# Contents

## Idea

A notion of stratified space is supposed to be a notion of topological space that is not necessarily a manifold, but which is filtered into “strata” that are.

Examples include polyhedra, algebraic varieties, orbit spaces of many group actions on manifolds and mapping cylinders of maps between manifolds.

## Definitions

There are various ways in the literature to make this precise, including:

## References

A notion of purely topologically stratified sets was introduced in

• Frank Quinn, Homotopically stratified sets, J. Amer. Math. Soc. 1 (1988), 441–499. MR 89g:57050

Notions of smoothly stratified spaces were considered by

• H. Whitney, Local properties of analytic varieties, Differentiable and combinatorial topology (S. Cairns, ed.), Princeton Univ. Press, Princeton, 1965, pp. 205–244. MR 32:5924

• R. Thom, Ensembles et morphismes stratifiés, Bull. Amer. Math. Soc. 75 (1969), 240–282. MR 39:970

• J. Mather, Notes on topological stability, Harvard Univ., Cambridge, (1970)

Locally conelike stratified spaces have been considered in

• L. Siebenmann, Deformations of homeomorphisms on stratified sets, Comment. Math. Helv. 47 (1971), 123–165. MR 47:7752

These are all special cases of (Quinn’s definition).

Discussion of differential operators on stratified spaces is in

• R. B. Melrose, Pseudodifferential operators, corners and singular limits, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), 217–234, Math. Soc.

• Pierre Albin, Eric Leichtnam, Rafe Mazzeo, Paolo Piazza, The signature package on Witt spaces, I. Index classes (arXiv:0906.1568v2)

• Bruce Hughes, Geometric topology of stratified spaces (pdf)

Discussion of the fundamental category of a (Whitney‑)stratified space is in

• Jonathan Woolf, Transversal homotopy theory (arXiv:0910.3322)

• M. Banagl, Topological invariants of stratified spaces, Springer Monographs in Math. 2000.

A homotopy hypothesis for stratified spaces is discussed in

• David Ayala, John Francis, Nick Rozenblyum, A stratified homotopy hypothesis, arXiv

whilst an earlier paper on exit paths is

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