# Tagged Questions

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

**9**

votes

**1**answer

311 views

### Stable moduli interpretation of $\mathbb{R}\mathrm{P}^\infty_{-1}$

I attended a talk recently which closed with the following tantalizing facts: there is a naturally occurring map of spectra $$K(\mathbb{S}) \to \Sigma \mathbb{C}\mathrm{P}^\infty_{-1},$$ which can be ...

**1**

vote

**0**answers

82 views

### Action of Landweber-Novikov algebra on infinite polynomial ring

Denote $\text{Aut}(\hat{{\mathbb A}}^1)$ be the affine group over ${\mathbb Z}$ that sends some ring $R$ to the strict automorphisms of $R[[t]]$, i.e. those of the form $X\mapsto X + r_1 X^2 + r_2 X^3 ...

**5**

votes

**1**answer

228 views

### Image of J splitting

The image of J space is defined for $p$ odd as the homotopy fibre $J_{(p)}$ of the self map $$\psi^k - 1: BU_{(p)} \to BU_{(p)}.$$ Here, $\psi^k$ is an Adams operation, and $k \in \mathbb{N}$ ...

**34**

votes

**2**answers

899 views

### Is there an analog of Sperner's lemma for the Hopf invariant?

Recall that Sperner's lemma is essentially a combinatorial version of the topological statement "A map from $S^n$ to $S^n$ with degree one cannot be nullhomotopic."
My question is, does there exist ...

**1**

vote

**0**answers

103 views

### When is equivariant cohomology generated by equivariant Euler classes?

Let $X$ be a smooth complex projective variety acted upon by algebraically by a complex torus $T$. Let $F_1,\ldots,F_n$ be the connected components of $X^T$ and assume that the restriction map ...

**4**

votes

**0**answers

69 views

### Computing the Product Structure in Equivariant Cohomology via a Stratification and Thom-Gysin

Let $X$ be a smooth complex projective variety acted upon algebraically by a complex torus $T$. Suppose that $\{X_{\beta}\}_{\beta\in B}$ is a finite $T$-invariant stratification of $X$ into smooth ...

**3**

votes

**0**answers

118 views

### Robust Invariants of a solution of systems of equations

Let $X$ be a $k$-dimensional triangulated manifold and $A:=\partial X$ be its boundary. Let $f:X\to R^d$ be a continuous function such that $g:=f|_{A\cup X^{(i-1)}}$ avoids zero and let $z_g\in Z^{i} ...

**1**

vote

**1**answer

100 views

### Image of the map induced on homology by a covering

I asked this question on math.se (http://math.stackexchange.com/questions/647930/image-of-the-map-on-homology-induced-by-a-covering), but it have not attracted much of attention.
Let $X$ and $Y$ are ...

**3**

votes

**0**answers

198 views

### Homotopy category of groupoids

The nlab Ho(Cat) page says: morphisms in the homotopy category of groupoids $Ho(Gpd)$, have two equivalent description:
iso-classes of functors.
formally invert equivalence functors (i.e. ...

**4**

votes

**1**answer

135 views

### Sullivan's $H$-space equivalence between $G/PL[1/2]$ and $BO[1/2]$

There is a theorem by Sullivan of the following form:
Theorem: There is an equivalence of $H$-spaces
$$ G/PL[\tfrac{1}{2}] \simeq BO_{\otimes}[ \tfrac{1}{2} ]\ . $$
It can be found for example ...

**3**

votes

**0**answers

115 views

### Topological interpretation for groups of type $FP_2$

A group $G$ is of type $FP_2$ if it admits a partial projective resolution of $\mathbb{Z}G$-modules $$ P_2 \rightarrow P_1 \rightarrow P_0 \rightarrow \mathbb{Z} \rightarrow 0$$ with each $P_i$ being ...

**1**

vote

**0**answers

63 views

### Homology of a higher-dimensional full torus with a certain disk removed

Let $X=S^2\times B^5$ and let us consider an embedding of an
open $5$-disk in $X$ with its boundary
glued by a map of degree $2$ to some $\{
s_0\} \times S^{4}\subseteq \partial X$ . The embedding ...

**6**

votes

**2**answers

232 views

### Moore decomposition, dual to Postnikov tower

Let X be a CW complex with given cohomologies H^n(X,Z)=G, H^m(X,Z)=H and other reduced cohomologies are zero. Which additional algebraic information/structures I need to identify homotopy type of X? ...

**3**

votes

**1**answer

121 views

### Gysin sequence for vector-bundle valued cohomology

Let $M$ be a closed smooth manifold and $E\longrightarrow M$ be a vector bundle with a flat connection
$$\nabla:\Gamma(E)\longrightarrow \Gamma(T^{*}M\otimes E).$$
Consider the space of differential ...

**3**

votes

**2**answers

266 views

### n-cocycles of finite abelian groups from cohomology group

Question: Given a generic finite abelian group $G=\mathbb{Z}_{N^{(1)}} \times \cdots \times \mathbb{Z}_{N^{(k)}}$.
(1) What is the explicit forms of its cohomology group (see my definition) in a ...

**3**

votes

**0**answers

85 views

### Equivariant Poincare Series of Based Loop Group of SU(2)

Let $\Omega SU(2)$ denote the based loop group of $SU(2)$, and consider the action of $S^1$ on $\Omega SU(2)$ as a maximal torus of $SU(2)$. (This is not the "loop rotation" action.) Is there an ...

**2**

votes

**0**answers

77 views

### The transfer map $H_*(BSO(3))\rightarrow H_*(BO(2))$: reference request

All cohomology and homology will be $Z/2$ coefficient. The restriction map
$H^*(BSO(3))\rightarrow H^*(BO(2))$ is well-known to be the inclusion of
the Dickson invariant $Z/2[w_2,w_3]$ into the ...

