Geometric transfer and the homotopy type of the automorphism groups of a manifold. (English) Zbl 0489.57008

Burghelea, D.; Lashof, R.

Trans. Am. Math. Soc. 269, 1-38 (1982).

MSC:

57R99	Differential topology
57R50	Differential topological aspects of diffeomorphisms
58D05	Groups of diffeomorphisms and homeomorphisms as manifolds
57T20	Homotopy groups of topological groups and homogeneous spaces
57Q60	Cobordism and concordance in PL-topology
57N70	Cobordism and concordance in topological manifolds
18F25	Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)

Keywords:

homotopy type of the groups of diffeomorphisms and homeomorphisms; spaces of imbeddings; group of pseudoisotopies; homotopy functor with values in category of infinite loop spaces; monoid of simple homotopy equivalences; Waldhausen’s algebraic K-theory of spaces; groups of automorphisms of compact manifolds; stability for concordances

References:

[1]	Peter L. Antonelli, Dan Burghelea, and Peter J. Kahn, The concordance-homotopy groups of geometric automorphism groups, Lecture Notes in Mathematics, Vol. 215, Springer-Verlag, Berlin-New York, 1971. · Zbl 0222.57001
[2]	Dan Burghelea, Automorphisms of manifolds, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 347 – 371.
[3]	-, The structure of block automorphisms of \( M \times {S^1}\), Topology 16 (1977), 67-78. · Zbl 0344.57003
[4]	-, The rational homotopy groups of \( \operatorname{Diff} (M)\) and \( \operatorname{Homeo} (M)\) in the stability range, Proc. Conf. Algebraic Topology (Aarhus, 1978), Lecture Notes in Math., vol. 763, Springer, Berlin and New York, pp. 604-626.
[5]	D. Burghelea and R. Lashof, Stability of concordances and the suspension homomorphism, Ann. of Math. (2) 105 (1977), no. 3, 449 – 472. · Zbl 0393.55009 · doi:10.2307/1970919 · doi.org
[6]	Dan Burghelea, Richard Lashof, and Melvin Rothenberg, Groups of automorphisms of manifolds, Lecture Notes in Mathematics, Vol. 473, Springer-Verlag, Berlin-New York, 1975. With an appendix (”The topological category”) by E. Pedersen. · Zbl 0307.57013
[7]	A. E. Hatcher, Higher simple homotopy theory, Ann. of Math. (2) 102 (1975), no. 1, 101 – 137. · Zbl 0305.57009 · doi:10.2307/1970977 · doi.org
[8]	-, Concordance spaces, Proc. Sympos. Pure Math., vol. 32, Amer. Math. Soc., Providence, R. I., 1978.
[9]	W. C. Hsiang and B. Jahren, On the homotopy groups of the diffeomorphism groups of spherical space forms (preprint). · Zbl 0535.57015
[10]	R. Lashof, Embedding spaces, Illinois J. Math. 20 (1976), no. 1, 144 – 154. · Zbl 0334.57017
[11]	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. J. P. May, The geometry of iterated loop spaces, Springer-Verlag, Berlin-New York, 1972. Lectures Notes in Mathematics, Vol. 271. · Zbl 0285.55012
[12]	Friedhelm Waldhausen, Algebraic \?-theory of topological spaces. I, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 35 – 60. Friedhelm Waldhausen, Algebraic \?-theory of topological spaces. II, Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978), Lecture Notes in Math., vol. 763, Springer, Berlin, 1979, pp. 356 – 394.