Homotopy, stable homotopy, homology and cohomology, homotopical algebra.

**6**

votes

**1**answer

96 views

### Hochschild-Serre spectral sequence and non-trivial action on coefficients

Consider an extension\begin{equation}1\rightarrow N\rightarrow G\xrightarrow{\rho} K\rightarrow 1\end{equation}Let $K$ act on a $K$-module $A$ by $\phi_k: a\mapsto k\cdot a$. Define a $G$-action ...

**0**

votes

**1**answer

126 views

### Gauss Sums over “semisimple spherical tensor category”?

I read on the arXiv the following:
Let $\mathcal{\mathbf{C}}$ be a semisimple spherical tensor category with simple unit and let
$\mathbf{\Gamma}$ be the set of isomorphism classes of simple ...

**8**

votes

**0**answers

208 views

### Thom Spectra and Hopf-Galois Extensions of Ring Spectra

So I've been fiddling with this for a long time, so apologies to anyone that's already heard me talk about this ad nauseum. I haven't been able to get anywhere with it, and it seemed that as such, it ...

**8**

votes

**0**answers

93 views

### Is the $\infty$-topos $Sh(X)$ hypercomplete whenever $X$ is a CW complex?

It can be shown (see Is every paracompact, Hausdorff, locally contractible space homotopy equivalent to a CW complex?) that if $X$ is a locally contractible paracompact Hausdorff space such that the ...

**2**

votes

**0**answers

151 views

### Generator of $\pi_3(SU(4))$ in Mimura-Toda

In this paper of Mimura and Toda, tables are given for low-dimensional homotopy groups of $SU(3)$, $SU(4)$ and $Sp(2)$. As far as I understand it, Theorem 6.1 gives the generator of $\pi_3(SU(4))$ as ...

**23**

votes

**0**answers

556 views

### Finite complexes whose homotopy groups are not “finitely generated”

I'll say $K$ has "finitely generated" homotopy groups if there is a finite wedge of spheres $W = \bigvee S^{n_i}$ and a map $f: W\to K$ which induces a surjection on $\pi_*$.
It seems likely that ...

**0**

votes

**0**answers

84 views

### Dimension of category of sheaves [duplicate]

Let $k$ be a field. Consider the category $Shv(\mathbb{R}^n)$ of sheaves of $k$-vector spaces on $\mathbb{R}^n$. What is the cohomological dimension $d$ of $Shv(\mathbb{R}^n)$? I know that $d \in ...

**2**

votes

**1**answer

204 views

### How to calculate the first and second homotopy groups of the following space constructed from $U(4)$

In solving a physics problem, I came across a weird topological space constructed from $U(4)$, the group of $4\times4$ unitary matrices. I want to know the first two homotopy groups of it. Here is how ...

**4**

votes

**0**answers

225 views

### The general Smith homomorphism in bordism

The Smith homomorphisms are a family of homomorphisms between equivariant bordism groups in different dimensions.
One example that is known to be an isomorphism is the map
$$
\tilde\Omega_{d+1}^{\rm ...

**5**

votes

**0**answers

79 views

### A cdga for compactly supported cohomology (à la Sullivan's algebra of polynomial forms)

Let $M$ be a smooth manifold, and let $\Omega^\bullet(M)$ be the commutative dg-algebra of differential forms on $M$. It is quasi-isomorphic to the dg-algebra of singular cochains on $M$. If $M$ is no ...

**8**

votes

**1**answer

220 views

### Shriek push-forward for parameterized spectra

In May and Sigurdsson's Parameterized Homotopy Theory, Proposition 2.2.11, four isomorphisms of functors are given. For a pullback square of base spaces $C=holim(A\overset{f}\to B\overset{j}\leftarrow ...

**3**

votes

**3**answers

260 views

### Adams Spectral Sequence for Triangulated Categories

We have the Adams SS with
$$ E_2^{p,q} = Ext^{p,q} _{E^*(E)}([S,E],[S,E]) $$
where $E$ is the Eilenberg-Maclane Spectrum yielding $\mathbb{Z}/p$ coefficients.
I was wondering if there is a SS for ...

