Аннотация:
При изучении теории $SU$-бордизмов возникает промежуточная теория между ними и комплексными бордизмами — теория $c_1$-сферических бордизмов $W$. В докладе я расскажу об общем описании $SU$-линейных операций на комплексных кобордизмах, об $SU$-линейных умножениях на теории $W$ и $SU$-линейных проекторах из теории комплексных бордизмов на неё, а также о комплексных ориентациях на $W$ и некоторых вопросах о соответствующих формальных группах.
\noindentИ. М. Широков. Hopf-type theorems for $f$-neighbors
We work within the framework of a program aimed at exploring various extended versions for theorems from a class containing Borsuk–Ulam type theorems, some fixed point theorems, the KKM lemma, Radon, Tverberg, and Helly theorems. In this talk we will present variations of the Hopf theorem concerning continuous maps of a compact Riemannian manifold $M$ of dimension $n$ to $\mathbb{R}^n$. We investigate the case of maps $f\colon M \to \mathbb{R}^m$ with $n < m$ and introduce several notions of varied types of $f$-neighbors, which is a pair of distinct points in $M$ such that $f$ takes it to a СsmallТ set of some type. Next for each type, we ask what distances on $M$ are realized as distances between $f$-neighbors of this type and study various characteristics of this set of distances. One of our main results is as follows. Let $f\colon M \to \mathbb{R}^{m}$ be a continuous map. We say that two distinct points $a$ and $b$ in $M$ are visual $f$-neighbors if the segment in $\mathbb{R}^{m}$ with endpoints $f(a)$ and $f(b)$ intersects $f(M)$ only at $f(a)$ and $f(b)$. Then the set of distances that are realized as distances between visual $f$-neighbors is infinite. Besides we generalize the Hopf theorem in a quantitative sense.
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