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

**5**

votes

**1**answer

317 views

### Iterated Homotopy Quotient

If one has a normal Lie group inclusion $H\to G$, with quotient $G/H$, and a $G$-manifold $X$, one can take the quotient $X/H$. Then the $G$-action on $X/H$ factors thru a $G/H$-action, so one can ...

**6**

votes

**2**answers

261 views

### Seifert--van Kampen for the loop space dga

I recently noticed that I could mostly prove a special case of the following statement. I think it's true in general, though perhaps only for nice spaces.
Let $X$ be a topological space, and ...

**6**

votes

**1**answer

204 views

### canonical action of symmetric groups on orthogonal groups

There is a canonical faithful orthogonal representation of the symmetric group $S_{n+1}$, for $n\geq 1$:
$$
S_{n+1}\to O(n)
$$
given as follows.
(1). I regard $O(n)$ as the isometry group of the unit ...

**6**

votes

**3**answers

300 views

### Examples of Stiefel-Whitney classes of manifolds

Let $M$ by an compact, connected $n$-dimensional manifold without boundary.
Are there any other computable examples of the Stiefel-Whitney class $w(M)$ except for $M=S^m, \mathbb{R}P^m,\mathbb{C}P^m, ...

**5**

votes

**1**answer

109 views

### symmetric group of regular polyhedrons

Let $\Delta^n$ be the regular $n$-simplex spanned by $(n+1)$ vertices, equipped with an Riemannian metric such that all the edges are of equal length. For example,
$\Delta^2$:
$\Delta^3$:
Let ...

**12**

votes

**2**answers

471 views

### formula for Eta invariant

Hirzebruch's signature formula is not valid for manifolds with boundary.
An error term is introduced by Atiyah-Patodi-Singer to fix it.More precisely:
$$sign (M)=L(M)[M]+\eta(\partial M)$$
Yet ...

**1**

vote

**1**answer

192 views

### Banach algebraic proof of the Borsuk Ulam theorem

I am wondering whether there exists a proof of the Borsuk Ulam theorem
for the Euclidean n-sphere that is based only on the theory
of Banach algebras. I checked on MR but had no success.

**1**

vote

**0**answers

103 views

### Cofibre of the $n$-fold transfer $\mathbb{R}P_+^{\wedge n}\to S^0$

I want to know what is known about the cofibre of the $n$-fold transfer map $\mathbb{R}P^{\wedge n}_+\to S^0$, for $n>1$. I am happy to know of any specific example worked out. The case $n=1$ is ...

**6**

votes

**0**answers

93 views

### cohomology ring of stable configuration spaces

Let $M$ be a compact Riemannian manifold without boundary. Distinct $k$-points $x_1,\cdots,x_k\in M$ are called stable if the potential energy given by coulomb forces among $k$ electrons reaches ...

**16**

votes

**5**answers

818 views

### What are examples when the equality of some invariants is good enough in algebraic topology?

As far as my understanding goes, most of the tools of algebraic topology (homotopy groups, homology groups, cup product, cohomology operations, Hopf invariant, signature, characteristic classes, knot ...

**10**

votes

**2**answers

383 views

### Do there exist “topologically significant” (and not “algebraic”) triangulated categories killed by the multiplication by $p$?

I have a somewhat vague question: does there exist a prime $p$ and a triangulated category killed by the multiplication by $p$ that would be "interesting for topologists"? This category would probably ...

**19**

votes

**0**answers

391 views

### Finding the octonionic analog of the K3 surface, via (almost) hyperkahler geometry?

The K3 manifold is an amazing object in mathematics which plays an important role in several fields ranging from the study of smooth 4-manifolds to algebraic geometry to differential geometry and ...

**8**

votes

**1**answer

285 views

### electron configuration on manifolds

Let $M$ be a Riemannian manifold. For $k\geq 2$, suppose there are $k$ particles whose mass and volume can be regarded as zero and negatively charged with electricity equally. These $k$ particles move ...

