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

**4**

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**1**answer

163 views

### How many quadratic fields occur as trace fields of hyperbolic knot complements?

I am interested in when the trace field of a knot complement has the form $F(\sqrt{-d})$ for $F\subset\mathbb{R}$ and $d\in F^+$ (squarefree). Does this occur for infinitely many choices of pairs $(F,...

**2**

votes

**0**answers

71 views

### why is $\cap \mu_B:H^k(\mathbb{R}^n,\mathbb{R}^n\setminus B;R)\to H_{n-k}(\mathbb{R}^n;R)$ an isomorphism? [closed]

I asked this http://math.stackexchange.com/q/1694046/309968 question already on MSE, but received no answer and I hope it's ok if I ask here for once.
Let $R$ be commutative ring with $1_R$
Lemma: ...

**6**

votes

**3**answers

262 views

### For an arithmetic hyperbolic 3-manifold group, when is its trace field not its invariant trace field?

Edit: In my original post I failed to require the group to be a manifold group. The answer below from @BenLinowitz works in that case. I am really interested though in when the group is torsion-free, ...

**3**

votes

**1**answer

172 views

### How many non-commensurable non-arithmetic manifolds have a quaternion algebra like this?

I am interested in realizing commensurability classes of hyperbolic $3$-manifolds whose quaternion algebra (note: not invariant quaternion algebra) is isomorphic to one of the form $\Big(\frac{a,b}{F(\...

**1**

vote

**0**answers

105 views

### Behaviour of the Serre spectral sequence on a product of fibrations

Given fibration sequences $F\rightarrow E\rightarrow B$ and
$F'\rightarrow E'\rightarrow B'$,
consider the homology Serre spectral sequence $S$ for the product of fibrations
$F\times F'\rightarrow E\...

**6**

votes

**1**answer

291 views

### Classify $K(\pi,n)$ that are manifolds

Inspired by `Naturally occuring' $K(\pi, n)$ spaces, for $n \geq 2$. and When is a classifying space a topological manifold?, I'd like to formulate a precise question:
For which $n \in \mathbb{Z}...

**3**

votes

**0**answers

70 views

### Example of R-bad space

I have been looking around for examples of $R$-bad spaced in the sense of Bousfield and Kan. In their book "Homotopy limits, completions and localizations] they give several examples of such spaces ...

**5**

votes

**1**answer

162 views

### The properness of the special singular simplicial spaces

This is a question related to another one in MO
Background : A special simplicial space $X_{\cdot}$ is a simplicial space with $X_{0}=\ast$ and $X_{n}\simeq X_{1}^{n}$ via the simplicial map $v_{i}:[1]...

**6**

votes

**0**answers

132 views

### kernel of the mod $2$ Bockstein on the first cohomology group

Let $M$ be a path-connected finite $CW$-complex. Suppose the first integral homology group is $H_1(M;\mathbb{Z})= \mathbb{Z}_2^{\oplus r}\oplus A$ where $r\geq 1$ and $A$ is a finite abelian group of ...

**5**

votes

**1**answer

103 views

### Isomorphism classes of differential rank $k$ vectors bundles over $S^q$ [closed]

Could anybody provide a motivated sketch of why the isomorphism classes of the differentiable rank $k$ real vector bundles over the sphere $S^q$ are given by$$\text{Vect}_k(S^q) \simeq \pi_{q - 1}(\...

**11**

votes

**0**answers

101 views

### Is every simply connected finite complex the classifying space of a finite monoid

On page 323 of Fiedorowicz, "Classifying Spaces of Topological Monoids and Categories" it was stated that "it seems likely that any finite simply connected complex should [have the same weak homotopy ...

**2**

votes

**0**answers

112 views

### Connecting homomorphism in generalized cohomology theory

I have some compact manifold with boundary $(M,\partial M)$, and there is a long exact sequence
$$\cdots\to KO^{-1}(\partial M)\xrightarrow{\partial} KO^{0}(M,\partial M)\to KO^0(M)\to KO^0(\partial ...

**12**

votes

**1**answer

309 views

### Characteristic classes for odd $K$-theory

There are different models of odd $K$-theory. In one case,
one takes the group $U=\lim\limits_{\longrightarrow}U(n)$ as classifying space. Similarly, if $\mathcal U$ denotes the unitary group of a ...