**3**

votes

**1**answer

121 views

### Does every canonical decomposition of the intersection form come from a canonical homology basis?

Take a closed surface $X$ of genus $n$. By a canonical homology basis, I will mean a set of $2n$ homologically independent simple closed curves $\{\alpha_1,\ldots,\alpha_n,\beta_1,\ldots,\beta_n\}$, ...

**6**

votes

**3**answers

445 views

### Is every paracompact, Hausdorff, locally contractible space homotopy equivalent to a CW complex?

Milnor proved that any paracompact Hausdorff space which is equi-locally convex (and hence in particular locally contractible) is homotopy equivalent to a CW complex. However, unlike being paracompact ...

**4**

votes

**1**answer

198 views

### Generators of cohomology groups of higher push-forward sheaves

Let $\phi:S\rightarrow \mathbb{P}^1$ be an elliptic fibration of a compact complex surface. Assume that there is a multiple section $s$ of $\phi$. Is it true that $H^0(\mathbb{P}^1,R^2f_*\mathbb{R})$ ...

**2**

votes

**1**answer

151 views

### Are there analogs of String Homology structure in cyclic homology?

I was reading John D.S. Jones' paper "Cyclic homology and equivariant homology" where he introduces a variant of cyclic homology that is isomorphic (as modules over the ring $K[u]$) to equivariant ...

**3**

votes

**1**answer

146 views

### Condition on a Hopf operad for tensor product in the base categoy to be a (categorical) coproduct for algebras

A Hopf operad will be an operad endowed with a coproduct $P(n) \longrightarrow P(n) \otimes P(n)$ which is compatible in the obvious sens with operad laws (no more structure is assumed a priori. ...

**16**

votes

**2**answers

540 views

### What's the relationship between these two isomorphisms involving G and T?

Let $G$ be a compact connected Lie group with maximal torus $T$ and Weyl group $W$. Recall the following two isomorphisms.
Isomorphism 1: $R(G) \cong R(T)^W$, where $R(-)$ denotes the representation ...

**5**

votes

**1**answer

209 views

### How to embed genus 4 surface inside $\mathbb{C}P^2\# \mathbb{C}P^2$ representing nontrivial homology class

As the title says, I want to embed the genus 4 surface inside $\mathbb{C}P^2\# \mathbb{C}P^2$ representing a nontrivial homology class.
I know that $H_2(\mathbb{C}P^2 \# \mathbb{C}P^2; ...

**12**

votes

**1**answer

322 views

### Fundamental theorem of K-theory for loop groups over $\mathbb{F}_1$?

As the title says, I would like to know what the fundamental theorem of algebraic K-theory would say over the field with one element. Recall that the fundamental theorem of K-theory provides a ...

**7**

votes

**0**answers

119 views

### $v_1$-periodic homotopy and principal bundle classification

This question came back to my mind while pondering this MO question. The classification of principal bundles is seriously difficult because of our lack of understanding of homotopy groups of compact ...

**10**

votes

**3**answers

716 views

### Which submanifolds are zero sets of $\mathbb{R}^n$-valued maps?

If $M$ is a smooth, compact, orientable manifold, then any framed submanifold $N$ is the preimage $f^{-1}(y) $ for a smooth sphere-valued map $f$ transversal to $y$, with the framing of the normal ...

**3**

votes

**0**answers

162 views

### Action of the mapping class group on separating spheres in a connected sum of aspherical 3-manifolds

Let $M$ be a connected sum of $g$ closed aspherical 3-manifolds $M_1, \ldots, M_g$. [Update: I also assume that all the $M_i$-s are diffeomorphic, i.e. $M$ is a connected sum of copies of the same ...

