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

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6
votes
2answers
214 views

The Thom space of a Whitney sum of vector bundles

Let $\xi$ and $\eta$ be vector bundles over the same base space $X$. Their Whitney sum is a bundle $\xi\oplus\eta$ over $X$. I read somewhere (without proof) that its Thom space is given by $$ ...
21
votes
1answer
650 views

Big list - Equivalent descriptions of Hodge conjecture?

I would like to know equivalent descriptions of the Hodge conjecture (with references). Dan Freed's Version: Consider a topological cycle (boundary less chains that are free to deform) on a ...
8
votes
0answers
108 views

Equivariant and orbifold Chern classes

Edit. After thinking about this problem a bit longer, I am not so sure anymore that the Bredon cohomology proposed by Adem and Ruan gives me the invariants I am looking for. I have therefore moved ...
3
votes
1answer
212 views

How to calculate the fundamental group of general configuration space

Define the configuration space of $n$ points in a general manifold $M$, where $\dim M=m$, as $K=(M^n-D)/S_n$ where $S_n$ is the permutation group and $D=\{(x_1,\cdots,x_n)| \exists i,j\ s.t. x_i=x_j ...
9
votes
2answers
281 views

Lower central series quotients in terms of (co)homology

Let $G$ be a group. It is well-known that $H_1(G,\mathbb{Z})=G/[G,G]$. Also (at least up to torsion) $[G,G]/[G,[G,G]]=\Lambda^2H^1(G,\mathbb{Z})/H_2(G,\mathbb{Z})$ as explained, for example, in this ...
3
votes
0answers
152 views

Complete Segal operads and dendroidal sets

There is a Quillen equivalence between the model category presenting Lurie's $\infty$-operads (which are inner fibrations $\mathcal{C}\to\mathrm{N}(\mathbf{F})$ satisfying certain conditions) and the ...
5
votes
1answer
179 views

What do globes (used to construct globular sets, $\omega$-categories, etc.) actually look like?

Nlab introduces the globular category as a geometrical model to construct certain higher categorical structures (e. g. strict $\omega$-categories), just as quasi-categories, for example, are modelled ...
0
votes
0answers
89 views

Generalization of the fiber changing trick for principal bundles?

We know that a principal bundle can induce a fiber bundle as follows: if $F$ is a space which admits a $G$-action then a principal $G$-bundle $p: E \to B$ induces a fiber bundle $p: E \times_G F \to ...
14
votes
0answers
318 views

Does the category of G-spectra know G?

I was recently in the situation of having access to the category of $G$-modules (for some group $G$ which I had forgotten), as just a category, i.e. no monoidal structure, together with the forgetful ...
8
votes
0answers
137 views

Two transfers for ramified or branched covers

Let $\pi: X \rightarrow Y$ be a 2-fold branched cover of complex varieties. I know of (at least) two types of pushforwards associated to this situation: If I'm not mistaken, there is a pushforward ...
3
votes
1answer
187 views

Is there a degree one map from a product $B\times S^1 \to \#_n S^2 \times S^1$ for any n

For any $n \geq 1$, let $\Sigma_n$ denote the closed orientable surface of genus n. In http://arxiv.org/abs/1202.6302, the authors showed that for any $n$, there is a degree two, $\pi_1$-surjective, ...
4
votes
1answer
114 views

Equivariant maps from simplicial complexes to spheres

Given a topological space $X$ with involution $\nu$, the $\mathbb Z_2$-index $\text{ind}(X)$ is the minimum integer $n$ such that there exists a map $f:X \to S^n$ which is equivariant with respect to ...
2
votes
1answer
101 views

Kernel of projection formula

For a closed embedding of compact complex manifolds $$ \iota : Y \hookrightarrow X $$ and any $\alpha \in H^*(X,\mathbb Q)$, we have trivially: $$ \iota^*(\alpha)=0\quad \Rightarrow ...
23
votes
2answers
870 views

What is the relation between sphere spectrum and supersymmetry?

In this this google+ post of Urs Schreiber, he says: "Grading over the sphere spectrum is supersymmetry" and then he redirect us to the abstract idea of superalgebra (in nLab). Are there some ...
3
votes
1answer
202 views

(Geometric) Proof for the projective bundle formula in K-theory

I'm trying to piece together a proof of the projective bundle formula from several incomplete sources. Here's the statement I'd like to prove: Projective bundle formula: Let $\pi: E \to X$ be a ...
5
votes
0answers
176 views

Does $\#_n S^2×S^1$ really admit a map of non-zero degree from $B×S^1$?