**1**

vote

**0**answers

59 views

### Possible directions of saddle connections

Let's consider a Riemann surface $X$ of genus $g\ge 2$ and $q$ a holomorphic quadratic differential on $X$. A natural parameter on $X$ is a chart for which $q=dz^2$. A $\theta$-trajectory is a maximal ...

**28**

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**0**answers

541 views

### Atiyah-Singer theorem-a big picture

So far I made several attempts to really learn Atiyah-Singer theorem. In order
to really understand this result rather broad background is required: you need
to know analysis (pseudodifferential ...

**10**

votes

**1**answer

384 views

### What was a cusp to Hurwitz in 1892?

Let $d\in\mathbb{N}$
be squarefree.
Let $\mathcal{O}_d$
be the ring of integers of $\mathbb{Q}(\sqrt{-d})$.
Let $\Gamma_d=\mathrm{PSL}_2(\mathcal{O}_d)$.
Let $\mathcal{H}^3$
be the upper half-space ...

**5**

votes

**0**answers

99 views

### cohomology ring of configuration spaces on $S^2$ and the projective plane

For a manifold $M$ and a positive integer $n$, the unordered configuration space $B(M,n)$ is the space consisting of all unordered collections of $n$ distinct points on $M$. Precisely,
$$
B(M,n)=\{(...

**17**

votes

**1**answer

403 views

### A spectral sequence for computing cohomology of a space from that of its strata

Let $X$ be a smooth complex variety (not necessarily compact) and let $D$ be a normal crossings divisors with components $D_1$, $D_2$, ..., $D_N$. For a set of indices $I$, let $D_I = \bigcap_{i \in I}...

**10**

votes

**3**answers

537 views

### What are parabolic bundles good for?

The question speaks for itself, but here is more details: Vector bundles are easy to motivate for students; they come up because one is trying to do "linear algebra on spaces". How does one motivate ...

**3**

votes

**0**answers

67 views

### Have locally principal crossed homomorphisms been studied?

Take a (multiplicative finite) group $H$ acting on the left (by automorphisms) on an (additive finite) abelian group $A$, and recall that the abelian (additive) group of crossed homomorphisms from $H$ ...

**9**

votes

**2**answers

212 views

### non-triviality of the underlying real vector bundle of the complexification of a real vector bundle

Let $M$ be a given manifold and $\xi$ be a given $k$-dimensional vector bundle over $M$. How to determine whether the underlying real vector bundle of $\xi\otimes\mathbb{C}$, i.e. the Whitney sum $\xi\...

**7**

votes

**0**answers

163 views

### “Flat” and “Affine” morphisms of smooth manifolds

Let $f: X \to Y$ be a qcqs morphism of qcqs schemes. We have the following characterizations:
$f$ is flat $\iff$ $f^*:QCoh(Y) \to QCoh(X)$ is exact.
$f$ is affine $\iff$ $f_*: QCoh(X) \to QCoh(Y)$ ...

**8**

votes

**0**answers

211 views

### When does an $E_\infty$ algebra come from a commutative differential graded algebra?

Suppose that $K$ is an $E_\infty$-algebra on a space $X$ (more generally, any ringed topos; also, feel free to assume that $X$ is a point). That is, $K$ is a cochain complex of sheaves on $X$, endowed ...

**3**

votes

**0**answers

84 views

### Eilenberg-Moore spectral sequence for path-loop fibration over Q\Sigma X (reference request)

Related to the question here, here is another question. Consider the kernel of the map $H_*(QY;Z/p)\rightarrow H_{*+1}(Q\Sigma Y;Z/p)$. restricted to
$PH_*(QY)$, and let's say $Y$ itself is a ...

**8**

votes

**2**answers

300 views

### Examples of calculating perverse sheaves on algebraic varieties with easy stratification

I have been learning intersection homology and perverse sheaves in the following way. I started by reading the first $7$ chapters of Kirwan and Woolf's book
http://www.amazon.com/Introduction-...

**11**

votes

**1**answer

310 views

### Dimension of a homotopy type

What is the state of knowledge about the dimension of homotopy types? By the latter I mean the minimal number which is the dimension of a topological space representing the homotopy type. The open ...

