Chain homotopy

In the mathematical subfield of homological algebra , a chain homotopy is an abstraction of the topological concept of a homotopy .

definition

Let there be and coquette complexes and two chain images, i.e. H. Systems of morphisms which are compatible with the differentials in the sense that holds. ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle f,g\colon X\to Y}$ ${\displaystyle f^{k}\colon X^{k}\to Y^{k}}$ ${\displaystyle f^{k+1}\circ d_{X}^{k}=d_{Y}^{k}\circ f^{k}}$

Then a chain homotopy is a sequence of morphisms such that , or more detailed ${\displaystyle D\colon f\simeq g}$ ${\displaystyle D^{k}\colon X^{k}\to Y^{k-1}}$ ${\displaystyle f-g=Dd+dD}$

${\displaystyle f^{k}-g^{k}=D^{k+1}\circ d_{X}^{k}+d_{Y}^{k-1}\circ D^{k}}$for all , ${\displaystyle k}$

applies.

${\displaystyle f}$and are called homotop if there is a chain homotopy . Homotopy is an equivalence relation compatible with the composition on the set of all chain mappings. ${\displaystyle g}$ ${\displaystyle D\colon f\simeq g}$

Homotopias of mappings between chain complexes (and not coquette complexes) are defined analogously. Two chain mappings and between chain complexes and are called homotop if there is a sequence of morphisms such that ${\displaystyle f=(f_{k})_{k}}$ ${\displaystyle g=(g_{k})_{k}}$ ${\displaystyle X=((X_{k})_{k},(d_{k}^{X})_{k})}$ ${\displaystyle Y=((Y_{k})_{k},(d_{k}^{Y})_{k})}$ ${\displaystyle (D_{k})_{k}}$ ${\displaystyle D_{k}\colon X_{k}\rightarrow Y_{k+1}}$

${\displaystyle f_{k}-g_{k}=d_{k+1}^{Y}\circ D_{k}+D_{k-1}\circ d_{k}^{X}}$for everyone . ${\displaystyle k}$

Two chain complexes and are called chain homotopy equivalent if there are chain images and for which the sequential and in each case are homotopic to the identity. ${\displaystyle X=((X_{k})_{k},(d_{k}^{X})_{k})}$ ${\displaystyle Y=((Y_{k})_{k},(d_{k}^{Y})_{k})}$ ${\displaystyle f\colon X\rightarrow Y}$ ${\displaystyle g\colon Y\rightarrow X}$ ${\displaystyle fg}$ ${\displaystyle gf}$

meaning

• A mapping that is homotopic to the null mapping is called nullhomotopic. The category of coquette complexes modulo nullhomotopic images is the homotopy category.

• Homotopic chain maps induce the same mapping in the homology or cohomology.

• If in particular there is a coquette complex and a homotopy between the identity on and the null map on , then the cohomology of trivial, i.e. H. is exact . One then speaks of a contracting homotopy . ${\displaystyle C}$ ${\displaystyle D\colon \mathrm {id} _{C}\simeq 0}$ ${\displaystyle C}$ ${\displaystyle C}$ ${\displaystyle C}$ ${\displaystyle C}$

• If two continuous mappings and between topological spaces and are homotopic , then the assigned mappings and between the associated singular chain complexes are homotopic in the sense defined above. In particular, the induced mappings are the same between the singular homology groups . ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle Sf}$ ${\displaystyle Sg}$

• Two projective resolutions of a module over a ring are homotopic. In particular, the homologies of the resolutions are the same, which leads to the concept of the derived functor .

Individual evidence

1. ^ Peter Hilton and Urs Stammbach: A course in homological algebra , Springer-Verlag (1970), Graduate Texts in Mathematics, ISBN 0-387-90032-2 , Chapter IV, §3, Homotopy
2. ↑ Peter Hilton and Urs Stammbach: A course in homological algebra , Springer-Verlag (1970), Graduate Texts in Mathematics, ISBN 0-387-90032-2 , Chapter IV, §3, sentence 3.1
3. ↑ Saunders Mac Lane : Homology , Springer Grundlehren der Mathematischen Wissenschaften Volume 114 (1967), Theorem 8.2
4. ↑ Peter Hilton and Urs Stammbach: A course in homological algebra , Springer-Verlag (1970), Graduate Texts in Mathematics, ISBN 0-387-90032-2 , Chapter IV, §4, Sentence 4.3