In Proposition 4 on page 6 of this paper, http://arxiv.org/abs/1202.6302, the authors claim to produce a degree 2 $\pi_1$-surjective map $f$ between $M=S^1 \times \Sigma_2$ and $N=\#_2 S^2 \times S^1$ ...
5
votes
0answers
121 views

Two natural maps asssociated with the nerve of a cover

Let $X$ be a nice (e.g. paracompact, locally contractible) topological space, and let $\mathcal{U}=\{U_i\}_{i\in I}$ be an open cover of $X$. Also denote by $N$ the (topological realization of) the ...
7
votes
0answers
127 views

Topological localization of (infinite) inverse limits

The classical localization of topological spaces at a given set of primes $\mathcal{P}$ is a functor $\mathcal{T}\xrightarrow{(-)_{(\mathcal{P)}}}\mathcal{T}$ from a suitable category of topological ...
11
votes
1answer
173 views

Passing from T-equivariant to G-equivariant cohomology

Let G=GLn(ℂ) and let T be a maximal torus. Let X be a topological space with a G-action. My question is: when is the canonical map $$H^*_G(X;\mathbb{Z})\to H^*_T(X;\mathbb{Z})$$ injective? Some ...
3
votes
2answers
85 views

Fixed point set for a subcircle of torus actions

Let $T=S^{1}\times S^{1}\times ...\times S^{1}$ ($n$ times) be $n$ dimensional torus and $X$ be a $T$-space. Lemma: If $X$ has finitely many connected orbit type, then there is a subcircle ...
6
votes
1answer
144 views

Doubt regarding the definition of slice filtration

Voevodsky defined the slice filtration on the motivic stable homotopy category $SH(S)$ over a Noetherian scheme $S$. In the article Open Problems in the Motivic Stable Homotopy Theory, I, Section 2, ...
11
votes
0answers
190 views

Cohomology of a configuration space of points on $\mathbb C^\times$ with an additional restriction

Let $Conf_{1,n}^3$ be the configuration space of collections of $n$ distinct numbered points on the annulus $\mathbb C^\times$ with an imposed restriction: for any $r\in \mathbb R^+$ the circle ...
1
vote
0answers
67 views

Compact Vertical Cohomology and Euler Class of CP1

First of all, please excuse my English. I'm not native Englsih speaker, so you will see so many grammer mistakes, I just only hope that my mistkae wouldn't effect what I want to mean. Hi, recently ...
11
votes
2answers
225 views

Homotopy property of constructible sheaves on stratified spaces

Let $X$ be a stratified topological space (in my case $X$ is a compact space presented as a finite union of locally closed topological manifolds of finite dimension (strata) such that the closure of ...
9
votes
1answer
256 views

Elements of infinite order in the topological mapping class group

Let $M$ be a closed topological manifold, and let $\operatorname{MCG}(M):=\operatorname{Homeo}(M)/\operatorname{Homeo}_0(M)$ denote the topological mapping class group of $M$ ...
12
votes
2answers
372 views

H-space structures on non-sphere suspensions?

It is well known that $S^n$ admits an H-space structure if and only if $n=0,1,3,7$. I'm interested in whether there are other suspensions $\Sigma X$ that admit H-space structures: Question 1 For ...
13
votes
2answers
418 views

Discrete Morse theory: how do zig-zag paths determine homotopy type?

Let $\Delta$ be a simplicial complex (or more generally, a regular CW complex). Let $\mathcal{M}$ be a Morse matching (or equivalently, a discrete Morse function) on $\Delta$. By Forman's theorems, ...
12
votes
1answer
230 views

From relative categories to marked simplicial sets

Both relative categories and marked simplicial sets (over Δ^0) present the ∞-category of ∞-categories. Naturally, one could ask whether there is a reasonably direct way to pass between these two ...
2
votes
0answers
72 views

Fibre bundle and Borel contruction of compact groups

if $G$ is any compact group and $H$ is closed subgroup of $G$, then $G/H\rightarrow X_{H}\rightarrow X_{G}$ is a fibre bundle? ($X_G=X\times _{G}E=\left( X\times E\right) /G $ is orbit space where ...
7
votes
1answer
236 views

Closed formulas for topological K-theory?

Let $X$ be a compact manifold. I'm interested in whether any of the following cases admits a general closed formula for (complex)-$K$-theory. Let $E$ be a complex vector bundle with a given line ...
3
votes
0answers
141 views

The homotopy type of the mapping space $Map_{B\rho}(BS^1,BG)$? for $G$ a compact Lie group

Given a homomorphism $\rho:S^1\rightarrow G$ with $G$ a compact Lie group there is an induced map of classifying spaces $B\rho:BS^1\rightarrow BG$. What is known about the homotopy type of the mapping ...
3
votes
1answer
178 views

Fiber bundle and fibration of classifying space [closed]

Let $BG$ is classifying space of $G$ topological group. If $G$ is any compact group and $H$ is a closed subgroup of $G$, then the inclusion map $i:H\rightarrow G$ induces \begin{equation*} ...
13
votes
2answers
379 views

When do colimits agree with homotopy colimits?

