## New answers tagged at.algebraic-topology

2
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 ...

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3
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 ...

Community wiki

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{...

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 $...

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\...

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 ...

7
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 ...

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
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 ...

10
votes

Accepted

### Does the Atiyah-Hirzebruch spectral sequence for $E^\ast(X)$ collapse whenever $E$ is complex-oriented and $X$ has even cells?

Yes, this is true.
As you commented, $X$ has finitely generated free homology, and so the $E_2$-term of the AHSS can be identified with
$$
H^*(X; E^*) \cong E^* \otimes H^*(X).
$$
To show collapse, we ...

5
votes

### What is the center of Morava $K$-theory?

As others pointed out in the comments, the topological Hochschild cohomology of $K(n)$ (i.e., the center of $K(n)$) was calculated by Angeltveit. Here is the paper: https://doi.org/10.2140/gt.2008.12....

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4
votes

### Derivations in the Steenrod algebra

I have a guess for question 1. Fix $n \geq 0$ and let $E(n)$ be the Hopf subalgebra dual to $\mathbb{F}_2 [\xi_{n+1}, \xi_{n+2}, \dots] / (\xi_i^{2^{n+1}})$. Every $x\in E(n)$ satisfies $x^2=0$, and ...

13
votes

Accepted

### Derivations in the Steenrod algebra

The $D$ with $D(xy) = xD(y) + D(x)y$ are the primitives in the Steenrod algebra $A$, which are dual to the indecomposables $\xi_i$ in $A_* = F_2[\xi_i \mid i\ge1]$, so there is one such $D$ in each ...

3
votes

Accepted

### homotopic to a constant map

Let $X = \Bbb{RP}^n$ for $n \geq 2$. I claim that the map $X \to X^3 / M$ is nontrivial on mod-2 cohomology.
Here is some general material.
For $1 \leq i \leq 3$, let $p_i: X^3 \to X^2$ be the ...

11
votes

Accepted

### Approximate classifying space by boundaryless manifolds?

Yes, this is possible. The construction below is fairly standard and I'm not sure where I learned it.
The aim is to show that for each compact Lie group $G$, there is a closed $n$-connected manifold $...

5
votes

### Does there exist a GRR-like generalization of the AS Index Theorem?

I'm sorry for the self-citation. But your question is largely answered in the monograph Coherent Sheaves, Superconnections, and Riemann-Roch-Grothendieck, or the arxiv version, joint work of Jean-...

4
votes

### Algebraic K-theory of a ring

The comments point out that this question is very similar to another question, but an attempt to close as a duplicate failed. So, rather than leave this question on the unanswered queue, I will try to ...

0
votes

### Does the classification diagram localize a category with weak equivalences?

Sorry to be (very) late to the party. As was already noticed by OP in the comments, the assignment $(C,W)\mapsto N(C,W)$ can naturally be extended to a functor
$$N:\mathsf{sSet}^+\to\mathsf{sSet}$$
...

15
votes

Accepted

### Identifying two definitions of orientation on a vector space

Here's a direct way to relate the two:
One more structure is additivity of orientations. For $V, W$ of dimensions $n,m$, we have a canonical pairing
$$
\Lambda^n V \otimes \Lambda^m W \cong \Lambda^{n+...

3
votes

Accepted

### (Derived category of) sheaves over an infinite union

An easier argument would be that $R^q\pi_\ast \mathbf Q$ vanishes for $q\notin \{0,3\}$ and has rank $1$ for $q \in \{0,3\}$. Indeed, just check this on stalks. Then the cone of $\mathbf Q\to R\pi_\...

6
votes

Accepted

### What is the group completion of finite sets with respect to cartesian product?

As already addressed in the comments:
Group completing the groupoid of finite pointed sets under the smash product gives a contractible space.
The groupoid of finite sets under the cartesian product ...

3
votes

Accepted

### Comparing Kummer maps to étale homotopy at finite level

The key problem here is something so subtle I didn't notice it the first time I read your question - apologies.
of course $\pi_1^{\et}(\mathbb{G_m},1)\cong \widehat{\mathbb{Z}}$
Of course, "of ...

2
votes

### Is cohomology with local coefficients a representable functor?

Floris van Doorn's Ph.D. thesis On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory gives a very nice account of parametrized cohomology (which generalizes cohomology with ...

4
votes

Accepted

### Extending curves on a surface to a basis for its first homology satisfying intersection criteria

If I understand your question correctly, what you’re looking for is Lemma A.3 in my paper here.

