12
votes
What are some good examples of spectral sequences which degenerate after the first nontrivial differential?
Some examples with one nonzero family of differentials:
The classical Adams spectral sequence for $j/p$, the connective image-of-J spectrum reduced mod $p$, collapses at $E_3$, by Theorems 4.5 (at $p=...
10
votes
What are some good examples of spectral sequences which degenerate after the first nontrivial differential?
The Serre spectral sequence for the path-loop fibration for an $n$-sphere is a positive answer to question 1, a negative answer to question 2. More generally, a fibration in which either the base or ...
9
votes
What are some good examples of spectral sequences which degenerate after the first nontrivial differential?
One example is a construction that is often used in the passage from smooth projective varieties to arbitrary varieties. There are various variants of this:
For a proper variety $X$, take a ...
9
votes
Accepted
Does there exist a Bousfield localization of the category of spectra which makes the sphere unbounded below?
As discussed in the comments, the opposite of your premise seems to be true: Basically all interesting localisations seem to take non-bounded below values.
To substantiate this, let's classify the ...
- 8,729
8
votes
Accepted
Does a ring spectrum with even homotopy and even cells always have a polynomial algebra of homotopy groups?
Let $G$ be any discrete group, and let $MU[G] = MU \otimes \Sigma^\infty_+ G$ be the associated group algebra over $MU$. Additively, $MU[G] \simeq \bigoplus_{g \in G} MU$, and so it has both even ...
- 51.5k
7
votes
What are some good examples of spectral sequences which degenerate after the first nontrivial differential?
If $X$ is a smooth projective variety of dimension $r$ over $\mathbf{C}$ then the Leray spectral sequence of the (ordered) configuration space $F^n X$ of $n$ points on $X$ including into $X^n$ has ...
7
votes
What are some good examples of spectral sequences which degenerate after the first nontrivial differential?
Any DG algebra $A$ over a field has a minimal model, which is a minimal $A_\infty$-algebra $(H,m_3,m_4,\dots)$. It consists of a graded algebra $B=H^*(A)$ equipped with multi-linear operations of ...
7
votes
Accepted
Linking number and intersection number
$\DeclareMathOperator\tX{\widetilde{X}}\DeclareMathOperator\tB{\widetilde{B}}\DeclareMathOperator\tD{\widetilde{D}}\DeclareMathOperator\Z{\mathbb{Z}}$
In fact, $B$ must intersect $D$ at least $|\text{...
- 43.6k
6
votes
Cover the $n$-disc irredundantly with $n+1$ open sets. Suppose that the $(n+1)$-fold intersection is empty. Then is some $n$-fold intersection empty?
It is not true in general. Let $n=3$, and take a piece of a brick wall (thickness one brick) that is homeomorphic to a 3-disc. For example, just take a largish rectangular wall. There are bricks of ...
- 3,421
6
votes
Are Chern classes always vertical?
abx's counterexample is correct. It might be worth remembering the splitting principle, though: Let $E$ be any rank $n$ vector bundle on $M$, and let $F(E)$ be the bundle of complete flags in $E$, so $...
- 151k
5
votes
Accepted
Are Chern classes always vertical?
For a counter-example (with real coefficients), take for $M$ the Grassmannian $\mathbb{G}(p,p+q)$ with $p\neq q$, and $p,q\geq 2$. If I computed correctly:
$$c_2(M) = \frac{1}{2}\left[(p-q)^2-(p-q)+2\...
- 37.3k
5
votes
Accepted
Cover the $n$-disc irredundantly with $n+1$ open sets. Suppose that the $(n+1)$-fold intersection is empty. Then is some $n$-fold intersection empty?
Too long for a comment.
In the case $n=2$ and if the $U_i$ are connected, we can prove that $U_1\cap U_2\cap U_3$ is nonempty if the $U_i\cap U_j$ are nonempty: letting $p_1,p_2,p_3$ be in $U_2\cap ...
- 8,076
4
votes
Accepted
Relationship between quotient CW-complexes after attaching cells
If I understand the question correctly, you have a CW complex $Y'$ which is the union of two subcomplexes $Y$ and $X'$ whose intersection is the subcomplex $X$. We can first collapse $X$ to a point ...
- 19.4k
2
votes
Accepted
Pullback morphism of a hyperplane inclusion is zero in the derived category
You are asking whether a specific element of ${\rm Hom}_{D^b({\rm Ab})}(\mathbb Z[-2n], \mathbb Z[-2n-2])$ is zero. But this group is ${\rm Ext}^2_{\rm Ab}(\mathbb Z, \mathbb Z) = 0$. (Because $\...
- 3,673
2
votes
Is there an ∞-categorical interpretation of the Quillen S⁻¹S construction?
A published reference for the claim (right after the question in boldface) is the proof given by Thomason on pages 1657-1658 of "First quadrant spectral sequences in algebraic K-theory via ...
- 8,857
1
vote
Is there an ∞-categorical interpretation of the Quillen S⁻¹S construction?
Consider an $E_n$-monoid X. We can deloop $X$ to an $\infty$-category $\mathbf{B}X$. There's a natural functor $X^\circlearrowleft : \mathbf{B}X \rightarrow \text{Spc}$ given by the left action of $X$ ...
- 1,973
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