конференция памяти Е.Г.Скляренко
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Dmitry V. Gugnin
Oct 19, 2025, 8:51:41 AMOct 19
to Dmitry V. Gugnin
Уважаемые коллеги!
В ближайшую среду, 22 октября, вместо Учебно-научного и Постниковского семинаров состоится миниконференция "Теории гомологий", посвященная 90-летию со дня рождения Заслуженного профессора Московского университета
Евгения Григорьевича Скляренко (1935-2009).
Конференция пройдет с 16:45 до 20:05 в ауд. 12-24 ГЗ МГУ.
Приглашенные докладчики:
Е.В.Щепин (МИАН) "Выпуклость и ацикличность гиперплоских сечений"
А.С.Мищенко (МГУ) "Когомологии симплициальных пространств с промежуточными носителями в смысле Е.Г.Скляренко"
С.А.Мелихов (МИАН) "Пространства, гомологии которых ведут себя аналогично когомологиям локальных компактов"
А.С.Мищенко (МГУ) "Когомологии симплициальных пространств с промежуточными носителями в смысле Е.Г.Скляренко"
С.А.Мелихов (МИАН) "Пространства, гомологии которых ведут себя аналогично когомологиям локальных компактов"
Аннотации докладов:
1) Е.В.Щепин (МИАН) "Выпуклость и ацикличность гиперплоских сечений"
Доклад посвящен следующему вопросу: при каких условиях на подмножество линейного пространства ацикличность его гиперплоских сечений влечет выпуклость? Будут представлены известные результаты и поставлены нерешенные
проблемы.
проблемы.
2) А.С.Мищенко (МГУ) "Когомологии симплициальных пространств с промежуточными носителями в смысле Е.Г.Скляренко"
Аннотация в приложении.
3) С.А.Мелихов (МИАН) "Пространства, гомологии которых ведут себя аналогично когомологиям локальных компактов"
Spaces whose homology behaves similarly to cohomology of local compacta
Abstract: Locally compact separable metrizable spaces are
characterized among all metrizable spaces as those that admit a
cofinal sequence
K_1\subset K_2\subset ... of compact subsets. Their Čech cohomology is
well-understood due to the following short exact sequence, which first
appeared in a paper by S. V. Petkova, a student of E. G. Sklyarenko
[3; Proposition 4]: 0 -> lim^1 H^{n-1}(K_i) -> H^n(X) -> lim H^n(K_i)
-> 0.
(Apparently this paper [3] was the main part of her dissertation
written under Sklyarenko's guidance. Its main results are improved
versions
of the main results of Sklyarenko's paper [2].) It should be noted
that Petkova's short exact sequence was later rediscovered by W.
Massey
[4; Theorem 4.22]. In fact, it was almost discovered by A.
Grothendieck in 1957 [1; Proposition 3.10.2], but not quite, because
he was not aware
of the lim^1 functor at that time.
We study a dual class of spaces, which we show to satisfy a dual short
exact sequence in homology [5]. We call a metrizable space X
a "coronated polyhedron" if it contains a compactum K such that X-K is
a polyhedron. These include, apart from compacta and polyhedra,
spaces such as the topologist's sine curve (or the Warsaw circle) and
the comb (=comb-and-flea) space. The complement of every
locally compact subset of S^n is a coronated polyhedron. We prove that
a metrizable space X is a coronated polyhedron if and only if
it admits a countable polyhedral resolution; or, equivalently, a
sequential polyhedral resolution ... -> R_2 -> R_1. ("Resolution" is
in
the sense of Mardešić.) In the latter case, we establish a short exact
sequence 0 -> lim^1 H_{n+1}(R_i) -> H_n(X) -> lim H_n(R_i) -> 0
for Steenrod-Sitnikov homology and also for any (possibly
extraordinary) homology theory satisfying Milnor's axioms of map
excision and
\prod-additivity. On the other hand, Quigley's short exact sequence 0
-> lim^1 \pi_{n+1}(R_i) -> \pi_n(X) -> lim \pi_n(R_i) -> 0
for Steenrod homotopy of compacta fails for Steenrod-Sitnikov homotopy
of coronated polyhedra, at least when n=0.
