# References on a non-homeomorphism that “folds” a manifold

As part of my small research, I am looking at a certain problem, where I want to study images of manifolds $M \subset \mathbb{R}^n$, under a family of non-homeomorphisms $f_{k, A}(\vec{x}) :\mathbb{R}^n \to \mathbb{R}^n$.

$f_{k, A}$ has the following specific form:

Definition of $f_{k, A}$.

First, let $\mathbf{I}_{(-k)}(x_1, \ldots, x_n) = ( y_1, \ldots, y_n)$, with to be defined by

$$y_j := \begin{cases} x_j & j \neq k\\ |x_j| & j = k \end{cases},\qquad (k = 1, \ldots, n)$$

With the above, set $$f_{k, A}(\vec{x}) := \mathbf{I}_{(-k)}(A \vec{x}).$$ Where $A \in \mathbb{R}^{2n}$ is a non singular square matrix.

Overall the functions in $f_{k,A}$ "rotates" and "stretches" the space $\mathbb{R}^n$ and then folds one of the coordinates to the positive side.

Illustration

To quickly illustrate how $f_{k,A}$ works, pick $A$ to be an identity matrix, the function $f_k$ folds the k-$th$ coordinate to the positive side. For $n=2$ and $k = 2$ the image of a circle $S$ under mapping $f_{2, I}$ is given in the the figure below)

$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$

A concrete question

I want to be able to understand how this map is able to change the topology of the manifold. For example, to say that any manifold with Betti numbers $\{\beta_i\}_{i=0}^{n-1}$ can be reduced to contractible manifold by at most $C(\{\beta_i\}_{i=0}^{n-1})$ consecutive applications of functions from $f_{k, A}$.

General references

I have googled the internet for quite some time but was not able to find any research that can be related to this, probably I'm using wrong terminology in my search query. Is there a theory that studies "foldings", what would be the proper topological notions to search for? and any references to material that can be related to my problem is appreciate.

• I'm not aware of any research in the literature on this problem. – Lee Mosher Mar 24 '17 at 14:58
• It is very unlikely that the Betti numbers are the right invariants here; for instance, your manifold could be a homology sphere with complicated fundamental group. – Moishe Kohan Mar 24 '17 at 16:29
• @MoisheCohen Thanks! what in you opinion \ intuition are better characteristics of the shape to investigate? – them Mar 24 '17 at 16:41
• I do not know, maybe Lusternik–Schnirelmann category: en.wikipedia.org/wiki/Lusternik%E2%80%93Schnirelmann_category – Moishe Kohan Mar 24 '17 at 16:48