Let us be open-minded first, in order to read the question below.
I would like to have some mathematical tools/formulations/theories to understand the ordering of submanifolds in a $d$-sphere $S^d$ or $\mathbb{R}^d$.
Take the following figure for examples,
Let us limit to the $S^d = S^3$ first.
(1) Imagine the origin $X$ is a 0-dimensional point, while the submanifold with a radius $a$ is a $S^2$, called $S^2{}_a$, linking with/bounding the point $X$. There is another submanifold with a radius $b$ is a $S^2$, called $S^2{}_b$, linking with/bounding the point $X$. Here $b>a>0$, so submanifolds $X$, $S^2{}_a$, and $S^2{}_b$ all are in the $S^3$ . We see that there is an ordering of submanifolds, that
$S^2{}_b$ bounds $S^2{}_a$, while
$S^2{}_a$ bounds $X$,
we cannot re-order $S^2{}_a$ and $S^2{}_b$, without making these two submanifolds touching each other.
Namely we cannot exchange the positions of $S^2{}_a$ and $S^2{}_b$ surfaces, without letting them touch each other by a smooth deformation. (Let us call this case as there is an ordering of submanifolds.)
(2) Now, imagine the origin of the above figure becomes an $L$, as a 1-dimensional closed line (or a circle), while the submanifold with a radius $a$ is a $S^1$, called $S^1{}_a$, linking with the line $Y$. There is another submanifold with a radius $b$ is a $S^1$, called $S^1{}_b$, linking with the line $Y$. Here $b>a>0$, so submanifolds $Y$, $S^1{}_a$, and $S^1{}_b$ all are in the $S^3$ (or $\mathbb{R}^3$). We see that there is a no natural ordering of submanifolds, that
$S^1{}_b$ links with $Y$, while
$S^2{}_a$ links with $Y$.
But we CAN re-order $S^1{}_a$ and $S^1{}_b$, without making these two submanifolds touching each other.
Namely we CAN exchange the positions of $S^1{}_a$ and $S^1{}_b$ submanifold in $S^3$, without letting them touch each other by a smooth deformation. (Let us call this case as there is an NO ordering of submanifolds.)
- question: what is the proper mathematical tools to address the above ordering issues of submanifolds inside $\mathbb{S}^d$ or $\mathbb{R}^d$ or any $M^d$ with all homotopy group $\pi_i(M^d)=0$ trivial? I suppose we need to understand the deformations of submanifolds, and the homology/cohomology of the spaces. (But what precisely math formulas can we check whether there is a natural ordering for these submanifolds?)
Note add: When the submanifolds $\Sigma^{d-1}_i$ are $d-1$ dimensional, where $i=1,\dots,N$ are $N$ of them, all linking with a 0-dim point (the origin $X$ in the above). There is a natural ordering of this submanifolds. (The total space is $d$-dimensional $M^d$ with all homotopy group trivial $\pi_i(M^d)=0$.) But I wonder whether there can be more natural ordering of submanifolds even for lower dimensional submanifolds $\Sigma^{j}$ for $j< d-1$?
Many thanks!