# Tagged Questions

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

**4**

votes

**2**answers

410 views

### Generalized Jordan theorem and winding number

By the generalized Jordan theorem any continuous injective map
$S^{n-1} \hookrightarrow R^n$ splits $R^n$ into two regions, one being bounded (interior) and the other one unbounded (exterior). It ...

**6**

votes

**0**answers

190 views

### Reference request: Whitehead product and the Borel construction

This is a question about signs.
Fix
a based space $(X,x_0)$,
a topological group $G$
acting on $X$ from the left, so that the basepoint $x_0$ is fixed,
a based map $\alpha\colon S^p\to G$ ...

**0**

votes

**1**answer

172 views

### Proper actions and diffeomorphism groups

Since the diffeomorphism group is not locally compact; is it true that there is no proper action of an infinite-dimensional diffeomorphism group on a finite-dimensional smooth manifold?
Edit: The ...

**20**

votes

**1**answer

267 views

### Primary obstruction to the existence of a cross-section of $V_{n - q}(\omega)$ is a cohomology class in $H^{2q+2}(B, \pi_{2q+1} V_{n - q}(F))$?

Let $V_{n - q}(\mathbb{C}^n)$ denote the complex Stiefel manifold consisting of all complex $(n - q)$-frames in $\mathbb{C}^n$, where $0 \le q < n$. This manifold is $2q$-connected, and$$\pi_{2q + ...

**1**

vote

**0**answers

68 views

### Rational homotopy groups of unordered configuration spaces of the torus

Is there any computations or investigation about the rational homotopy groups of unordered configuration spaces of the torus?
Any help is welcome

**11**

votes

**3**answers

387 views

### $A_{\infty}$-structure on closed manifold

Is there an exmaple of a closed smooth connected manifold $M$ having a structure of $A_{\infty}$-space (with unit) but $M$ is not homeomorphic to a compact connectd Lie group as space ?
Edit: First, ...

**8**

votes

**1**answer

239 views

### “Lagrangian” subalgebra of cohomology, with respect to Poincare duality?

Let $M$ be a compact oriented $n$-manifold, and let $H^*(M)$ denote
its cohomology ring with coefficients in $\mathbb{R}$.
Let's say that a graded subalgebra $K^\bullet \subset H^\bullet(M)$ is a
...

**13**

votes

**1**answer

250 views

### Do vanishing characteristic classes of the tangent bundle imply a manifold is stably frameable?

Suppose we know that all stable characteristic classes of the tangent bundle of a manifold $M$ vanish, i.e. the map $f:M\rightarrow BO$ stably classifying the tangent bundle is trivial on cohomology, ...

**1**

vote

**0**answers

111 views

### Question concerning computing $\pi_1(\mathbb R^{3}-B)$ in Alexander Horned Sphere

I was studying an example of the Alexander Horned Sphere on page 171 of Allen Hatcher's book. The example computes the fundamental group $\pi_1(\mathbb R^{3}-B)$ of the complement of the sphere in ...

**4**

votes

**0**answers

79 views

### Example request: seriously deficient homogeneous spaces

In a previous post, I cite a dimension condition commonly satisfied by homogeneous spaces and claim that a counterexample must have deficiency at least $3$. For convenience, I reproduce the definition ...

**4**

votes

**0**answers

64 views

### A dimension condition on the cohomology of a homogeneous space

The rational cohomology of a homogeneous space $G/K$ admits a homomorphism from $H^*(BK)$ induced from the classifying map $G/K \to BK$ of the principal $K$-bundle $G \to G/K$. Assume the Lie group is ...

**26**

votes

**1**answer

417 views

### Is there an explicit description of a cobordism between $\mathbb{CP}^n$ and $\mathbb{RP}^n\times\mathbb{RP}^n$?

With a little bit of work, one can show that $\mathbb{CP}^n$ and $\mathbb{RP}^n\times\mathbb{RP}^n$ have the same Stiefel-Whitney numbers, so by a theorem of Thom, they are (unorientedly) cobordant.
...