**3**

votes

**2**answers

264 views

### Is it possible to compute coefficients of the formal group of an elliptic curve?

This may be trivial but I cannot find it. Given an elliptic curve, $C$ over $R$ with chosen parametrization $R\to \mathbb{A}^5_{\mathbb{Z}}=Spec(A)$, is there a way to compute coefficients of the ...

**14**

votes

**1**answer

264 views

### Moduli space of boundary maps with prescribed chain and homology groups?

Let $R$ be a reasonable ring (maybe I mean a PID, or $\mathbb{Z}$, and when sufficiently desperate, a field). Now consider fixed sequences $C_n$ and $H_n$ of $R$-modules, which are tame in every ...

**3**

votes

**1**answer

165 views

### Reference: Betti Numbers of the free loop space are finite

let $M$ be a compact, simply connected Riemannian manifold with dimension $< \infty$. I'm looking for a reference that
$$ \dim H_k(\Lambda M, \mathbb{Z}) < \infty, $$
is true in that case. Here ...

**0**

votes

**0**answers

49 views

### Three-manifolds related by degree 1 map, whose products with the two-sphere are diffeomorphic

Suppose $M$ and $N$ are compact oriented smooth 3-manifolds such that there is an orientation-preserving diffeomorphism between the products $F:M \times S^2 \to N \times S^2$. Further, there are ...

**2**

votes

**0**answers

110 views

### Do complex schemes locally deformation retract onto closed subschemes in the analytic topology?

Let $X$ be a scheme of finite type over $\mathbb{C}$ and let $Z \hookrightarrow X$ be a closed subscheme. Consider the associated closed inclusion $Z_{an} \hookrightarrow X_{an}$ between their ...

**4**

votes

**1**answer

73 views

### distinct multiple points in a space with at least one point lying in a subspace

Let $X$ be a topological space and $A$ a subspace of $X$. Given $k\geq 2$, let the unordered configuration space be
$$
B(X,k)=\{(x_1,x_2,\cdots,x_k)\in X^k\mid x_i\neq x_j \text{ for any } i\neq j\}
...

**4**

votes

**1**answer

171 views

### Kunneth formula of Cartesian product modulo orders of coordinates

Let $X$ be a topological space and $F$ a field. Let the $n$-th permutation group $\Sigma_n$ act on
$$
\prod_n X
$$
by
$$
\sigma(x_1,\cdots,x_n)=(x_{\sigma(1)},\cdots,x_{\sigma(n)}), \sigma\in ...

**5**

votes

**0**answers

138 views

### On a very weak notion of fibration (of topological spaces)

Suppose that $f:Y \to X$ is a map of topological spaces, and lets assume for simplicity that $X$ is connected. For the fibers of $f$ to compute the homotopy fibers, one would usually want to demand ...

**5**

votes

**1**answer

147 views

### “Database” of simplicial polytopes/spheres

Reading through various papers on polytopes I have come across really interesting examples of simplical polytopes and non-shellable (or non-PL) simplicial spheres but sometimes it is hard to keep ...

**36**

votes

**1**answer

569 views

### Is there an octonionic analog of the K3 surface, with implications for stable homotopy groups of spheres?

The infamous K3 surface has many constructions in many fields ranging from algebraic geometry to algebraic topology. Its many properties are well known. For this question I am really interested in the ...

**20**

votes

**2**answers

819 views

### Structure of Hopf algebras - trouble understanding an old paper

UPDATE: I am grateful to Peter May for the accepted answer, which makes most of the details below irrelevant. However, I will leave them in place for the record.
I am trying to understand the proof ...

**2**

votes

**2**answers

212 views

### Is the “inverse” (i.e., the “cohomological”) numeration for singular (i.e., $H\mathbb{Z}$-)homology of spectra “acceptable”? [closed]

I have recently realized that in one of my (published) papers I have used the "inverse" numeration for the $H\mathbb{Z}$-homology of the objects of the stable homotopy category (so, if we consider ...

