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

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4
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1answer
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
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0answers
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
3answers
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
1answer
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
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0answers
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
1answer
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
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0answers
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
1answer
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
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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
1answer
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
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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
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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
1answer
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
0answers
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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0answers
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
1answer
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
0answers
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
1answer
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
3answers
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
0answers
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
2answers
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
0answers
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
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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
0answers
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
2answers
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
1answer
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
1answer
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
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0answers
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
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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
1answer
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
0answers
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
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0answers
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
1answer
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
0answers
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
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0answers
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
1answer
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
2answers
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
1answer
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
2answers
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
0answers
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
1answer
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
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1answer
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 + ...
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vote
0answers
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
3answers
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
1answer
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
1answer
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, ...
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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
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0answers
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
0answers
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
1answer
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. ...