I'm wondering about when the colimit and the homotopy colimit agree with diagrams of simplicial sets. I know that hocolim$(F)=$colim$(F_c)$ where $F_c$ is the cofibrant replacement of $F$. However, it ...
13
votes
1answer
324 views

Homotopy fiber of a map between classifying spaces

I'm looking for a reference (and precise hypothesis if more are needed) for the following facts (or a correction, if I'm just plain wrong): Let $G$ and $H$ be topological groups and $f : G \to H$ be ...
4
votes
1answer
159 views

On push-forward of the constant sheaf for fibrations

Let $f\colon E\to B$ be a fiber bundle with a connected fiber $F$, $f$ is proper. Let $\underline{\mathbb{C}}_E$ be the constant sheaf on $E$. Let $f_*(\underline{\mathbb{C}}_E)$ denote its direct ...
4
votes
0answers
114 views

Relative Hurewicz Theorem

For a given zero-reduced simply connected simplicial set $X$, one can define simplicial group $GX$ representing the loop space of $X$, its Abelianization $AX = GX/[GX,GX]$ and show that the map ...
2
votes
0answers
94 views

How many linear independent vector fields can be constructed on a general manifold with $\chi(M)=0$?

We have known how many linear independent vector fields can be constructed on $S^n$:https://en.wikipedia.org/wiki/Vector_fields_on_spheres So how many linear independent vector fields can be ...
7
votes
0answers
105 views

Are infinite simplicial complexes all manifolds?

Are infinite dimensional simplicial complexes manifolds locally modeled on $\mathbb R^\infty=\operatorname{colim}\mathbb R^n$? If they are homotopy equivalent, are they homeomorphic? Of course not. ...
2
votes
1answer
108 views

Universal space for the family of subgroups of a finite cyclic group

Let $G$ be a compact Lie group and let $\mathcal{P}_G$ denote the family of proper subgroups of $G$. The universal space for the family $\mathcal{P}_G$ is a cofibrant $G$-space which does not have ...
4
votes
0answers
115 views

Bockstein morphism of spectral sequences

Given an omega spectrum $E$, there is a type of chern character map given by its rationalization $$r:E\to E\wedge M\mathbb{R}\;,$$ where $M\mathbb{R}$ denotes a Moore spectrum. The cofiber of the map ...
7
votes
1answer
190 views

Source request for $H^*(B\mathrm{TOP},\mathbb{Q}) \cong H^*(BO,\mathbb{Q})$

Let $B\mathrm{TOP}$ denote the classifying space for microbundles, i.e. $B\operatorname{Homeo}(\mathbb{R}^n,0)$. Now we get a map from $BO$ to $B\mathrm{TOP}$ via the inclusion. Let $f$ denote the ...
30
votes
5answers
2k views

Does $\mathbb C\mathbb P^\infty$ have a group structure?

Does $\mathbb C\mathbb P^\infty$ have a (commutative) group structure? More specifically, is it homeomorphic to $FS^2$, (the connected component of) the free commutative group on $S^2$? $\mathbb ...
10
votes
1answer
281 views

A symmetric embedding of manifolds

Assume that $M$ is a manifold. Is there an embedding of $M$ in some $\mathbb{R}^{n}$ such that the image of $M$ in $\mathbb{R}^{n}$ is invariant under each reflection $(x_{1},x_{2},\ldots ...
0
votes
0answers
152 views

About the homotopy type of diffeomorphism groups

In this paper by Antonelli, Burghelea and Kahn (Topology, 1972), a homomorphism $L :\pi_i(\operatorname{Diff}(S^n, D_+^{n})) \rightarrow \Gamma^{n+i+1}$ was used as a tool to detect non-triviality of ...
20
votes
0answers
292 views

Does the Tate construction (defined with direct sums) have a derived interpretation?

Any abelian group M with an action of a finite group $G$ has a Tate cohomology object $\hat H(G;M)$ in the derived category of chain complexes. There are several ways to define this. One is as the ...
3
votes
0answers
36 views

Understanding SFH of a product sutured manifold without stabilization

I'd like to understand Juhasz's proposition that If $(M,\gamma)$ is a product sutured manifold, then $$ SFH(M,\gamma) \cong \mathbf{Z} $$. The main sticking point is this -- we know $(M,\gamma) = ...
13
votes
2answers
271 views

Classification of $O(2)$-bundles in terms of characteristic classes

I had asked this question in stackexchange but there seems to be no consensus in the answer It is well-known that $SO(2)$-principal bundles over a manifold $M$ are topologically characterized by ...
12
votes
3answers
458 views

K-theory of non-compact spaces

This is a question on nomenclature of $K$-theory in the topological category. The $K$-theory of a compact space $X$ is defined as the Grothendieck group of the vectorbundles on $X$. The Atiyah-Jänich ...
2
votes
0answers
84 views

relation between representations of homology class

Let X be a topological space, for its homology class [f], we can alway construct a simplicial complex K_f by gluing "cancelling boundary pairs" of f and an induced continuous map f' from K_f to X. ...
3
votes
1answer
171 views

The space of homotopy classes of maps of products of spheres

Proposition 17.6.1 of "Differential form in Algebraic Topology" by Bott and Tu proves the following beautiful result: $[S^{q}, X]\simeq \frac{\pi_{q}(X,x)}{\pi_{1}(X,x)}$ where $S^{q}$ is the ...