**3**

votes

**0**answers

130 views

### isomorphism of Chern character in kk-theory

Suppose we work with Fréchet algebras. Cuntz defined kk-theory for those algebras and hence we have the notions of K-theory and K-homology for those algebras. Now suppose Chern character is ...

**7**

votes

**1**answer

240 views

### Cap product on Leray-Serre spectral sequences

Let $\ p:X\to B\ $ be a fibration of spaces with fiber $F$. Then there are homological and cohomological Leray-Serre spectral sequences $E^{r}_{pq}$ and $E_r^{pq}$ that converge to $H_*(X)$ and ...

**5**

votes

**1**answer

218 views

### Graph spectra and topology

This is a somewhat vague question, but I'm wondering if there has been any research into connections between the spectrum of a graph and some notion of the "topology" of that graph.
To give an ...

**10**

votes

**2**answers

289 views

### Rational homology sphere that is not Seifert manifold

I wonder if there is an example of rational homology sphere that is not a Seifert manifold. If there is, how can one construct such a rational homology sphere from a surgery of a knot in $S^3$?

**6**

votes

**0**answers

165 views

### Bott-Samelson theorem for simplicial sets

Let $X\in \mathrm{sSet}$ and $FX$ be the Milnor's construction (model for $\Omega\Sigma |X|$) - in each dimension $n$ this is the free group on $X_n$ with one relation $*=1$. I'm interested in ...

**8**

votes

**2**answers

229 views

### Topological Derivation of Leray Spectral Sequence

I'm interested in computing - to the extent possible - the Leray spectral sequence for a particular map which is almost, but not quite, a fiber bundle (e.g. a Seifert fiber space). The hardest step ...

**16**

votes

**1**answer

306 views

### Is the restriction map for embeddings of manifolds with boundary a fibration?

Let $M$ and $W$ be smooth manifolds (possibly with boundary) and $V\subseteq W$ a submanifold. We have a map between embedding spaces
$$Emb(W,M)\rightarrow Emb(V,M)$$ given by restriction.
Richard ...

**7**

votes

**1**answer

197 views

### Chern-Simons forms, characteristic numbers, and boundary terms?

For any principal $G$-bundle $P \to M$ with principal connection $\omega$, given a $G$-invariant polynomial $p: \mathfrak{g} \to \mathbb{R}$ we can construct a form $p(F_\omega)$ on $P$ which descends ...

**10**

votes

**2**answers

320 views

### Stable homotopy groups of $RP^{\infty}$

Are the stable homotopy groups $\pi^s_i(\mathbb R P^{\infty})$ known for small $i$? In particular, I would be interested in the values for $i = 5,6$. A quick Internet search did not lead to anything.

**2**

votes

**1**answer

160 views

### Holonomic splitting

I am reading the book "Introduction to the h-Principle" by Eliashberg and Mishachev. At the moment I try to understand the Section 1.7 Holonomic splitting on page 12 but without success. I do not ...

**11**

votes

**2**answers

213 views

### Schur multiplier of $Sp(2g, \mathbb{Z}/2)$ for $g \geq 3$

This question is about the computation of $H_2(Sp(2g, \mathbb{Z}/2), \mathbb{Z})$, where $Sp(2g, \mathbb{Z}/2)$ is the group of symplectic $2g \times 2g$ matrices over $\mathbb{Z}/2$.
With respect to ...

**8**

votes

**1**answer

250 views

### $K$ theory and singular cohomology

For cell complexes${}^1$ $X$ we have an isomorphism
$$
K^*(X)\otimes \mathbb{Q}\cong H^{*}(X;\mathbb{Q}),
$$
which is induced by the Chern character.
What is the analogous statement for $KO(X)$?
...

