raising property for the rooms with met. ${\displaystyle X=\left[0,1\right]^{n}}$ ${\displaystyle n=0,1,2,\ldots }$

Quasi-fibers

A quasi-grain is a continuous mapping for which ${\displaystyle p\colon E\longrightarrow B}$

${\displaystyle p_{*}:\pi _{i}(E,p^{-1}(x);y)\rightarrow \pi _{i}(B,x)}$

is an isomorphism for each and every one. ${\displaystyle x\in B,y\in p^{-1}(x)}$ ${\displaystyle i\geq 0}$

If the base is path-contiguous, then all fibers are weakly homotopy equivalent .

Every Serre fiber is a quasi fiber.

Examples

• Let be any topological space and be ${\displaystyle F}$

${\displaystyle p:B\times F\to B}$

a projection onto the first factor, then there is a grain. ${\displaystyle p}$

• Every overlay is a grain.
• More generally, each fiber bundle is Serre fiber. In this case, the archetypes of different points are not only equivalent to homotopy, but are even homeomorphic .
• There are examples of fiber bundles that are not Hurewicz fibers. But fiber bundles over paracompact spaces are always also Hurewicz fibers (Huebsch-Hurewicz theorem).
• A fiber that does not have to be a fiber bundle is the path fiber of a topological space.

Long exact homotopy sequence

For Serre fibers (and also more generally for quasi-fibers) one has a long exact sequence of homotopy groups ${\displaystyle p:E\rightarrow B}$

${\displaystyle \ldots \rightarrow \pi _{n+1}(B,y)\rightarrow \pi _{n}(F,x)\rightarrow \pi _{n}(E,x)\rightarrow \pi _{n}(B,y)\rightarrow \pi _{n-1}(F,x)\rightarrow \ldots }$.

Here is and the fiber. ${\displaystyle x\in E,y=p(x)\in B}$ ${\displaystyle F=p^{-1}(x)}$

Example: Hopf fibers with fibers . As is well known, it is for everyone , it follows for everyone , in particular . ${\displaystyle p:S^{3}\rightarrow S^{2}}$ ${\displaystyle S^{1}}$ ${\displaystyle \pi _{n}(S^{1})=0}$ ${\displaystyle n\geq 2}$ ${\displaystyle \pi _{n}(S^{3})\cong \pi _{n}(S^{2})}$ ${\displaystyle n\geq 3}$ ${\displaystyle \pi _{3}(S^{2})=\mathbb {Z} }$

Fibers homology groups

The homology groups of Serre fibers can often be calculated with the help of spectral sequences.

literature

• Edwin H. Spanier: Algebraic Topology. McGraw-Hill, New York NY et al. 1966 ( McGraw-Hill Series in Higher Mathematics ).
• Allen Hatcher : Algebraic topology. Cambridge University Press, Cambridge, 2002. ISBN 0-521-79160-X pdf
• Jean-Pierre Serre : Homologie singulière des espaces fibers. Applications. Ann. of Math. (2) 54, (1951). 425-505. pdf
• Albrecht Dold , René Thom : Quasi-fibers and infinite symmetrical products. Ann. of Math. (2) 67 1958 239-281. pdf
• JP May .: Weak equivalences and quasifibrations. In Groups of self-equivalences and related topics (Montreal, PQ, 1988), volume 1425 of Lecture Notes in Math., Pages 91-101. Springer, Berlin, 1990.