# On the topology of cyclic products of spheres

@article{Liao1954OnTT, title={On the topology of cyclic products of spheres}, author={S. Liao}, journal={Transactions of the American Mathematical Society}, year={1954}, volume={77}, pages={520-551} }

We propose to study certain topological properties of cyclic products of spheres. Applications may be made in the theory of fiber bundles('). Our success is as follows. We shall introduce first the notion of "regular imbedding of a space into its r-products" (this notion appears in a more general situation). Let X be a locally compact, paracompact Hausdorff space. By the r-product X" of X we mean the orbit space over a q-fold Cartesian product Xq of X on which a group r of permutations of… Expand

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#### References

SHOWING 1-10 OF 17 REFERENCES

A Theorem on Periodic Transformations of Homology Spheres

- Mathematics
- 1952

In his theory of periodic transformations in a topological space' Smith proved the following theorem: THEOREM A2. Let X be a compact Hausdorff space of finite dimension which is a homology n-sphere… Expand

On the Theory of Obstructions

- Mathematics
- 1951

N. E. Steenrod has solved1 the homotopy classification problem for maps K -+ S', where n > 2 and K is an (n + 1)-dimensional polyhedron. His solution is stated in terms of separation cochains and… Expand

Vector Fields on the n-Sphere.

- Physics, Medicine
- Proceedings of the National Academy of Sciences of the United States of America
- 1951

This chapter discusses vector fields on the n-sphere, where any set of 2 k continuous vector fields tangent to S n are somewhere dependent. Expand

A Certain Exact Sequence

- Mathematics
- 1950

Page INTRODUCTION 51 CHAPTER I. THE SEQUENCE 2(C, A) 53 1. Definition of 2(C, A) 53 2. The secondary modular boundary operator 55 3. Induced homomorphisms of 2 55 4. Combinatorial realizability 57… Expand

Homology Groups of Symmetric Groups and Reduced Power Operations.

- Mathematics, Medicine
- Proceedings of the National Academy of Sciences of the United States of America
- 1953

The Iteration of the Steenrod Squares in Algebraic Topology.

- Computer Science, Medicine
- Proceedings of the National Academy of Sciences of the United States of America
- 1952

5«?- les puissances de Steenrod

- Colloques de Topologie de Strasbourg,
- 1951

The (n+3)-dimensional homotopy group of the n-sphere

- Bull. Amer. Math. Soc. vol
- 1951