V. A. Rokhlin and D. B. Fuks :
Nachal’nyj kurs topologii: Geometricheskie glavy
[Beginner’s course in topology: Geometric chapters ].
Nauka (Moscow ),
1977 .
A French translation was published in 1981 , an English translation in 1984 .
Zbl
0417.55002 book
People
BibTeX
Vladimir Abramovich Rokhlin
Related
@book {key0417.55002z,
AUTHOR = {Rokhlin, V. A. and Fuks, D. B.},
TITLE = {Nachal\cprime nyj kurs topologii: {G}eometricheskie
glavy [Beginner's course in topology:
{G}eometric chapters]},
PUBLISHER = {Nauka},
ADDRESS = {Moscow},
YEAR = {1977},
PAGES = {488},
NOTE = {A French translation was published in
1981, an English translation in 1984.
Zbl:0417.55002.},
}
V. A. Rokhlin and D. B. Fuks :
Premier cours de topologie: Chapitres geométriques
[Beginner’s course in topology: Geometric chapters ].
Mir (Moscow ),
1981 .
French translation of 1977 Russian original .
Zbl
0453.55001 book
People
BibTeX
Vladimir Abramovich Rokhlin
Related
@book {key0453.55001z,
AUTHOR = {Rokhlin, V. A. and Fuks, D. B.},
TITLE = {Premier cours de topologie: {C}hapitres
geom\'etriques [Beginner's course in
topology: {G}eometric chapters]},
PUBLISHER = {Mir},
ADDRESS = {Moscow},
YEAR = {1981},
PAGES = {493},
NOTE = {French translation of 1977 Russian original.
Zbl:0453.55001.},
}
D. B. Fuks and V. A. Rokhlin :
Beginner’s course in topology: Geometric chapters Universitext .
Springer (Berlin ),
1984 .
English translation of 1977 Russian original .
MR
759162 Zbl
0562.54003 book
People
BibTeX
Vladimir Abramovich Rokhlin
Related
@book {key759162m,
AUTHOR = {Fuks, D. B. and Rokhlin, V. A.},
TITLE = {Beginner's course in topology: {G}eometric
chapters},
SERIES = {Universitext},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1984},
PAGES = {xi+519},
URL = {https://www.springer.com/gp/book/9783540135777},
NOTE = {English translation of 1977 Russian
original. MR:759162. Zbl:0562.54003.},
ISSN = {0172-5939},
ISBN = {9783540135777},
}
V. I. Arnol’d, A. M. Vershik, O. Ya. Viro, A. N. Kolmogorov, S. P. Novikov, Ya. G. Sinaj, and D. B. Fuks :
“Vladimir Abramovich Rokhlin (obituary) Usp. Mat. Nauk
41 : 3(249)
(1986 ),
pp. 159–163 .
An English translation was published in Russ. Math. Surv. 41 :3(1986)
MR
854242 Zbl
0604.01010 article
People
BibTeX
@article {key854242m,
AUTHOR = {Arnol\cprime d, V. I. and Vershik, A.
M. and Viro, O. Ya. and Kolmogorov,
A. N. and Novikov, S. P. and Sinaj,
Ya. G. and Fuks, D. B.},
TITLE = {Vladimir {A}bramovich {R}okhlin (obituary)},
JOURNAL = {Usp. Mat. Nauk},
FJOURNAL = {Uspekhi Matematicheskikh Nauk [N.S.]},
VOLUME = {41},
NUMBER = {3(249)},
YEAR = {1986},
PAGES = {159--163},
URL = {http://mi.mathnet.ru/eng/umn2084},
NOTE = {An English translation was published
in \textit{Russ. Math. Surv.} \textbf{41}:3(1986).
MR:854242. Zbl:0604.01010.},
ISSN = {0042-1316},
}
V. I. Arnol’d, A. M. Vershik, O. Ya. Viro, A. N. Kolmogorov, S. P. Novikov, Ya. G. Sinaj, and D. B. Fuks :
“Vladimir Abramovich Rokhlin (obituary) Russ. Math. Surv.
41 : 3
(June 1986 ),
pp. 189–195 .
English translation of Russian original published in Usp. Mat. Nauk 41 :3(249) (1986)
Zbl
0608.01032 article
People
BibTeX
@article {key0608.01032z,
AUTHOR = {Arnol\cprime d, V. I. and Vershik, A.
M. and Viro, O. Ya. and Kolmogorov,
A. N. and Novikov, S. P. and Sinaj,
Ya. G. and Fuks, D. B.},
TITLE = {Vladimir {A}bramovich {R}okhlin (obituary)},
JOURNAL = {Russ. Math. Surv.},
FJOURNAL = {Russian Mathematical Surveys},
VOLUME = {41},
NUMBER = {3},
MONTH = {June},
YEAR = {1986},
PAGES = {189--195},
DOI = {10.1070/RM1986v041n03ABEH003331},
NOTE = {English translation of Russian original
published in \textit{Usp. Mat. Nauk}
\textbf{41}:3(249) (1986). Zbl:0608.01032.},
ISSN = {0036-0279},
}
O. Ya. Viro and D. B. Fuks :
“Homology and cohomology 123–240
in
Topologiya–2
[Topology–2 ].
Edited by S. P. Novikov, V. A. Rokhlin, and R. V. Gamkrelidze .
Seriya sovremennye problemy matematiki: Fundamental’nye napravleniya 24 .
VINITI (Moscow ),
1988 .
