Celebratio Mathematica — Kister — Complete Bibliography

Complete Bibliography

Works connected to Lawrence N. Mann

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J. M. Kister and L. N. Mann: “Isotropy structure of compact Lie groups on complexes,” Mich. Math. J. 9 : 1 (1962), pp. 93–96. MR 132120 Zbl 0111.35702 article

In this paper we prove the following conjecture of Floyd [Borel et al. 1960, p. 95]:

If \( G \) is a compact Lie group operating on a finite complex \( K \), then there are only finitely many distinct conjugate classes of isotropy subgroups.

In the proof we use a decomposition of \( K \) into a finite number of invariant open manifolds, each of which has an orientable covering manifold whose integral cohomology (with compact supports) is finitely generated. Lifting the action of \( G \) to the covering manifolds, we apply a result of Mann [1962] to establish the theorem. The theorem is false when \( K \) is locally-finite complex having finitely generated integral cohomology. To see this, consider the 2-complex consisting of a line and a sequence of closed discs, with centers on the line and going off to infinity. Define an action of the circle group \( S^1 \) on this complex by defining \( \theta \) in \( S^1 \) (\( 0 < \theta < 2\pi \)) to act as the rotation \( j\theta \) on the \( j \)th disc. Then \( Z_j \) (the subroup of \( S^1 \) isomorphic to the integers, modulo \( j \)) leaves the \( j \)th disc point-wise fixed, and therefore \( Z_j \) is an isotropy subgroup for each \( j \). On the other hand, the one-point compactification of this complex is of the same homotopy type as the circle, and therefore the finitely generated integral cohomology condition is satisfied.

J. M. Kister and L. N. Mann: “Equivariant imbeddings of compact abelian Lie groups of transformations,” Math. Ann. 148 : 2 (April 1962), pp. 89–93. MR 141727 Zbl 0209.44104 article