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Apr
10
comment The image of the Hurewicz map for rational loop spaces
I expect that is a counterexample, but if she didn't specifically point you to it, I doubt that it is the simplest counterexample.
Apr
10
comment Reference request: linearly independent cycles in a manifold
Is this a topology question or a linear algebra question? There are two ingredients to reduce to linear algebra: that the manifolds give homology and cohomology classes (so that you have a maps from $k^n$ to $H_j$ and $H^j$); and that transverse intersection is cap product. Now you have a linear algebra question about the rank of a bilinear form, specifically that a bilinear form with invertible determinant has full rank. So it detects every element of $k^n$, yet the form factors through $H_j$, so there was no kernel.
Apr
6
comment Topological $n$-manifolds have the homotopy type of $n$-dimensional CW-complexes
Actually, Tom knows from Milnor that a topological manifold has the homotopy type of a CW complex, which gives the comparison theorem.
Apr
6
comment Topological $n$-manifolds have the homotopy type of $n$-dimensional CW-complexes
Wall's theorem requires singular cohomology as input, so you need another theorem, probably also in Bredon, comparing singular cohomology to sheaf cohomology, just using the manifold hypothesis.
Apr
5
comment Topological $n$-manifolds have the homotopy type of $n$-dimensional CW-complexes
That is overkill and probably circular.
Apr
4
comment The image of the Hurewicz map for rational loop spaces
Jeff knows that it is not finitely generated in the commutative dga case. mathoverflow.net/questions/182437/…
Apr
3
comment Pseudomanifolds and Poincaré duality
No, that doesn't work. It is necessary that the parameterizing variety have $\chi=0$.
Mar
31
comment Pseudomanifolds and Poincaré duality
Here's a modification of the algebraic example that feels less "finite." You can interpret the elliptic curve example saying that elliptic curves have a canonical polarization, so the canonical bundle of curves on $M_{1,1}$ has a projective structure, thus there is an associated bundle of affine cones. Same for $M_g$. But there are more interesting complete curves in $M_g$, so those support a bundle of affine cones, so the singularities twist more.
Mar
28
comment Pseudomanifolds and Poincaré duality
(but if you are only interested in homology with $\mathbb Q$ coefficients, 2,3,4 are fine)
Mar
28
comment Pseudomanifolds and Poincaré duality
@DavidC maybe this example can be made algebraic. Replace the the hyberbolic automorphism with the order 6 (not 2,3,4) automorphism of the right elliptic curve. And replace the cone on an elliptic curve with an affine cone. The link of its singularity is not the elliptic curve, but the $S^1$ bundle over it. And replace the circle with a complete variety, say, another elliptic curve. But the surgery is hopeless, so twisted coefficients are out.
Mar
27
comment Pseudomanifolds and Poincaré duality
Also, the third bullet point in David's definition of pseudomanifold is common, but I think it's a silly axiom. It amounts to the normalization being connected. I guess the point is to reduce the number of orientations, just as a connected manifold has at most one. Connectedness hypotheses are usually a bad idea, but this one is also unwieldy.
Mar
27
comment Pseudomanifolds and Poincaré duality
I continue to doubt that there are any algebraic examples, but this one has a nice dualizing sheaf, so my earlier suggestion was inadequate to eliminate them.
Mar
27
answered Pseudomanifolds and Poincaré duality
Mar
22
comment Pseudomanifolds and Poincaré duality
My guess is that the semisimplicity of perverse sheaves shows that if the dualizing sheaf of a variety is not a local system (ie, if it is not a homology manifold), then it does not satisfy duality. At least with char zero coefficients: $\mathbb Q$-PD $\implies$ $\mathbb Q$-homology manifold. . . . Normal: good question.
Mar
22
comment Pseudomanifolds and Poincaré duality
How much do you care about the complex hypothesis? Here is a non-complex example: take a manifold of dimension at least 3, embed an interval, and glue the interval to itself by an involution that reverses the endpoints.
Mar
14
awarded  Yearling
Jan
15
comment Does every embedded 2-sphere in $\mathbb{R}^n$ bound an embedded ball?
Haefliger's theorem is optimal in the smooth category, but PL or Top, codimension 3 is all that is needed to unknot.
Dec
31
comment Example of a saturated class of morphisms which is not _obviously_ saturated?
I think that simple homotopy equivalences satisfy 2-out-of-3, but are not saturated in the category of finite complexes.
Dec
30
comment Lefschetz fixed notation
Is the word "stuff" supposed to indicate that while the local term is consistently not L, it is not consistently any other letter? Or perhaps that there is more to notation than the choice of letter?
Dec
21
awarded  Nice Answer