On the group $\textrm {SSF}(G),\;G$ a cyclic group of prime order
HTML articles powered by AMS MathViewer
- by M. Maller and J. Whitehead PDF
- Trans. Amer. Math. Soc. 290 (1985), 725-733 Request permission
Abstract:
We extend the definition of the obstruction group ${\text {SSF}}(G)$ in the case where $G$ is a cyclic group of prime order. We show that an endomorphism of a free $ZG$-module is a direct summand of a virtual permutation if its characteristic polynomial has the appropriate form. Among these endomorphisms the virtual permutations are detected by ${K_0}$. The main application is in detecting Morse-Smale isotopy classes.References
- Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. I, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1981. With applications to finite groups and orders. MR 632548
- John Franks and Carolyn Narasimhan, The periodic behavior of Morse-Smale diffeomorphisms, Invent. Math. 48 (1978), no. 3, 279–292. MR 508988, DOI 10.1007/BF01390247
- John Franks and Michael Shub, The existence of Morse-Smale diffeomorphisms, Topology 20 (1981), no. 3, 273–290. MR 608601, DOI 10.1016/0040-9383(81)90003-3
- H. W. Lenstra Jr., Grothendieck groups of abelian group rings, J. Pure Appl. Algebra 20 (1981), no. 2, 173–193. MR 601683, DOI 10.1016/0022-4049(81)90091-8
- Michael Maller, Fitted diffeomorphisms of nonsimply connected manifolds, Topology 19 (1980), no. 4, 395–410. MR 584563, DOI 10.1016/0040-9383(80)90022-1
- Michael Maller, Algebraic problems arising from Morse-Smale dynamical systems, Geometric dynamics (Rio de Janeiro, 1981) Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 512–521. MR 1691597, DOI 10.1007/BFb0061432
- Michael Maller and Jennifer Whitehead, Virtual permutations of $\textbf {Z}[\textbf {Z}^{n}]$ complexes, Proc. Amer. Math. Soc. 90 (1984), no. 1, 162–166. MR 722437, DOI 10.1090/S0002-9939-1984-0722437-4
- M. Shub, Morse-Smale diffeomorphisms are unipotent on homology, Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971) Academic Press, New York, 1973, pp. 489–491. MR 0331439
- M. Shub and D. Sullivan, Homology theory and dynamical systems, Topology 14 (1975), 109–132. MR 400306, DOI 10.1016/0040-9383(75)90022-1
- Richard G. Swan, $K$-theory of finite groups and orders, Lecture Notes in Mathematics, Vol. 149, Springer-Verlag, Berlin-New York, 1970. MR 0308195, DOI 10.1007/BFb0059150
Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 290 (1985), 725-733
- MSC: Primary 58F09; Secondary 20C99
- DOI: https://doi.org/10.1090/S0002-9947-1985-0792823-0
- MathSciNet review: 792823