Homotopy fiber

In algebraic topology , a branch of mathematics , the homotopy fiber of a map is a useful term in homotopy theory .

definition

For every continuous mapping of topological spaces

${\displaystyle f\colon X\to Y}$

there is a homotopy equivalence such that ${\displaystyle \phi \colon X^{\prime }\to X}$

${\displaystyle f\circ \phi \colon X^{\prime }\to Y}$

is a fiber . The fiber of this fiber is called homotopy fiber of . It is only uniquely determined except for homotopy equivalence. ${\displaystyle f}$

construction

Inclusions

We first consider the simpler case that is an injective mapping . In this case one can construct as a set of all paths in which end in. ${\displaystyle \iota \colon X\to Y}$ ${\displaystyle X^{\prime }}$ ${\displaystyle Y}$ ${\displaystyle X}$

${\displaystyle X^{\prime }=\left\{\sigma \in Y^{I}\colon \sigma (1)\in X\right\}}$.

${\displaystyle X}$can be embedded in as a set of constant paths and one then has a homotopy equivalence . The figure defines a fiber and for a fixed point the fiber is the set of all paths in that start and end in the fixed base point . ${\displaystyle X^{\prime }}$ ${\displaystyle X^{\prime }\to X}$ ${\displaystyle \sigma \to \sigma (0)}$ ${\displaystyle X^{\prime }\to Y}$ ${\displaystyle x_{0}\in Y}$ ${\displaystyle F}$ ${\displaystyle Y}$ ${\displaystyle x_{0}}$ ${\displaystyle X}$

${\displaystyle F=\left\{\sigma \in Y^{I}\colon \sigma (0)=x_{0},\sigma (1)\in X\right\}}$

example

The product of two circles is a torus , the one-point union of the circles maps into the torus.

As an example, consider the inclusion of the one-point union in the product . The homotopy fiber, as described above, is the union along the intersection . (Here denotes the path space and the loop space .) ${\displaystyle X\vee Y}$ ${\displaystyle X\times Y}$ ${\displaystyle PX\times \Omega Y\cup \Omega X\times PY}$ ${\displaystyle \Omega X\times \Omega Y}$ ${\displaystyle PX}$ ${\displaystyle \Omega X}$

If and have the homotopy type of CW complexes , this homotopy fiber is weakly homotopy equivalent to the connection of the two loop spaces. ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle \Omega X*\Omega Y}$

General illustrations

For a unnecessarily injective map, consider ${\displaystyle f\colon X\to Y}$

${\displaystyle X^{\prime }=\left\{(x,\sigma )\in X\times Y^{I}\colon f(x)=\sigma (1)\right\}}$.

${\displaystyle X}$can be embedded in means for the respectively constant path and one then has a homotopy equivalence . The figure defines a fiber and for a fixed point is the fiber ${\displaystyle X^{\prime }}$ ${\displaystyle x\to (x,\sigma _{x})}$ ${\displaystyle \sigma _{x}}$ ${\displaystyle X^{\prime }\to X}$ ${\displaystyle (x,\sigma )\to \sigma (0)}$ ${\displaystyle X^{\prime }\to Y}$ ${\displaystyle x_{0}\in Y}$

${\displaystyle F=\left\{(x,\sigma )\in X\times Y^{I}\colon \sigma (0)=x_{0},f(x)=\sigma (1)\right\}}$

Long exact sequence

Let be a continuous map and its homotopy fiber. Then you have a long exact sequence of homotopy groups ${\displaystyle f\colon X\to Y}$ ${\displaystyle F}$

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

Here is and is the way in that is constant . ${\displaystyle x\in X,y=f(x)}$ ${\displaystyle \sigma _{x}\in F}$ ${\displaystyle Y}$ ${\displaystyle f(x)}$

From the knowledge of the homotopy fiber one obtains connections between the homotopy groups of and . ${\displaystyle X}$ ${\displaystyle Y}$

literature

• R. Bott, L. Tu: Differential forms in Algebraic Topology , Graduate Texts in Mathematics, Springer, 1982. (Pages 249–250)

Individual evidence

1. ^ T. Ganea: A generalization of the homology and homotopy suspension , Comm. Math. Helv. 39, 295-322, 1964.