**2**

votes

**0**answers

103 views

### The homology of $\varinjlim SO(p,q)$

Is there a way to explicitly compute the homology of the space
$$
\varinjlim_{(p,q)} SO(p,q)^+,
$$
where each $SO(p,q)$ is the indefinite special orthogonal group, and $SO(p,q)^+$ its identity ...

**4**

votes

**1**answer

103 views

### Lifting a differential operator

Let $D$ be a differential operator acting between the spaces of smooth sections of two vector bundles $E,F$ over compact manifold $M$. If $M$ is not simply connected one can construct the universal ...

**27**

votes

**0**answers

359 views

### What is the “real” meaning of the $\hat A$ class (or the Todd class)?

In the Atiyah-Singer index theorem as well as in the Grothendieck-Riemann-Roch theorem, one encounters either the $\hat A$-class or the Todd class, depending on the context. I want to focus on the ...

**3**

votes

**0**answers

104 views

### Intersection patterns of loops on surfaces

Let $a,b$ be to simple closed loops on a surface $S$ with homologically trivial intersection (more generally I'm also interested in the case when $b$ is 1-codimensional). Denote their intersection on ...

**3**

votes

**1**answer

257 views

### Differential characters, Chern-Simons forms, and differential cohomology

I've read through the classic Chern-Simons paper where they introduce the Chern-Simons forms. These are differential forms whose exterior derivative gives you the characteristic forms for any given ...

**1**

vote

**1**answer

272 views

### research articles in topology/geometry [closed]

There is a saying "Do you read the masters?"
I want to read some basic papers in Topology/geometry...
I can not clearly state what is basic as of now...
My back ground includes course in
...

**4**

votes

**1**answer

198 views

### The fibration map $Diff(M) \rightarrow Emb(N,M)$

Let $M$ be a non-compact manifold, equipped with a (closed?) submanifold $N\subset M$. The action of $Diff(M)$ on the set of embeddings $N\hookrightarrow M$ induces a map
$$
Diff(M) \rightarrow ...

**2**

votes

**0**answers

86 views

### A Künneth-Theorem version for relative singular cohomology

I'm not an expert in algebraic topology, but sometimes I need some results from this area, for example tools to determine singular cohomology groups of product spaces.
The Künneth-Theorem which I ...

**3**

votes

**0**answers

118 views

### Is the bar construction of a CDGA model a Hopf algebra model for the loop space?

By a theorem of Adams, if $A = C^*(X;\mathbb{Q})$ is the CDGA of rational cochains on $X$ then the cohomology of the bar complex of $A$ is isomorphic to $H^*(\Omega X; \mathbb{Q})$ as a coalgebra (see ...

**9**

votes

**0**answers

303 views

### Milnor-Stasheff Characteristic Classes Problem 7B, Borel 1953

There is the following Proposition 11.1 from Borel's 1953 paper La cohomologie mod 2 de certains espaces homogènes (see here).
Proposition 11.1 The classes $w^i$ and $\overline{w}^j$ are related ...

**11**

votes

**2**answers

548 views

### Intuition/idea behind a proof of the splitting principle?

The splitting principle is as follows.
Given a vector bundle $E \to X$ with $X$ compact Hausdorff, there is a compact Hausdorff space $F(E)$ and a map $p: F(E) \to X$ such that the induced map ...

**2**

votes

**0**answers

87 views

### Can additivity of the Euler characteristic be interpreted in terms of the Poincaré–Hopf theorem? [closed]

Whenever there is a long exact sequence in homology induced by a short exact sequence of chain complexes one finds that the corresponding Euler characteristics are additive. For example, if $Y \subset ...

**3**

votes

**2**answers

442 views

### Definition of E-infinity operad

What is the definition of $E_\infty$-operad in the category of chain complexes over $\mathbb{Z}/p\mathbb{Z}$? J. Smith in http://arxiv.org/abs/math/0004003 define it for complexes over $\mathbb{Z}$ ...

**1**

vote

**0**answers

138 views

### Braids with an infinite number of strings

Has anyone developed a theory for braids with an infinite number of strings?