**5**

votes

**1**answer

208 views

### a question about Bockstein spectral sequence

I find the following theorem for Bockstein spectral sequence at http://pages.vassar.edu/mccleary/files/2011/04/MC10.fin_.pdf, page 459:
Question. for a fixed $k$, if $\beta$ does not hit $H_k(X;\...

**6**

votes

**0**answers

141 views

### Intuition behind small object argument and cofibrantly generated model categories?

With regards to model categories, what is the intuition behind the small object argument and cofibrantly generated model categories?

**5**

votes

**0**answers

69 views

### The behaviour of the suspension homomorphism on $H_*(QX;Z/p)$ for odd $p$ (Reference request)

The mod $p$ homology of $QX=\Omega ^{\infty}\Sigma ^{\infty}X$ for connected $X$ was computed by Dyer-Lashof Homology of Iterated Loop Spaces, Amer. J. of Math., vol.84, No.1 pp 35-88. 1962 It follows ...

**0**

votes

**1**answer

267 views

### How do Schubert classes form a basis for $H^{*}(Gr(k, n))$?

I've gone through many texts in algebraic geometry, specifically, Schubert calculus. They all claim that the Schubert classes $[\Omega_{\lambda}]$ form a basis for the cohomology ring of the complex ...

**3**

votes

**0**answers

265 views

### A question about cofibrations

Let $(X, A)$ be a cofibration, with $X$ compactly generated. This is equivalent to the fact that $A$ is a NDR of $X$, i.e., there exist two functions $\phi \colon X \rightarrow I$ e $H \colon X \times ...

**7**

votes

**0**answers

119 views

### Associated graded of double Koszul dual

Let $k$ be a field, and let $A$ be a graded, connected, augmented, locally finite $k$-algebra. If $\Omega^* A$ denotes the cobar complex of $A$ (i.e., the dual $Hom_k(B_*(A), k)$ of the bar complex ...

**8**

votes

**1**answer

196 views

### Is the action of $G$ on $H_1(T^n, \mathbb{Z}) = \mathbb{Z}^n$ faithful?

Let $G$ be a finite group of diffeomorphisms of the torus $T^n$ fixing some point $p$, i.e. $p$ is fixed by every element of $G$. I have two questions.
Is the action of $G$ on $H_1(T^n, \mathbb{Z}) =...

**6**

votes

**0**answers

187 views

### Is there an obstruction which classifies “quasi-isomorphism but not chain equivalence”?

Fix a ring $R$ and let $C_\bullet$, $D_\bullet$ be (possibly unbounded) chain complexes of $R$-modules. Assume that $f_\bullet:C_\bullet \to D_\bullet$ is a quasi-isomorphism: that is to say, $f$ is a ...

**1**

vote

**0**answers

79 views

### coefficient of homology of configuration spaces over real projective spaces

In the slides Characteristic Classes of Surface Bundles
and Configuration Spaces, Miguel A. Xicot'encatl, page 38, what is the coefficient of the following homology?
Could the coefficient be an ...

**16**

votes

**1**answer

530 views

### Finiteness Conjecture (New Doomsday conjecture)

This is completely out of curiosity.
I wonder if there has been any recent progress reported on the Finiteness or New Doomsday conjecture, in the form of a talk, preprint or possibly a paper?
Just ...

**13**

votes

**2**answers

669 views

### Torsion-freeness of two groups with 2 generators and 3 relators and Kaplansky Zero Divisor Conjecture

Let $G_1$ and $G_2$ be the groups with the following presentations:
$$G_1=\langle a,b \;|\; (ab)^2=a^{-1}ba^{-1}, (a^{-1}ba^{-1})^2=b^{-2}a, (ba^{-1})^2=a^{-2}b^2 \rangle,$$
$$G_2=\langle a,b \;|\; ...

**10**

votes

**1**answer

248 views

### real and complex vector spaces as topological categories

Let $Vect_{\mathbb{R}}$ be the category of (say, finite dimensional) vector spaces over $\mathbb{R}$. The automorphism group of the object $\mathbb{R}^n\in Vect_{\mathbb{R}}$, is $GL_n(\mathbb{R})$. ...

**4**

votes

**2**answers

412 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 ...

**7**

votes

**0**answers

196 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 $\alpha\colon S^p\to G$ ($p\geq1$...

**0**

votes

**1**answer

173 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

269 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

69 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

396 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

241 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

254 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

114 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

427 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.
...