On the Wall finiteness obstruction for the total space of certain fibrations

Munkholm, Hans J.; Pedersen, Erik Kjaer

doi:10.1090/S0002-9947-1980-0580901-6

On the Wall finiteness obstruction for the total space of certain fibrations
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by Hans J. Munkholm and Erik Kjaer Pedersen PDF

Trans. Amer. Math. Soc. 261 (1980), 529-545 Request permission

Abstract:

The problem of computing the Wall finiteness obstruction for the total space of a fibration $p: E \to B$ in terms of that for the base and homological data of the fiber has been considered by D. R. Anderson and by E. K. Pedersen and L. R. Taylor. We generalize their results and show how the problem is related to the algebraically defined transfer map ${\varphi ^{\ast }}: {K_0}({\textbf {Z}}{\pi _1}(B)) \to {K_0}({\textbf {Z}}{\pi _1}(E))$, $\varphi = {p_{\ast }}: {\pi _1}(E) \to {\pi _1}(B)$, whenever the latter is defined.

References

Douglas R. Anderson, Generalized product theorems for torsion invariants with applications to flat bundles, Bull. Amer. Math. Soc. 78 (1972), 465–469. MR 293636, DOI 10.1090/S0002-9904-1972-12947-1
Douglas R. Anderson, The obstruction to the finiteness of the total space of a flat bundle, Amer. J. Math. 95 (1973), 281–293. MR 334227, DOI 10.2307/2373786
Douglas R. Anderson, The Wall invariant of certain $S^{1}$ bundles, Proc. Amer. Math. Soc. 31 (1972), 529–535. MR 287545, DOI 10.1090/S0002-9939-1972-0287545-5
Hyman Bass, Algebraic $K$-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0249491
Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480

Finiteness obstruction of fiber spaces

Karel Ehrlich, Fibrations and a transfer map in algebraic $K$-theory, J. Pure Appl. Algebra 14 (1979), no. 2, 131–136. MR 524182, DOI 10.1016/0022-4049(79)90002-1
P. J. Hilton and D. Rees, Natural maps of extension functors and a theorem of R. G. Swan, Proc. Cambridge Philos. Soc. 57 (1961), 489–502. MR 124377, DOI 10.1017/s0305004100035544
V. J. Lal, Wall obstruction of a fibration, Invent. Math. 6 (1968), 67–77. MR 231380, DOI 10.1007/BF01389834
Saunders Mac Lane, Homology, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1975 edition. MR 1344215
John Milnor, Introduction to algebraic $K$-theory, Annals of Mathematics Studies, No. 72, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971. MR 0349811
Erik Kjaer Pedersen and Lawrence R. Taylor, The Wall finiteness obstruction for a fibration, Amer. J. Math. 100 (1978), no. 4, 887–896. MR 509078, DOI 10.2307/2373914
Irving Reiner, Class groups and Picard groups of group rings and orders, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 26, American Mathematical Society, Providence, R.I., 1976. MR 0404410
Jean-Pierre Serre, Cohomologie des groupes discrets, Prospects in mathematics (Proc. Sympos., Princeton Univ., Princeton, N.J., 1970) Ann. of Math. Studies, No. 70, Princeton Univ. Press, Princeton, N.J., 1971, pp. 77–169 (French). MR 0385006
Richard G. Swan, $K$-theory of finite groups and orders, Lecture Notes in Mathematics, Vol. 149, Springer-Verlag, Berlin-New York, 1970. MR 0308195
Richard G. Swan, Periodic resolutions for finite groups, Ann. of Math. (2) 72 (1960), 267–291. MR 124895, DOI 10.2307/1970135
Richard G. Swan, Projective modules over Laurent polynomial rings, Trans. Amer. Math. Soc. 237 (1978), 111–120. MR 469906, DOI 10.1090/S0002-9947-1978-0469906-4
C. T. C. Wall, Finiteness conditions for $\textrm {CW}$-complexes, Ann. of Math. (2) 81 (1965), 56–69. MR 171284, DOI 10.2307/1970382