**1**

vote

**2**answers

135 views

### Pontryagin square of Stiefel-Whitney classes and Pontryagin classes

On a four-manifold, there is apparently a relation between the first Pontryagin class modulo 4 and the Pontryagin square of the second Stiefel-Whitney class:
$\mathfrak{P}(w_2) = p_1 \; {\rm mod} \; ...

**1**

vote

**0**answers

219 views

### Functors with Mayer-Vietoris Sequences

Let $F$ be a contravariant functor from some category of spaces (e.g. smooth manifolds or (compact?) topological Hausdorff spaces), to Abelian groups. Assume that for any open sets $U, V \subseteq X$ ...

**1**

vote

**0**answers

107 views

### Reference needed: Homology of the blow-up

Given an algebraic variety $X$ over the complex numbers.
Let $V$ be a subvariety of $X$ and $\pi_V \colon X' \rightarrow X$
be the blow-up of $X$ at $V$.
It is posible in general to compute the ...

**1**

vote

**1**answer

81 views

### Second cohomology on an open subset with complement of codimension 2

Let $X$ be a compact Kahler manifold of complex dimension $n$ and let $Y\subset X$ be an open subset such that $V:=X\setminus Y$ is of complex codimension 2.
I know that by Hartogs' theorem follows ...

**3**

votes

**1**answer

263 views

### Realizing homomorphisms between fundamental groups

Let $X,Y$ be compact connected manifolds and $\varphi\colon\pi_1(X)\to\pi_1(Y)$ be a homomorphism between their fundamental groups. Under what conditions on $X$, $Y$ and $\varphi$ is it true that ...

**8**

votes

**3**answers

468 views

### References for the “nerve of an algebraic variety”

Let's do algebraic geometry over an arbitrary base ring $k$.
I've frequently seen a definition of the algebraic $n$-simplex, as follows:
$$\Delta^n = ...

**7**

votes

**0**answers

223 views

### diameter as a Morse function

Consider the space $X_1$ of closed subsets not containing a pair of antipodal points of the unit circle. Here we have a kind of degenerate Morse function, defined by the diameter of the pointset. ...

**2**

votes

**1**answer

136 views

### Hodge isometry sending the Kahler class to its opposite

i would like to ask you a question i can not answer myself, i hope this is not too trivial and i'm not missing something too basic.
Let's suppose we have $X$ and $Y$ Kahler manifolds and ...

**12**

votes

**2**answers

361 views

### What is the coefficient ring of algebraic K theory of the discrete $\mathbb{C}$?

Ordinary (connective) complex $K$-theory is the algebraic $K$ theory of the topological ring $\mathbb{C}$ with analytic topology. One can also study the $K$ theory of $\mathbb{C}$ with discrete ...

**5**

votes

**3**answers

604 views

### Whitehead theorem for cohomotopy

Recall that the cohomotopy set $\pi^k(\mathcal{M})$ is $[\mathcal{M},S^k]$, i.e., the set of pointed homotopy classes of continuous mappings $\mathcal{M}\to S^k$. Recall also the Whitehead theorem:
...

**0**

votes

**1**answer

206 views

### What is the delooping of a looping?

What is $\mathbf{B}\Omega A$, where $A$ is a pointed object of an $(\infty,1)$ category with point $*\to A$, $\Omega A$ is the loop space of $A$, and $\mathbf{B}X$ is the delooping of $X$?
The ...

**5**

votes

**1**answer

158 views

### Pontryagin number for 4-dim surface bundle

In paper arXiv:math/0701247
"Divisibility of the stable Miller-Morita-Mumford classes" by Soren Galatius, Ib Madsen, Ulrike Tillmann, it was shown
that the Pontryagin numbers for a 4-dim surface ...

**3**

votes

**1**answer

193 views

### Postnikov towers in bounded t-structures

If $\mathcal{H}$ is the heart of a bounded t-structure in a triangulated category $\mathcal{T}$, then for every object $E$ in $\mathcal{T}$ there exists a finite sequence of integers ...