An English translation appeared in Topology II (2004)
MR
987943 Zbl
0674.55002 incollection
People
BibTeX
@incollection {key987943m,
AUTHOR = {Viro, O. Ya. and Fuks, D. B.},
TITLE = {Homology and cohomology},
BOOKTITLE = {Topologiya -- 2 [Topology -- 2]},
EDITOR = {Novikov, S. P. and Rokhlin, V. A. and
Gamkrelidze, R. V.},
SERIES = {Seriya sovremennye problemy matematiki:
{F}undamental\cprime nye napravleniya},
NUMBER = {24},
PUBLISHER = {VINITI},
ADDRESS = {Moscow},
YEAR = {1988},
PAGES = {123--240},
URL = {http://mi.mathnet.ru/eng/intf101},
NOTE = {An English translation appeared in \textit{Topology
II} (2004). MR:987943. Zbl:0674.55002.},
ISSN = {0233-6723},
}
O. Ya. Viro and D. B. Fuchs :
“Introduction to homotopy theory 1–93
in
Topology II .
Edited by S. P. Novikov and V. A. Rokhlin .
Encyclopaedia of Mathematical Sciences 24 .
Springer (Berlin ),
2004 .
Translated from the Russian by C. J. Shaddock.
Translation of Russian original published in Seriya sovremennye problemy matematiki 24 (1988)
MR
2054456 incollection
People
BibTeX
@incollection {key2054456m,
AUTHOR = {Viro, O. Ya. and Fuchs, D. B.},
TITLE = {Introduction to homotopy theory},
BOOKTITLE = {Topology {II}},
EDITOR = {Novikov, S. P. and Rokhlin, V. A.},
SERIES = {Encyclopaedia of Mathematical Sciences},
NUMBER = {24},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {2004},
PAGES = {1--93},
DOI = {10.1007/978-3-662-10581-8_1},
NOTE = {Translated from the Russian by C.~J.
Shaddock. Translation of Russian original
published in \textit{Seriya sovremennye
problemy matematiki} \textbf{24} (1988).
MR:2054456.},
ISSN = {0938-0396},
ISBN = {9783642080845},
}
O. Ya. Viro and D. B. Fuchs :
“Homology and cohomology 95–196
in
Topology II .
Edited by S. P. Novikov and V. A. Rokhlin .
Encyclopaedia of Mathematical Sciences 24 .
Springer (Berlin ),
2004 .
Translation of Russian original published in Seriya sovremennye problemy matematiki 24 (1988)
MR
2054457 Zbl
1078.57013 incollection
Abstract
People
BibTeX
A complex is a sequence of Abelian groups and homomorphisms of the form
\[ \dots \stackrel{\partial_3}{\rightarrow} C_3 \stackrel{\partial_2}{\rightarrow} C_1 \stackrel{\partial_1}{\rightarrow}C_0 \stackrel{\partial_0}{\rightarrow} C_{-1} \rightarrow \dots, \]
in which \( \partial_q \circ \partial_{q+1} = 0 \) for each \( q \) . A complex is positive if \( C_q = 0 \) for \( q < 0 \) , free if all the \( C_q \) are free Abelian groups, and a complex of finite type if the sum \( \bigoplus_q C_q \) is finitely generated. (In dealing with positive complexes, the sequence \( \{C_q \) , \( \partial_q\} \) is often cut short, and we speak of the sequence
\[ \dots\to C_1 \stackrel{\partial_1}{\rightarrow} C_0 \,.) \]
The homomorphisms \( \partial_q \) are called the differentials or boundary operators ; elements of \( C_q \) are called the \( q \) -chains of the complex; chains that are annihilated by the differential are called cycles , and cycles whose difference lies in the image of the differential are said to be homologous .
@incollection {key2054457m,
AUTHOR = {Viro, O. Ya. and Fuchs, D. B.},
TITLE = {Homology and cohomology},
BOOKTITLE = {Topology {II}},
EDITOR = {Novikov, S. P. and Rokhlin, V. A.},
SERIES = {Encyclopaedia of Mathematical Sciences},
NUMBER = {24},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {2004},
PAGES = {95--196},
DOI = {10.1007/978-3-662-10581-8_2},
NOTE = {Translation of Russian original published
in \textit{Seriya sovremennye problemy
matematiki} \textbf{24} (1988). MR:2054457.
Zbl:1078.57013.},
ISSN = {0938-0396},
ISBN = {9783642080845},
}
D. B. Fuchs :
“Classical manifolds 197–252
in
Topology II .
Edited by S. P. Novikov and V. A. Rokhlin .
Encyclopaedia of Mathematical Sciences 24 .
Springer (Berlin ),
2004 .
MR
2054458 incollection
Abstract
People
BibTeX
This article contains a variety of information (mainly topological, but also geometric and analytic) about classical manifolds, such as spheres, Stiefel and Grassmann manifolds, Lie groups, and lens spaces (a fuller list of the manifolds considered can be found in the table of contents). A considerable part of the material here will undoubtedly appear in other volumes on topology in the Encyclopaedia of Mathematical Sciences , but a systematic exposition in a single article seems useful.
@incollection {key2054458m,
AUTHOR = {Fuchs, D. B.},
TITLE = {Classical manifolds},
BOOKTITLE = {Topology {II}},
EDITOR = {Novikov, S. P. and Rokhlin, V. A.},
SERIES = {Encyclopaedia of Mathematical Sciences},
NUMBER = {24},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {2004},
PAGES = {197--252},
DOI = {10.1007/978-3-662-10581-8_3},
NOTE = {MR:2054458.},
ISSN = {0938-0396},
ISBN = {9783642080845},
}
