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

**-1**

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**0**answers

24 views

### Free cocompact action of discrete group gives a covering map

I'd like to find a short proof of the following seemingly basic fact encountered on the second page of Atiyah's paper "Elliptic operators, discrete groups, and von Neumann algebras." ...

**3**

votes

**0**answers

32 views

### Chern-Simons form and Rarita-Schwinger operator

The Rarita-Schwinger (RS) operator naturally generalizes the Dirac operator and in Physics it describes particles with spin-3/2.
I was wondering if there exists any reference concerning the ...

**5**

votes

**1**answer

139 views

### Cohomology ring of a fiberwise join

I am very interested in the cohomology ring of the following construction. Let $f: Y\to X$ be a map between (connected) topological spaces. Suppose that the image of the map $f^*:H^*(X) \to H^*(Y)$ is ...

**4**

votes

**1**answer

108 views

### “Small” simplicial complex with torsion trees

I am giving an expository talk soon about Duval-Klivans-Martin's paper Simplicial Matrix Tree Theorems, and I've been struggling to find a good example to do at the board. An important aspect of the ...

**7**

votes

**0**answers

124 views

### Are there genera for algebraic cobordism?

For real and complex manifolds, we can form the (oriented) cobordism ring $\Omega$, and a genus is defined to be a ring homomorphism
$$\varphi:\Omega\otimes\mathbb{Q}\to R$$
where $R$ is any ...

**5**

votes

**1**answer

93 views

### Homotopy Type of the Based Mapping Space $Map_*^{(k,l)}(\mathbb{C}P^2,BU(2))$

Path components of the based mapping space $Map_*(\mathbb{C}P^2,BU(2))$ are indexed by a pair of integers $(k,l)$ determined by the values of the first two Chern classes that a map ...

**9**

votes

**2**answers

262 views

### Representation viewpoint on Chern Weil (cohomology computations done with rep theory?)

Let $G$ be a compact lie group. Chern-Weil theory tells us that there's a homomorphism:
$$H^{*}(BG;\mathbb{R}) \to (Sym^{\bullet} \mathfrak{g^*})^G$$
Which in our case is an isomorphism since $G$ ...

**0**

votes

**0**answers

64 views

### Use of algebraic topology in geometry(differential and complex analytic) [on hold]

I'm curious in seeing connections of algebraic topology with other areas of mathematics.
As far as I know, it's not really used in scheme-theoretic algebraic geometry and arithmetics(aside from ...

**7**

votes

**1**answer

376 views

### what is the universal cover of GL(2,R)?

In the theory of Bridgeland stability conditions one has an action of the universal cover $G'$ of $G = GL^+(2,\mathbb R)$.
What is G'?
I know there is concrete description in terms of pairs ...

**3**

votes

**2**answers

178 views

### Equivalence of different cohomology groups

Let $X$ be a topological space (may be assumed to be locally compact). Let $A$ be either a field or $\mathbb{Z}$. One can consider various cohomology groups:
(1) singular cohomology ...

**2**

votes

**2**answers

222 views

### CW 4 manifolds with single 4 cell

Let $M$ be a connected compact closed 4 manifold. Then $H_4(M)=\mathbb{Z}$. If we assume it is smooth, from Morse theory we know that $M$ has a CW structure. But can we find a CW structure of $M$ with ...

**1**

vote

**1**answer

154 views

### Applications of topology to discrete dynamical systems?

I'd like to know some of the applications of topology to discrete dynamics. By discrete dynamics I loosely mean studying maps between discrete sets.
I mean cases where adding a topology to the sets ...

**5**

votes

**1**answer

69 views

### smoothing locally-finite (Borel-Moore chains)

Let $M$ be a smooth manifold. As is recorded in (for example) Lee's book, de Rham proved that one can calculated singular homology, $H_*(M)$ using smooth simplices. Does the result extend to ...

**2**

votes

**1**answer

225 views

### Local “pathologies” in spaces arising naturally in algebraic topology

I have been thinking about methods for constructing continuous paths locally in a space. These paths have domain the unit interval and map into "small" neighborhoods of points in a space. Moreover ...

**6**

votes

**1**answer

233 views

### “structure group” for fibration

Regarding "fibration" as a homotopy analogue of "fiber bundle",I want to see parallel notions of "structure group" and "fiber change" in "fibration".
Does it make sense to talk about "structure ...

**2**

votes

**1**answer

218 views

### Using Lefschetz duality in algebraic geometry

I am reading the paper of Fulton and Lazarsfeld on the connectivity of degeneracy loci of morphisms of vector bundles, but there is a comment in the article that I don't quite understand.
Let $G$ be ...

**1**

vote

**0**answers

55 views

### Adjunction of Crossed Module Functors

I am wondering about the following two related questions and don't know if they have already clear answers or not.
1) Suppose that we already know the functor $F \colon \mathcal{C} \to \mathcal{D}$ ...

**6**

votes

**2**answers

206 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
$$
...

**22**

votes

**1**answer

617 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

**0**answers

97 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

**1**answer

204 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

**2**answers

268 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

**0**answers

143 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

**1**answer

171 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

**0**answers

82 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 ...

**13**

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**0**answers

304 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 ...

**7**

votes

**0**answers

132 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

**1**answer

173 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, ...

**2**

votes

**0**answers

63 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

**1**answer

91 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 ...

**24**

votes

**2**answers

829 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

**1**answer

195 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

**0**answers

171 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

**0**answers

101 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

**0**answers

126 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

**1**answer

171 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 ...

**2**

votes

**2**answers

66 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

**1**answer

134 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

**0**answers

189 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

**0**answers

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 ...

**9**

votes

**1**answer

166 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

**1**answer

233 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

**2**answers

369 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

**2**answers

387 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

**1**answer

219 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

**0**answers

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 ...

**6**

votes

**1**answer

231 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

**0**answers

140 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

**1**answer

171 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

**2**answers

366 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 ...