**0**

votes

**0**answers

274 views

### A Generalized De Rham cohomology

Edit According to the comment of Alex Degtyarev, I deleted the last part of the previous version.
Let $E$ be a real vector space. The complex valued $k$- tensors on $E$ is denoted by ...

**1**

vote

**0**answers

232 views

### Kunneth spectral sequence

In Rotman's Homological Algebra, 1st edition, there is written:
Is every detail of 11.31-11.35 correct? Isn't the spectral sequence in 11.35 1st quadrant and not 3rd quadrant? Do 11.34-35 also ...

**2**

votes

**0**answers

62 views

### Relations between Stiefel-Whitney classes on mapping torus

In question
Relations between Stiefel-Whitney classes
the relations between Stiefel-Whitney classes on manifold are obtained.
My question is that do we have additional relations between ...

**3**

votes

**1**answer

169 views

### Stable homotopy of classifying space for nilpotent groups

Let $BG$ denote the classifying space of a (discrete) group and $BG_+$ its disjoint union with a point.
Question: What is known about the stable homotopy groups $\pi^S_*(BG_+)$ ?
If $G$ is finite ...

**2**

votes

**1**answer

163 views

### Serre spectral sequence for cobordism

If I have a fibration, perhaps with twisting data respecting the fibration, is there a Serre spectral sequence computing cobordism of the total space?
An example that I'm particularly interested in ...

**4**

votes

**1**answer

348 views

### What was Seifert's contribution to the Seifert-van Kampen theorem?

The Seifert-van Kampen theorem is the classical theorem of algebraic topology that the fundamental group functor $\pi_1$ preserves pushouts; more often than not this is referred to simply as the van ...

**4**

votes

**0**answers

109 views

### Oriented cobordism group generated by mapping torus

The 4-dimensional oriented cobordism group of closed manifolds is $\Omega^{SO}_4=Z$, and we know that it cannot be generated by a mapping torus, since the Pontryagin for $p_1$ is zero for any mapping ...

**0**

votes

**3**answers

171 views

### Lifts across covering maps

Let $X,Y,Z$ be connected topological spaces, $f\colon X\to Y$ be a continuous map and $p\colon Z\to Y$ be a covering map. The problem is the existence of a continuous lift of $f$ across $p$. A ...

**7**

votes

**1**answer

226 views

### Are 4-dimensional mapping tori always spin?

We know that all compact orientable manifolds of dimension 3 are spin.
In 4 dimensions, $CP^2$ is not spin. I would like to ask if
all 4-dimensional compact orientable mapping tori are spin?
See ...

**1**

vote

**1**answer

112 views

### Relation between Stiefel-Whitney class and Chern class

A complex vector bundle of rank $n$ can be viewed as a real vector bundle of rank $2n$. From nLab, we have that the second Stiefel-Whitney class of the real vector bundle is given by the first Chern ...

**4**

votes

**2**answers

138 views

### Homology exponents for $QX$

We say that a space $X$ has a homology $p$-exponent if some power of $p$ annihilates the $p$-torsion in $H_\ast(X;\mathbb{Z})$.
I am interested in the homology exponents of the free infinite loop ...

**1**

vote

**1**answer

135 views

### Relation between Chern characteristic and Pontryagin characteristic

A 2-dim complex manifold can be viewed as a 4-dim real manifold.
What is the relation between the Chern characteristic and the Pontryagin characteristic of the tangent bundle? It should be $p_1=n_1 ...

**5**

votes

**1**answer

278 views

### Spectral Sequences reference

What is the best reference for spectral sequences for mathematicians who are not experts at the subject, but would just like to open a book and find the SS they need, without going in to deep.
I'm ...

**6**

votes

**1**answer

288 views

### First Cech cohomology of manifolds

Let $X$ be a compact connected manifold (with or without boundary) and let $H_1(X)$ denote its first Cech integral cohomology group or, equivalently, its first cohomotopy group. Is it true that ...