In the definition of an $n$-dimensional stratified pseudomanifold one demands the following filtration $X=X_n \supset X_{n-1}=X_{n-2} \supset X_{n-3}\supset ... \supset X_0 \supset X_{-1}=\emptyset$. where all $X_i$'s are closed subspaces of X. Why is it required that $X_{n-1}=X_{n-2}$?
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2$\begingroup$ Levi -- this condition, together with the requirement that the 0-codimensional stratum is connected and orientable, ensures that we can define the fundamental class of $X$. $\endgroup$– algoriCommented Jun 3, 2011 at 14:05
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I think the reason must be that a pseudomanifold $V$ has singular locus $\Sigma V$ of codimension 2 or greater. (Stratifications of varieties are obtained by letting $X_{k-1}$ be the singular locus of $X_k$, or some refinement of that to get the Whitney conditions.) This codimension condition is reflected topologically by the nonbranching condition in the definition of pseudomanifold.
answered Jun 3, 2011 at 11:01
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It's not!
Okay, that's a little too glib. You're right that it's often required that $X^{n-1}=X^{n-2}$, but it depends somewhat on your purpose. In most of my recent papers, including those I'm current working on with Jim McClure, and in the papers of Saralegi (which look at intersection homology from an analytic point of view), there's freedom to include codimension one strata, but you need to be more careful about your definition of intersection (co)homology.
In even more detail: there are a variety of connected reasons why one would classically not work with codimension one strata. One reason has to do with the properties of classical intersection homology theory. Many of the results of intersection homology, including the ones that are approached sheaf theoretically, depend, at heart, on having a nice formula for the intersection homology on cones. If there are no codimension one strata around and you're working with perversities as defined by Goresky and MacPherson, then this cone formula works out, the sheaf of intersection chains is quasi-isomorphic to the Deligne sheaf, and away you go. However, with codimension one strata, and Goresky-MacPherson perversities, the cone formula doesn't quite work out. I have a detailed exposition about this in the following paper:
Greg Friedman, An introduction to intersection homology with general perversity functions, in Topology of Stratified Spaces; Greg Friedman, Eugénie Hunsicker, Anatoly Libgober, Laurentiu Maxim (editors) Mathematical Sciences Research Institute Publications 58, Cambridge University Press 2011, 177-222
Here's the link to the copy on my web site:
https://faculty.tcu.edu/gfriedman/papers/MSRI-revised-2.pdf
One argument that I make there looks carefully at what the perversity should be on such a stratum: if the perversity is 0 (as one might expect), then the perversity is greater than the top perversity allowed by Goresky and MacPherson (which would have to be -1 for a codimension one stratum), and that leads to the cone formula issues. On the other hand, if the perversity assigned is $\leq -1$, now you're below the bottom perversity $\bar 0$ allowed by Goresky and MacPherson, and that causes other issues.
Another way to think about codimension one strata is that they could arise if you took a manifold with boundary and let the boundary be a codimension one stratum. Then, as algori notes, you have a problem getting your fundamental class as a geometrically defined cycle (unless you sufficiently modify your definition of intersection homology). And that causes trouble with Poincare duality.
So, after all that, in summary: a pseudomanifold can have codimension one strata, but much of intersection homology breaks down. Unless you modify your intersection homology a bit, and then everything does work! See my papers or feel free to e-mail if you have further questions.
