On dimensionally restricted maps

On dimensionally exotic maps

We call a value $y=f(x)$ of a map $f:X\to Y$ dimensionally regular if $\dim X\le \dim(Y\times f^{-1}(y))$. It was shown in \cite{first-exotic} that if a map $f:X\to Y$ between compact metric spaces does not have dimensionally regular values, then $X$ is a Boltyanskii compactum, i.e. a compactum satisfying the equality $\dim(X\times X)=2\dim X-1$. In this paper we prove that every Boltyanskii compactum $X$ of dimension $\dim X \geq 6$ admits a map $f:X\to Y$ without dimensionally regular values. Also we exhibit a 4-dimensional Boltyanskii compactum for which every map has a dimensionally regula…

Maps with dimensionally restricted fibers

We prove that if $f\colon X\to Y$ is a closed surjective map between metric spaces such that every fiber $f^{-1}(y)$ belongs to a class of space $\mathrm S$, then there exists an $F_σ$-set $A\subset X$ such that $A\in\mathrm S$ and $\dim f^{-1}(y)\backslash A=0$ for all $y\in Y$. Here, $\mathrm S$ can be one of the following classes: (i) $\{M:\mathrm{e-dim}M\leq K\}$ for some $CW$-complex $K$; (ii) $C$-spaces; (iii) weakly infinite-dimensional spaces. We also establish that if $\mathrm S=\{M:\dim M\leq n\}$, then $\dim f\triangle g\leq 0$ for almost all $g\in C(X,\mathbb I^{n+1})$.

3维复射影空间上特殊通用映射不存在性的证明

Proofs of the non-existence of special generic maps on the $3$-dimensional complex projective space

该研究通过多种方法证明3维复射影空间上不存在特殊通用映射，聚焦含该空间的闭单连通流形上的这类Morse函数推广映射，补充了相关存在性问题的研究成果。