An algebraic study of the Klein Bottle | SpringerLink

A background reference for this subject is given in [8]. Suggested references for chain complexes and homological algebra are [5] and [6]. For chain complexes and homology, see [7].

2.1

Strong deformation retracts

Let X and Y be chain complexes over a ring k, ,
be chain maps and let
be a degree one k-linear map such that
\(f\nabla = 1_X\) and
\(d\phi +\phi d = 1 – \nabla f\). Thus, f and
\(\nabla \) compose to the identity, but the composition the other way around is only chain homotopic to the identity. When these conditions hold, we say that this collection of data forms a strong deformation retraction (SDR) and we write

$$\begin{aligned} X Y \nabla f \phi . \end{aligned}$$

(1)

Important for certain computations are the side conditions [1]

$$\begin{aligned} \phi ^2 =0,\quad \phi \nabla =0,\quad \mathrm {and},\quad f\phi =0. \end{aligned}$$

(2)

In fact, these may always be assumed to hold: if the last two do not hold, replace
\(\phi \) by
\(\phi ‘ = D(\phi )\phi D(\phi )\) where
\(D(\phi ) = \phi d + d\phi \) and the last two conditions will now hold with respect to
\(\phi ‘\). If the first condition does not hold for
\(\phi ‘\), replace it by
\(\phi ” = \phi ‘ d \phi ‘\) and all three conditions will hold for the chain homotopy
\(\phi ”\). We will always assume that the side conditions hold when we talk about an SDR.

2.2

The perturbation Lemma

Given the SDR (1) and, in addition, a second differential
\(d_Y’\) on Y, let
\(t = d_Y’-d_Y\). The perturbation lemma, [2–4, 8] states that if we set
\(t_n = (t\phi )^{n-1}t, \ n \ge 1\) and, for each n, define new maps on X,

$$\begin{aligned} \partial _n&= d + f (t_1 + t_2 + \cdots + t_{n-1}) \nabla \\ \nabla _n&= \nabla + \phi (t_1 + t_2 + \cdots + t_{n-1}) \nabla , \end{aligned}$$

and on Y:

$$\begin{aligned} f_n&= f + f (t_1 + t_2 + \cdots + t_{n-1}) \phi \\ \phi _n&= \phi + \phi (t_1 + t_2 + \cdots + t_{n-1}) \phi , \end{aligned}$$

then in the limits, provided they exist, we have new SDR data

Note that the limits will certainly exist if
\(t\phi \) is nilpotent in each degree. Examples relevant to this paper may be found in [9–13].

Remark 2.1

In setting up a situation as above, it is useful to think of the perturbation as transferring the new differential
\(d’\) on a larger complex to a smaller one. In this sense, the data is said to be a transference problem and t is called the initiator. This notation is from [4] and its use is continued here.