3
votes

Accepted

### Bar construction in commutative algebras is calculated by pushout

This is just to be explicit about the role of the bar construction in David's answer.
If $I$ is the category $a \leftarrow b \rightarrow c$ parametrizing pushout diagrams, then there is a functor $f: ...

3
votes

### Does coproduct preserve cohomology in differential graded algebra category

The coproduct in the category of non-unital dg algebras is maybe easier to think about. Indeed note that the two relations in Jardine's note only have to do with the units in the two algebras. The non-...

9
votes

### Besides $F_q$, for which rings $R$ is $K_i(R)$ completely known?

In the ten years since this question was asked, there has been a lot of progress in algebraic $K$-theory. For example, Achim Krause, Ben Antieau, and Thomas Nikolaus came up with an algorithm to ...

7
votes

### Homotopy theory and algebraic topology last 10 years. Is it a dying field?

If the criterion is “results using algebraic topology which shocked the mathematical community in the last 10 years”, then how about Abouzaid and Blumberg’s proof of the Arnol’d conjecture using ...

Community wiki

2
votes

### Bisimplicial spaces as a coequalizer of maps between "simpler" bisimplicial spaces

I certainly respect going back to primary sources. But, in this case, it's helpful to remember that a LOT has been written about bisimplicial sets since Quillen's 1973 paper. For example, a reference ...

2
votes

### Vector bundles over a homotopy-equivalent fibration

As indicated in the comments, this question ended up being accidentally rather trivial.
Specifically, the following three facts are fairly well-known and rather easy to establish:
For homotopic ...

8
votes

### Homotopy theory and algebraic topology last 10 years. Is it a dying field?

No, it's not dying at all. If anything, now is the best time to do homotopy theory. Thanks to the recent work of Lurie and others, homotopy theory is easier than ever to get into (advances have ...

Community wiki

7
votes

### Bar construction in commutative algebras is calculated by pushout

A way to see this which doesn't dive into the specifics of the simplicial diagram "$C\otimes D^{\otimes n}\otimes E$" is to apply 3.2.4.7 to the symmetric monoidal $\infty$-category $Mod_D(\...

6
votes

### Bar construction in commutative algebras is calculated by pushout

Welcome to MathOverflow! First, let me point out that what you're asking is already true at the 1-categorical level. The pushout in the category of commutative rings is computed by the tensor product. ...

5
votes

Accepted

### $\operatorname{Spaces}/BG$ $\sim$ $\operatorname{Spaces}^G$ $\sim$ $??(\Omega G)$

If $A$ is a braided ∞-group, the delooping $\def\B{{\sf B}}\B A$ is an ∞-group.
Consider the ∞-category of spaces equipped with an action of the ∞-group $\B A$.
Since $\B Ω G≃G$, this ∞-category is ...

3
votes

Accepted

### Regular sequence in cohomology of Grassmannians

Modulo any prime ideal of the quotient ring, the product of polynomials
$$(1 + x_1 t + x_2 t^2 + \dots + x_m t^m) (1 + y_1 t + y_2 t^2 + \dots + y_n t^n ) $$ $$= 1 + (x_1+y_1) t + (x_2 + x_1 y_1 + y_2)...

2
votes

Accepted

### Homology of independence complex after removing a vertex

There is a fairly standard splitting technique as suggested in the comment, although not necessarily using $v$.
Let $G$ be a graph and $v$ is a simplicial vertex. Let $w$ be any neighbor of $v$. Then ...

1
vote

### coset poset of reflection subgroup

I assume that $W$ is finit (not just $S$) and I take parabolic subgroups rather than reflection subgroups. Then the coset poset is indeed Cohen-Macaulay.
In the recent work Cluster Parking Functions, ...

3
votes

Accepted

### Isn't every algebraic operad equipped with a trivial weight?

The comment of Tom Goodwillie is correct. In the book, a connected weight graded operad is one that decomposes as:
$$\mathcal{P} = \mathbb{K} \mathrm{id} \oplus \mathcal{P}^{(1)} \oplus \mathcal{P}^{(...

1
vote

### Space of the trivial long knot in the thickened surface

Let us show that $\mathcal E=Emb_0(I,F\times I)\sim\Omega_0(F,x_0)$.
We start with R. Budney's remark.
Proposition. Let $F$ be a connected compact 2-manifold and $P(F)$ the pseudoisotopy group, i.e ...

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