**6**

votes

**1**answer

274 views

### Construction of invariants of 4-manifolds with the Kirby calculus

I'm an undergraduate student, interested in the low dimensional topology, in particular, the 4-manifold theory.
I have a question.
In the knot theory, the Reidemeister moves play fundamental roles.
...

**3**

votes

**0**answers

157 views

### Cohomology algebra generated by $n$ Steifel whitney classes and and $k$ dual classes subject only to $n+k$ defining relations? [closed]

Is the cohomology algebra $H^*(G_n(\mathbb{R}^{n+k}))$ over $\mathbb{Z}/2$ generated by the Steifel-Whitney classes $w_1, \dots, w_n$ of $\gamma^n$ and the dual classes $\overline{w}_1, \dots, ...

**2**

votes

**0**answers

117 views

### Reference request for a “truncated version” of the de Rham algebra

Let's start on the $n$-torus for sake of simplicity.$\newcommand{\T}{\mathbb T}$
If I understand the relevant definitions correctly, the usual de Rham algebra of smooth differential forms on $\T^n$ is ...

**6**

votes

**0**answers

762 views

### Ordinary cohomology groups of $(\mathbb{C}^3\times T^2)/\mathbb{Z}_k$ that I need for my string theory research

Let $X=(B^6\times T^2)/\mathbb{Z}_k\subset (\mathbb{C}^3\times T^2)/\mathbb{Z}_k$ where $k=2,3,4,6$, where the generator of $\mathbb{Z}_k$ acts on $\mathbb{C}^3$ by the multiplication by a primitive ...

**0**

votes

**0**answers

131 views

### Is there a t-structure on the homotopy category of spectra that has the sphere spectrum in its heart?

Maybe the heart of such a t-structure should be the category of abelian groups, and the t-homology functor should be given by taking usual homology groups of the spectrum. Is it impossible?

**10**

votes

**2**answers

550 views

### A homology theory which satisfies Milnor's additivity axiom but not the direct limit axiom?

Let us agree on the following: a "homology theory" means a functor $h_*$ from the category of pointed CW complexes to the category of graded abelian groups, together with natural isomorphisms ...

**4**

votes

**0**answers

61 views

### The metric gives the optimal element in a class

In geometry there is plenty of examples in which the following happens:
Some elements are considered equivalent, in some topological or algebraic sense
We take the quotient
The metric is usually not ...

**0**

votes

**1**answer

169 views

### group actions on fibre bundles

Suppose that we have a group $G$ acting on the spaces $E$ and $B$. Suppose moreover that we have fibre bundles $\xi$ and $\eta$ in the following commutative diagram
If $\xi$ is a trivial bundle, ...

**7**

votes

**1**answer

158 views

### (Non)-equivariant equivalence in $G$-spectra

In HHR, an important part is the periodicity theorem. For proving the theorem, they invert a carefully defined class $D \in \pi^{C_8}_{19\rho_8}(N^8_2MU_{\mathbb{R}})$ and they can find an element in ...

**8**

votes

**1**answer

315 views

### Non-Cartesian Monoidal Model Structure on a Slice Category

Given a monoidal model category $(M,\otimes, 1)$, and a monoid therein $A$, one can take the slice model category $M_{/A}$. This category has a natural monoidal structure induced by taking fibered ...

**0**

votes

**1**answer

64 views

### Conjugation Cells [equivariant cohomology]

I'm studying conjugation spaces, I have found in many sources that a conjugation cell is a conjugation space (without a proof). The widest approach that I have found so far is this paper (example 3.5)
...

**3**

votes

**0**answers

177 views

### some terminologies on limiting mixed hodge structures or rather Derived categories

$f: X\rightarrow S$ is proper surjective homomorphism map from connected complex manifold to unite disk. $Y=f^{-1}(0)$ is algebraic and normal crossing in X, f is smooth away from 0, $X^*=X\setminus ...