**17**

votes

**2**answers

468 views

### Why study the p-completions of a space?

Given a nice topological space $X$ there are various notions of a 'completion' at a set of primes. Some of the most common constructions may be found in Bousfield-Kan's, May's, Neisendorfer's or ...

**4**

votes

**1**answer

239 views

### Does the (singular)cohomology of any acyclic spectrum vanish?

I am interested in those objects in the ("topological") stable homotopy category $SH$(I call them spectra) whose homology (with integral coefficients; should I call it singular or stable, or ...

**6**

votes

**0**answers

177 views

### How do the direct and inverse image sheaf functors interact with homotopy?

This is a crosspost of this MSE question.
The direct image sheaf functor $f_\ast$ and inverse image sheaf functor $f^\ast$ (here I mean the usual inverse image sheaf functor often denoted by ...

**3**

votes

**0**answers

40 views

### group action on Stiefel manifolds [duplicate]

In the paper The cohomology rings of real Stiefel manifolds
with integer coefficients, it is stated that
Question:
Suppose the permutation group $\Sigma_k$ acts on $V_{n,k}$ by permuting the order ...

**17**

votes

**1**answer

504 views

### stable homotopy groups and zeta function

I have heard during a discussion that there is a well known relation between the stable homotopy groups of a sphere (more precisely the order of stable homotopy groups of localized sphere spectrum ...

**2**

votes

**1**answer

96 views

### cohomology ring of the fundamental group of unordered configuration space

From the lecture notes INTRODUCTION TO CONFIGURATION SPACES AND THEIR
APPLICATIONS, p. 18, I find:
Os it possible to derive the cohomology ring $H^*(Conf(S,k)/\Sigma_k;\mathbb{Z}_2)$ from the above ...

**2**

votes

**1**answer

98 views

### Minimal model (resolution) for a specific colored operad

We know that for the operad $As:=\mathcal{F}(\mu)/(\mu\circ_1\mu-\mu\circ_2\mu)$, its minimal model is the free operad $\mathcal{F}(E)$ where $E=\mathbb{k}<\mu_2,\mu_3,\dots,\mu_n,\dots>$ is the ...

**1**

vote

**2**answers

86 views

### Retract embedding of $S^{n}$ in its unit tangent bundle

Acording to the comment of Mark Grant and the answer of Ryan Budney, I revise the question:
For what even $n$, there is a retract embedding of of $S^n$ in its unit tangent bundle?

**12**

votes

**1**answer

292 views

### Fibrant-cofibrant models of Eilenberg-MacLane spectra

There are many models for spectra, by which I mean a model category whose homotopy category is triangulated-equivalent to the stable homotopy category. In each model, there are ways to construct ...

**7**

votes

**1**answer

127 views

### Obstructions to Picard-graded groups of maps

Suppose $(C,\odot,\Bbb I)$ is an additive category with a compatible symmetric monoidal structure and $Pic(C)$ is the group of isomorphism classes of objects which have an inverse under $\odot$. For ...

**7**

votes

**1**answer

316 views

### Homotopy types of schemes

Let $X$ be a scheme over $\mathbb{C}$.
When does the topological space $X\left(\mathbb{C}\right)$ of $\mathbb{C}$-points have the homotopy type of a finite CW-complex?
When does the topological ...

**7**

votes

**0**answers

104 views

### How stable is the top cell of Lie group?

It is well know fundamental class of a compact lie group $G$ is stably spherical (see "H-Spaces and Duality" by Browder and Spanier, or "Thom Complexes" by Atiyah), and there is a stable equivalence ...