[1] A. Grothendieck, Sur quelques points d’algèbre homologique, Tohoku
Math. J. 9 (1957), 119–221; Russian transl., А. Гротендик, О некоторых вопросах
гомологической алгебры, ИЛ, Москва, 1961.
[2] E. G. Sklyarenko, Uniqueness theorems in homology theory, Mat.
Sb. 85 (1971), 201–223; English transl., Math. USSR–Sb. 14 (1971),
199–218
[3] S. V. Petkova, On the axioms of homology theory, Mat. Sbornik 90
(1973), 607–624; English transl., Math. USSR-Sb. 90 (1974), 597–614
[4] W. S. Massey, Homology and Cohomology Theory, Marcel Dekker, New
York, 1978; Russian transl., У. Масси, Теория гомологий и когомологий,
Мир, Москва, 1981.
[5] S. A. Melikhov, Coronated polyhedra and coronated ANRs, arXiv:2211.09951
Abstract: Locally compact separable metrizable spaces are
characterized among all metrizable spaces as those that admit a
cofinal sequence
K_1\subset K_2\subset ... of compact subsets. Their Čech cohomology is
well-understood due to the following short exact sequence, which first
appeared in a paper by S. V. Petkova, a student of E. G. Sklyarenko
[3; Proposition 4]: 0 -> lim^1 H^{n-1}(K_i) -> H^n(X) -> lim H^n(K_i)
-> 0.
(Apparently this paper [3] was the main part of her dissertation
written under Sklyarenko's guidance. Its main results are improved
versions
of the main results of Sklyarenko's paper [2].) It should be noted
that Petkova's short exact sequence was later rediscovered by W.
Massey
[4; Theorem 4.22]. In fact, it was almost discovered by A.
Grothendieck in 1957 [1; Proposition 3.10.2], but not quite, because
he was not aware
of the lim^1 functor at that time.
We study a dual class of spaces, which we show to satisfy a dual short
exact sequence in homology [5]. We call a metrizable space X
a "coronated polyhedron" if it contains a compactum K such that X-K is
a polyhedron. These include, apart from compacta and polyhedra,
spaces such as the topologist's sine curve (or the Warsaw circle) and
the comb (=comb-and-flea) space. The complement of every
locally compact subset of S^n is a coronated polyhedron. We prove that
a metrizable space X is a coronated polyhedron if and only if
it admits a countable polyhedral resolution; or, equivalently, a
sequential polyhedral resolution ... -> R_2 -> R_1. ("Resolution" is
in
the sense of Mardešić.) In the latter case, we establish a short exact
sequence 0 -> lim^1 H_{n+1}(R_i) -> H_n(X) -> lim H_n(R_i) -> 0
for Steenrod-Sitnikov homology and also for any (possibly
extraordinary) homology theory satisfying Milnor's axioms of map
excision and
\prod-additivity. On the other hand, Quigley's short exact sequence 0
-> lim^1 \pi_{n+1}(R_i) -> \pi_n(X) -> lim \pi_n(R_i) -> 0
for Steenrod homotopy of compacta fails for Steenrod-Sitnikov homotopy
of coronated polyhedra, at least when n=0.
[1] A. Grothendieck, Sur quelques points d’algèbre homologique, Tohoku
Math. J. 9 (1957), 119–221; Russian transl., А. Гротендик, О некоторых вопросах
гомологической алгебры, ИЛ, Москва, 1961.
[2] E. G. Sklyarenko, Uniqueness theorems in homology theory, Mat.
Sb. 85 (1971), 201–223; English transl., Math. USSR–Sb. 14 (1971),
199–218
[3] S. V. Petkova, On the axioms of homology theory, Mat. Sbornik 90
(1973), 607–624; English transl., Math. USSR-Sb. 90 (1974), 597–614
[4] W. S. Massey, Homology and Cohomology Theory, Marcel Dekker, New
York, 1978; Russian transl., У. Масси, Теория гомологий и когомологий,
Мир, Москва, 1981.
[5] S. A. Melikhov, Coronated polyhedra and coronated ANRs, arXiv:2211.09951
Если вам нужен пропуск для прохода на конференцию, пишите мне не позднее понедельника 20 октября.
Будет организована трансляция в ZOOM. Ссылка для подключения:
С уважением,
Дмитрий Гугнин
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