Abstract:
The main result is the following lemma whose special case for $k = 2l$
was proved by S. Avvakumov, I. Mabillard, A. Skopenkov, and U. Wagner in 2015.
Assume $k > l \geqslant 1$. Let $T := S^l \times S^l$ be the
$2l$-dimensional torus with meridian $m := S^l \times \cdot$ and parallel
$p := \cdot \times S^l$, and let $S^k_p$ and $S^k_m$ be copies of $S^k$.
Then there are no continuous maps $f : T \sqcup S^k_p \sqcup S^k_m \to
\mathbb{R}^{k+l+1}$ satisfying the following three properties:
- the $f$-images of the components are pairwise disjoint;
- $fS^k_p$ is linked modulo 2 with $fp$ and is not linked modulo 2
with $fm$, and
- $fS^k_m$ is linked modulo 2 with $fm$ and is not linked modulo 2
with $fp$.
For a proof we use a natural result stating that ‘in general position,
the preimage of a cycle is a cycle’.
The required general position argument is non-trivial.
As a corollary, we obtain $\mathbf{NP}$-hardness
of recognition of almost embeddability of finite $k$-dimensional complexes
in $\mathbb{R}^d$ for $d,k \geqslant 2$ such that $k+2 \leqslant d
\leqslant\frac{3k}2+1$.
A map $f: K \to \mathbb{R}^d$ of a simplicial complex is an almost
embedding if $f(\sigma) \cap f(\tau) = \varnothing$ whenever $\sigma,
\tau$ are disjoint simplices of $K$.
Link for connecting to the seminar: https://mian.ktalk.ru/j1xwg956wc7a
"PIN code": The number of homotopy classes of maps from the Russian word 쬄 to the Russian word ¨Æ (where a word is understood as the subset of the plane formed by its letters)
|
