American Mathematical Society

Book Review

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MathSciNet review: 1568129
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Book Information:

Author: V. A. Vassiliev
Title: Complements of discriminants of smooth maps: Topology and applications
Additional book information: American Mathematical Society, Providence, RI, 1992, 208 pp., US$164.00. ISBN 0-8218-4555-1.

D. W. Anderson, A generalization of the Eilenberg-Moore spectral sequence, Bull. Amer. Math. Soc. 78 (1972), 784–786. MR 310889, DOI 10.1090/S0002-9904-1972-13034-9

Tatsuji Kudo and Shôrô Araki, Topology of $H_n$-spaces and $H$-squaring operations, Mem. Fac. Sci. Kyūsyū Univ. A 10 (1956), 85–120. MR 87948

V. I. Arnol′d, The cohomology ring of the group of dyed braids, Mat. Zametki 5 (1969), 227–231 (Russian). MR 242196

[A2]

-, On a class of algebraic functions and cohomology of swallow tails, Uspekhi Mat. Nauk 23 (1968), 247-248.

V. I. Arnol′d, Topological invariants of algebraic functions. II, Funkcional. Anal. i Priložen. 4 (1970), no. 2, 1–9 (Russian). MR 0276244

Martin Bendersky and Sam Gitler, The cohomology of certain function spaces, Trans. Amer. Math. Soc. 326 (1991), no. 1, 423–440. MR 1010881, DOI 10.1090/S0002-9947-1991-1010881-8

Joan S. Birman and Xiao-Song Lin, Knot polynomials and Vassiliev’s invariants, Invent. Math. 111 (1993), no. 2, 225–270. MR 1198809, DOI 10.1007/BF01231287

C.-F. Bödigheimer, Stable splittings of mapping spaces, Algebraic topology (Seattle, Wash., 1985) Lecture Notes in Math., vol. 1286, Springer, Berlin, 1987, pp. 174–187. MR 922926, DOI 10.1007/BFb0078741

C.-F. Bödigheimer, F. R. Cohen, and R. J. Milgram, Truncated symmetric products and configuration spaces, Math. Z. 214 (1993), no. 2, 179–216. MR 1240884, DOI 10.1007/BF02572399

C.-F. Bödigheimer, F. Cohen, and L. Taylor, On the homology of configuration spaces, Topology 28 (1989), no. 1, 111–123. MR 991102, DOI 10.1016/0040-9383(89)90035-9

Fred Cohen, Homology of $\Omega ^{(n+1)}\Sigma ^{(n+1)}X$ and $C_{(n+1)}X,\,n>0$, Bull. Amer. Math. Soc. 79 (1973), 1236–1241 (1974). MR 339176, DOI 10.1090/S0002-9904-1973-13394-4

[C2]

-, Artin's braid groups, classical homotopy theory and sundry other curiosities, Braids (Joan S. Birman and Anatoly Libgober, eds.), Contemp. Math., vol. 78, Amer. Math. Soc., Providence, RI, 1988, pp. 167-206.

F. R. Cohen, On the mapping class groups for punctured spheres, the hyperelliptic mapping class groups, $\textrm {SO}(3)$, and $\textrm {Spin}^c(3)$, Amer. J. Math. 115 (1993), no. 2, 389–434. MR 1216436, DOI 10.2307/2374863

F. R. Cohen, R. L. Cohen, B. M. Mann, and R. J. Milgram, The topology of rational functions and divisors of surfaces, Acta Math. 166 (1991), no. 3-4, 163–221. MR 1097023, DOI 10.1007/BF02398886

F. R. Cohen and L. R. Taylor, Computations of Gel′fand-Fuks cohomology, the cohomology of function spaces, and the cohomology of configuration spaces, Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977) Lecture Notes in Math., vol. 657, Springer, Berlin, 1978, pp. 106–143. MR 513543

Eldon Dyer and R. K. Lashof, Homology of iterated loop spaces, Amer. J. Math. 84 (1962), 35–88. MR 141112, DOI 10.2307/2372804

Edward Fadell and Lee Neuwirth, Configuration spaces, Math. Scand. 10 (1962), 111–118. MR 141126, DOI 10.7146/math.scand.a-10517

R. Fox and L. Neuwirth, The braid groups, Math. Scand. 10 (1962), 119–126. MR 150755, DOI 10.7146/math.scand.a-10518

D. B. Fuks, Cohomology of the braid group $\textrm {mod}\ 2$, Funkcional. Anal. i Priložen. 4 (1970), no. 2, 62–73 (Russian). MR 0274463

Yasuhiko Kamiyama, The modulo $2$ homology groups of the space of rational functions, Osaka J. Math. 28 (1991), no. 2, 229–242. MR 1132162

V. Ja. Lin, Artin braids and related groups and spaces, Algebra. Topology. Geometry, Vol. 17 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 1979, pp. 159–227, 308 (Russian). MR 584570

R. James Milgram and Peter Löffler, The structure of deleted symmetric products, Braids (Santa Cruz, CA, 1986) Contemp. Math., vol. 78, Amer. Math. Soc., Providence, RI, 1988, pp. 415–424. MR 975092, DOI 10.1090/conm/078/975092

J. M. Boardman and R. M. Vogt, Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Mathematics, Vol. 347, Springer-Verlag, Berlin-New York, 1973. MR 0420609

Dusa McDuff, Configuration spaces of positive and negative particles, Topology 14 (1975), 91–107. MR 358766, DOI 10.1016/0040-9383(75)90038-5

Minoru Nakaoka, Homology of the infinite symmetric group, Ann. of Math. (2) 73 (1961), 229–257. MR 131874, DOI 10.2307/1970333

Graeme Segal, Configuration-spaces and iterated loop-spaces, Invent. Math. 21 (1973), 213–221. MR 331377, DOI 10.1007/BF01390197

Graeme Segal, The topology of spaces of rational functions, Acta Math. 143 (1979), no. 1-2, 39–72. MR 533892, DOI 10.1007/BF02392088

Arnold Shapiro, Obstructions to the imbedding of a complex in a euclidean space. I. The first obstruction, Ann. of Math. (2) 66 (1957), 256–269. MR 89410, DOI 10.2307/1969998

Steve Smale, On the topology of algorithms. I, J. Complexity 3 (1987), no. 2, 81–89. MR 907191, DOI 10.1016/0885-064X(87)90021-5

F. V. Vaĭnšteĭn, The cohomology of braid groups, Funktsional. Anal. i Prilozhen. 12 (1978), no. 2, 72–73 (Russian). MR 498903

[W]

Wu Wen-Tsün, A theory of imbedding, immersion, and isotopy of polytopes in a Euclidean space, Science Press, Beijing, 1974.

Review Information:

Reviewer: Frederick R. Cohen

Journal: Bull. Amer. Math. Soc. 31 (1994), 258-265
DOI: https://doi.org/10.1090/S0273-0979-1994-00525-7

Bulletin of the American Mathematical Society

Bibliography