Doll series

In mathematics , the puppet sequence is a construction of the homotopy theory .

It was introduced by Dieteruppe in 1958 and is also known as the doll sequence .

definition

Let it be a continuous mapping . Let it be the mapping cone of , then is ${\displaystyle f\colon X\to Y}$ ${\displaystyle C(f)}$ ${\displaystyle f}$

${\displaystyle Y\to C(f)}$

a cofiber and

${\displaystyle C(f)/Y=SX}$

is the attachment of . By iterating, what is known as the doll sequence is obtained ${\displaystyle X}$

${\displaystyle X\to Y\to C(f)\to SX\to SY\to SC(f)\to S^{2}X\to S^{2}Y\to \ldots }$

application

For a continuous mapping and for every space , the homotopy classes of continuous mapping form an exact sequence ${\displaystyle f\colon X\to Y}$ ${\displaystyle Z}$

${\displaystyle \ldots \left[SC(f),Z\right]\to \left[SY,Z\right]\to \left[SX,Z\right]\to \left[C(f),Z\right]\to \left[Y,Z\right]\to \left[X,Z\right]}$

Individual evidence

1. ↑ Dieter Doll: Homotopiemengen and their induced illustrations , Part I, Mathematical Journal, Vol 69, 1958, pp 299-344
2. ↑ James C. Becker, Daniel Gottlieb: A history of duality in algebraic topology , pdf
3. ↑ Tammo tom Dieck: Topology , 2. completely revised. and exp. Edition, de Gruyter (2000), pp. 202ff, ISBN 3-11-016236-9