A split action associated with a compact transformation group

Author:
Sol Schwartzman

Journal:
Proc. Amer. Math. Soc. **83** (1981), 817-824

MSC:
Primary 57S05; Secondary 54H15

MathSciNet review:
630061

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Abstract: We associate with an effective action of a compact connected Lie group as a pathwise connected space a split action of a quotient group on the quotient space . One application of the main theorem states that if is a compact oriented manifold whose principal cohomology class is a cup product of one-dimensional classes then the action of on splits. We prove this in the differentiate case; the topological case has since been dealt with by Schultz.

**[1]**P. E. Conner and D. Montgomery,*Transformation groups on a 𝐾(𝜋,1). I*, Michigan Math. J.**6**(1959), 405–412. MR**0123333****[2]**P. E. Conner and Frank Raymond,*Actions of compact Lie groups on aspherical manifolds*, Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969), Markham, Chicago, Ill., 1970, pp. 227–264. MR**0271958****[3]**P. E. Conner and Frank Raymond,*Injective operations of the toral groups*, Topology**10**(1971), 283–296. MR**0281218****[4]**P. E. Conner and Frank Raymond,*Injective operations of the toral groups. II*, Proceedings of the Second Conference on Compact Transformation Groups (Univ. Massachusetts, Amherst, Mass., 1971) Springer, Berlin, 1972, pp. 109–123. Lecture Notes in Math., Vol. 299. MR**0646079****[5]**Dan Burghelea and Reinhard Schultz,*On the semisimple degree of symmetry*, Bull. Soc. Math. France**103**(1975), no. 4, 433–440 (English, with French summary). MR**0397756****[6]**Reinhard Schultz,*Group actions on hypertoral manifolds. I*, Topology Symposium, Siegen 1979 (Proc. Sympos., Univ. Siegen, Siegen, 1979), Lecture Notes in Math., vol. 788, Springer, Berlin, 1980, pp. 364–377. MR**585669**

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1981-0630061-4

Keywords:
Equivariant map,
split action

Article copyright:
© Copyright 1981
American Mathematical Society