Luffing

In mathematics , cofibers are an important term in algebraic topology .

definition

A continuous map is cofiber if it satisfies the homotopy expansion property, i.e. H. when it comes to continuous images ${\displaystyle i\colon A\to X}$

${\displaystyle f\colon X\to Y,h\colon A\times \left[0,1\right]\to Y}$

With

${\displaystyle f\circ i=h\circ i_{0}}$

(for the inclusive defined by ) always a continuous mapping ${\displaystyle i_{0}(x)=(x,0)}$ ${\displaystyle i_{0}\colon A\to A\times \left[0,1\right]}$

${\displaystyle {\overline {h}}\colon X\times \left[0,1\right]\to Y}$

With

${\displaystyle {\overline {h}}\circ (i\times id)=h}$

and

${\displaystyle {\overline {h}}|_{X\times \{0\}}=f\circ \pi _{X}}$

(for natural projection ) there. ${\displaystyle \pi _{X}:X\times \{0\}\to X}$

If the inclusion is a subspace , then this condition is equivalent to that there is a retraction ${\displaystyle i\colon A\to X}$ ${\displaystyle A\subset X}$

${\displaystyle p\colon X\times \left[0,1\right]\to A\times \left[0,1\right]\cup X\times \left\{0\right\}}$

gives.

Examples

• The inclusion

${\displaystyle S^{n-1}\to D^{n}}$

is a cofibre.

• For every CW complex and everyone is inclusion ${\displaystyle X}$ ${\displaystyle m\leq n}$

${\displaystyle X_{m}\to X_{n}}$

from the m-skeleton into the n-skeleton a fiber structure. In particular, CW complexes are cofibrant .

Cofiber

The homotopy cofiber of an (arbitrary) continuous mapping is its mapping cone . For any generalized theory of homology one has a long exact sequence ${\displaystyle f\colon A\to X}$ ${\displaystyle C_{f}}$ ${\displaystyle H_{*}}$

${\displaystyle \ldots \to H_{*+1}(C_{f})\to H_{*}(A)\to H_{*}(X)\to H_{*}(C_{f})\to H_{*-1}(A)\to \ldots }$

If the image is a cofiber, the homotopy cofiber is called a cofiber. ${\displaystyle f}$ ${\displaystyle C_{f}}$

If an inclusion is a cofiber, then the cofiber is homotopy-equivalent to the quotient space and it applies ${\displaystyle f\colon A\to X}$ ${\displaystyle C_{f}}$ ${\displaystyle X/A}$

${\displaystyle H_{*}(X,A)=H_{*}(C_{f})=H_{*}(X/A)}$.

literature

• Whitehead, George W .: Elements of homotopy theory. Graduate Texts in Mathematics, 61st Springer-Verlag, New York-Berlin, 1978. ISBN 0-387-90336-4