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Homotopy theory, homological algebra, algebraic treatments of manifolds.
3
votes
0
answers
273
views
Is there a spectral sequence for borel-moore homology associated to a whitney filtration?
Consider a Whitney stratified space
$$
\varnothing = X_{-1} \subseteq X_0 \subseteq X_1 \subseteq \cdots \subseteq X_n
$$
is there a spectral sequence for borel-moore homology which depends on the str …
3
votes
0
answers
425
views
Where should I look for computing the intersection homology of projective varieties?
I'm learning about intersection cohomology topologically through MacPherson's "New York Times Article". This is a very nice guide which gives a nice idea on how to use these methods for low-dimensiona …
4
votes
0
answers
226
views
Is there an analogue to the koszul complex for constructible sheaves?
Given a variety $X$ and a complete-intersection morphism
$$
Y \to X
$$
is there an analogue of the Koszul complex for $\mathcal{O}_Y \in \textbf{Coh}(X)$ in the setting of constructible sheaves? Meani …
3
votes
0
answers
229
views
How can I find the differential in the Serre spectral sequence for this sphere fibration?
Consider the assocaited sphere bundle $$S(E) \to \mathbb{P}^n$$ for the vector bundle $\mathcal{O}(k)\oplus \mathcal{O}(l) \to \mathbb{P}^n$. Is there a way to determine the differentials
$$
d_4^{p,m} …
10
votes
1
answer
618
views
Is the $E_\infty$-structure on the cochain complex of a $K(G,n)$ readily understandable?
One way to construct an $E_\infty$-algebra is to consider the cochain complex $C^*(X;M)$ for $X$ a topological space and $M$ a module over some ring $\Lambda$. From what I can recall, the $E_\infty$-a …
6
votes
1
answer
740
views
What tools can I use to compute the cohomology of the fibers of a Lefschetz Pencil?
I'm learning about Lefschetz pencils and vanishing cycles and have looked at a few sources:
http://www.math.purdue.edu/~dvb/preprints/sheaves.pdf
http://www3.nd.edu/~lnicolae/Morse2nd.pdf
Voisin's C …
4
votes
0
answers
198
views
How can I describe the monodromy of this variation of singular curves?
Consider the family of singular hyperelliptic curves
$$
y^2 - x(x-1)^2(x-2)(x-3)(x-4)(x-t)
$$
over $\mathbb{A}^1_t$. Over a generic point the fiber is a genus three curve where one of the genera comes …
8
votes
1
answer
676
views
Does an oriented $S^3$ fiber bundle admit the structure of a principal $SU(2)$-bundle?
Let $S \to X$ be an $S^3$-fiber bundle over a smooth manifold $X$. If $S$ is an oriented manifold does this fiber bundle admit the structure of an $SU(2)$-principal bundle?
There is a similar theorem …