**7**

votes

**1**answer

187 views

### Is every locally compactly generated space compactly generated?

[Parse it as (locally compact)ly generated.]
I stumbled across this one whilst supervising an undergraduate thesis. Convenient categories for homotopy theory (e.g. CGWH) have been discussed here ...

**23**

votes

**2**answers

844 views

### Why is there a duality between spaces and commutative algebras?

1) The category of affine varieties over $\mathbb{C}$ is equivalent to the opposite category of finitely generated algebras over $\mathbb{C}$. The equivalence associates to an affine variety its ...

**4**

votes

**0**answers

47 views

### Stiefel manifolds and “simplicial complex chromated Sitefel manifolds”

Let $K$ be a simplicial complex whose vertices are labelled by $1,2,\cdots,k$. I want to define a variant concept of the open Stiefel manifolds
$$
...

**2**

votes

**1**answer

100 views

### Which bordism classes fiber over the circle?

Let $\mathcal{G}$ denote a (stable) tangential structure such as $O$, $SO$, $Spin$, or $Pin^\pm$. Which bordism classes $[M,f]\in\Omega_*^\mathcal{G}(X)$ are represented by an $f:M\rightarrow X$ where ...

**3**

votes

**0**answers

72 views

### Cohomology of the classifying space of some Super Lie group

Are there any papers on the cohomology of the classifying space of the general linear supergroup $GL(n, m)$ or unitary supergroup $U(n, m)$?
I know basically nothing about supergeometry. It seems ...

**1**

vote

**0**answers

60 views

### Surjectivity of maps between spheres [closed]

I am wondering how to prove that a non-zero degree map from $S^n \to S^n$ is surjective. For example, identifying $S^1 \subset \mathbb{C}$, we can take $f:S^1 \to S^1$ via $f(z) = z^k$ with $k\neq 0$. ...

**4**

votes

**4**answers

458 views

### When is the quotient by an $n$-fold loop space an $m$-fold loop space?

Given a map of $n$-fold loop spaces $X\to Y$, we can take the homotopy cofiber, denote it $Y/X$ (all spaces here will also have a base point, and all maps pointed). I have some basic questions about ...

**2**

votes

**0**answers

109 views

### Serre spectral sequence to compute cup product of bundles

Could we use Serre spectral sequence to compute cup product $\cup$ for the fibration
$$
F\to E\to S^m?
$$
My supervisor said this is true and wrote
$$
H^*(S^m)\to H^*(E)\to H^*(F),\\
\alpha\to ...

**1**

vote

**0**answers

51 views

### cohomology ring of iterated loop-suspension of spheres

In the book The unstable Adams spectral sequence for free iterated loop spaces (http://www.ams.org/bookstore?fn=20&arg1=memoseries&ikey=MEMO-36-258), Corollary 3.14:
Question: given $n$, ...

**5**

votes

**1**answer

547 views

### Schematization of a topological space

I wanted to understand or at least to know if what follows make sense.
Given a connected toplogical space $X$, I want to associate a scheme. In the following way.
For a space $X$ and $A(X)$ the ...

**2**

votes

**1**answer

150 views

### generalized universal coefficient sequence

Take the familiar Universal Coefficient Theorem for ordinary homology with $\mathbb{Z}$-coefficients and ordinary cohomology with coefficients in some abelian group $A$:$$0\rightarrow ...

**7**

votes

**0**answers

144 views

### Which -icial sets produce the “standard” representations of symmetric groups?

Suppose you have a system of cell complexes (say, even convex polyhedra) $(P_n)_{n\geqslant0}$ which occur as faces of each other and are used to define the corresponding notion of "$P_*$-set". So ...

**2**

votes

**1**answer

89 views

### dimension of generators of cohomology ring of iterated loop-suspension

In the book The unstable Adams spectral sequence for free iterated loop spaces, R.J. Wellington, Mem. Amer. Math. Soc. 258, 1982, p. 32
Question: When $p=2$, $k\geq 1$, $n=0$ to $\infty$, what kind ...