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

learn more… | top users | synonyms (1)

6
votes
1answer
199 views

Are there familiar expressions for (finite) joins of finite groups?

Milnor construction of the classifying space of a topological group $G$ is given in terms of infinite joins of $G$. Schwarz then proved that the $k+1$ iterated self join of a group $G$ classifies ...
8
votes
1answer
298 views

Topology on the space of constructible sheaves

Let $X$ be a nice compact topological space with a fixed finite stratification by locally closed topological manifolds. At the beginning one may assume that $X$ is a complex algebraic manifold with ...
3
votes
0answers
193 views

N-periodic derived categories

I have some seemingly basic questions about $N$-periodic derived categories to which I have not found answers in any of the usual places. Let $R$ be a ring, and let $D(R)_{\mathbb Z/N\mathbb Z}$ ...
11
votes
1answer
252 views

Whitehead products and Framed Manifolds

The attaching map for the top cell of the torus $S^n \times S^n$ is a map $$ [x,y]: S^{2n-1} \to S^n \vee S^n $$ where the notation is such that $x,y : S^n \to S^n \vee S^n$ are the two ...
8
votes
1answer
242 views

Can the groupoid completion of a topological category be recovered from its classifying space?

Let $C$ be a category. The groupoid completion of $C$ is the free groupoid on $C$, i.e. the category $C[C^{-1}]$ obtained by localizing at everything. Recall that the classifying space $\mathbf{B}C$ ...
23
votes
2answers
818 views

Steenrod operations in etale cohomology?

For $X$ a topological space, from the short exact sequence $$ 0 \rightarrow \mathbb{Z}/2 \rightarrow \mathbb{Z}/4 \rightarrow \mathbb{Z}/2 \rightarrow 0 $$ we get a Bockstein homomorphism ...
0
votes
0answers
97 views

Mapping class groups acting on simple closed curves

Let $S_{g,d}$ be a genus $g$ compact Riemann surface with $d$ punctures. Let $\mathcal{M}_{g,d}$ be the moduli space of all such surfaces, i.e. genus $g$ compact Riemann surfaces with $d$ marked ...
5
votes
0answers
130 views

Push forward of the constant sheaf for a Serre's fibration

Let $f\colon X\to Y$ be a proper continuous map of topological spaces which is a Serre's fibration. $X$ and $Y$ may be assumed to be locally compact, $Y$ is connected topological manifold of finite ...
8
votes
1answer
184 views

Homotopy type of diffeomorphism which are the identity on and near the boundary

Let $M$ be a compact manifold with boundary. Denote by $Diff(M), Diff_\partial(M)$ and $Diff_{U\partial}(M)$ the groups of diffeomorphisms of $M$ and the subgroups of the ones that are the identity on ...
4
votes
1answer
160 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 ...
2
votes
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
254 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
171 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 ...
1
vote
0answers
103 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 ...
6
votes
1answer
284 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 ...
3
votes
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
134 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 ...
6
votes
0answers
131 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
97 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 - ...
11
votes
0answers
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
0answers
110 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
307 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 ...
26
votes
0answers
505 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
380 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
92 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, $$ ...
16
votes
1answer
382 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 ...
10
votes
3answers
528 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
65 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
206 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 ...
7
votes
0answers
160 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
0answers
207 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
82 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
292 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 ...
11
votes
1answer
307 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
198 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 ...
6
votes
0answers
136 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
0answers
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
265 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
votes
0answers
117 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
180 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
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
528 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
653 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
247 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
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
0answers
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
1answer
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 ...