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},
}
B. L. Feigin, D. B. Fuks, and V. S. Retakh :
“Massey operations in the cohomology of the infinite dimensional Lie algebra \( L_1 \) 13–31
in
Topology and geometry: Rohlin semininar .
Edited by O. Ya. Viro .
Lecture Notes in Mathematics 1346 .
Springer (Berlin ),
1988 .
MR
970070 Zbl
0653.17010 incollection
People
BibTeX
@incollection {key970070m,
AUTHOR = {Feigin, B. L. and Fuks, D. B. and Retakh,
V. S.},
TITLE = {Massey operations in the cohomology
of the infinite dimensional {L}ie algebra
\$L_1\$},
BOOKTITLE = {Topology and geometry: {R}ohlin semininar},
EDITOR = {Viro, O. Ya.},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {1346},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1988},
PAGES = {13--31},
DOI = {10.1007/BFb0082769},
NOTE = {MR:970070. Zbl:0653.17010.},
ISSN = {0075-8434},
ISBN = {9783540459583},
}
O. Ya. Viro and D. B. Fuks :
“Introduction to homotopy theory 6–121
in
Topologiya–2
[Topology–2 ].
Seriya sovremennye problemy matematiki: Fundamental’nye napravleniya 24 .
VINITI (Moscow ),
1988 .
An English translation appeared in Topology II (2004)
MR
987942 Zbl
0674.55001 incollection
People
BibTeX
@incollection {key987942m,
AUTHOR = {Viro, O. Ya. and Fuks, D. B.},
TITLE = {Introduction to homotopy theory},
BOOKTITLE = {Topologiya -- 2 [Topology -- 2]},
SERIES = {Seriya sovremennye problemy matematiki:
{F}undamental\cprime nye napravleniya},
NUMBER = {24},
PUBLISHER = {VINITI},
ADDRESS = {Moscow},
YEAR = {1988},
PAGES = {6--121},
URL = {http://mi.mathnet.ru/eng/intf100},
NOTE = {An English translation appeared in \textit{Topology
II} (2004). MR:987942. Zbl:0674.55001.},
ISSN = {0233-6723},
}
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},
}