**5**

votes

**0**answers

101 views

### Reference for a generalization of Γ-spaces to monoidal model categories

Γ-spaces were introduced by Segal in 1969 as models for what can be now described
as commutative ∞-monoids and ∞-groups in cartesian symmetric monoidal ∞-categories, e.g., E_∞-spaces and connective ...

**1**

vote

**1**answer

140 views

### Normal tubular neighborhood theorem for semi(or pseudo)-riemannian manifolds

Suppose you have a manifold $M$ and a closed sub-manifold $A$, and let $g$ be a semi-riemannian metric,ie, $g_x$ defines a quadratic form on $T_xM$ such that $g_x(v,v)\ge0$, but $g_x(v,v)=0$ not ...

**2**

votes

**0**answers

66 views

### Zeroth G-equivariant Stable Stem [duplicate]

Let G be a finite group. Can anyone give me a motivation and rigorous proof of the Burnside ring A(G) is isomorphic to the zeroth G-equivariant stable stem ?

**6**

votes

**1**answer

172 views

### Stiefel-Whitney classes of virtual vector bundles

Let $E=\xi-\eta$ be a virtual vector bundle over a compact base $B$, which we may assume is a CW complex. A quick and dirty way to define the total Stiefel-Whitney class $w(E)\in ...

**8**

votes

**1**answer

624 views

### What are simplicial topological spaces intuitively?

(This is a repost of a question from MSE. I hope there is more to say.)
I tend to imagine simplicial objects in a category as some kind of "topological objects", with a notion of homotopy. Simplicial ...

**4**

votes

**1**answer

131 views

### twisted poincare duality

Let $M$ be a closed smooth oriented manifold of dimension $n$. Suppose that $\pi:L\longrightarrow M$ is a line bundle with a flat connection $\nabla$. Consider the space of $L$-valued differential ...

**11**

votes

**1**answer

234 views

### Group cochains invariant under the action of the symmetric group

Let $G$ be a finite group and $A$ an abelian group. Recall the cochain groups
$$
C^k = \{f: G^k \to A\}
$$
and the coboundary map
$$
\delta : C^k \to C^{k+1}
$$
$$
(\delta f)(g_1, \ldots, g_{k+1}) ...

**4**

votes

**0**answers

182 views

### Limits and colimits of A_{\infty} categories

I have a question related to the discussion (Coequalizer in category of dg-algebras). How do you prove that the category of (small) dg-categories and the category of (small) A_{\infty} categories are ...

**2**

votes

**1**answer

186 views

### Computing Bredon Cohomology of Z/2-spheres?

Can anyone suggest me how to calculate explicitly the Bredon Cohomology of the sphere(at least for 2-dimensional case) with antipodal Z/2-action with constant coefficient system associated to ...

**-1**

votes

**2**answers

161 views

### Restriction of a line bundle to a two-cycle

I am reading a paper on Chiral Differential Operators
http://arxiv.org/pdf/hep-th/0604179v3.pdf
and it says on page 23 that a line bundle over a manifold C can be characterized completely by its ...

**1**

vote

**1**answer

186 views

### Massey product in Dual Steenrod Algebra

Let $\tau_0$ be the element of dual Steenrod algebra $A_p^{*}$ at a prime $p$ which is dual to Bockstein $\beta \in A_p$. It is well known $\tau_0^2 =0$. Is it true/known that the elemnet $\xi_1$ ...

**5**

votes

**0**answers

132 views

### Nilpotent Localization in Group Theory

Algebraic topologists have invented a very pretty technique of localizing nilpotent groups. (Garth Warner covers the topic in his book manuscript Topics in Topology and Homotopy Theory). For ...

**0**

votes

**0**answers

137 views

### Metalinear frame bundle on sphere or $\mathbb{C}P^n$

Let $M$ be a smooth manifold. A complex metalinear frame bundle $\tilde F(P)\to M$ of a rank $n$ complex vector bundle $P\to M$ is a principal $ML(n,\mathbb{C})$-bundle together with a covering map ...

**3**

votes

**1**answer

156 views

### How to compute the Hurewicz image of a stable map into real K theory

We know that there is a map from $h:\pi_{i}^{st}(pt)\rightarrow KO_{i}(pt)$ and we know all the $KO_{i}(pt)$ by Bott periodicy: they are $Z, Z_{2},Z_{2},0,Z,0,0,0$. We also know $\pi_{i}^{st}(pt)$ for ...

**0**

votes

**1**answer

177 views

### Birkhoff decomposition vanishing of the Chern numbers

Birkhoff decomposition vanishing of the Chern numbers of the holomorphic line bundles of the Birkhoff-Grothendieck decomposition, is some statement I read off in One of Connes papers. Without going ...

**4**

votes

**1**answer

163 views

### Complexes of sheaves with locally constant cohomology versus $C_{*}(\Omega M)$-modules

Let $M$ be a nice, connected topological space. Assume it is a manifold, if you like.
There are two rather similar looking differential-graded (dg) categories that one can associate to $M$ that ...

**46**

votes

**15**answers

3k views

### Teaching homology via everyday examples

What stories, puzzles, games, paradoxes, toys, etc from everyday life are better understood after learning homology theory?
To be more precise, I am teaching a short course on homology, from ...

**6**

votes

**1**answer

248 views

### What is the relation between Lefschetz fixed point theorem and Poincare-Hopf theorem on vector fields?

In Dubrovin/Fomenko/Novikov Modern geometry--Methods and applications, Part II, the (Poincare-)Hopf theorem is treated in section 15.2 (see theorem 15.2.7 on page 129), while the Lefschetz theorem on ...

**22**

votes

**2**answers

1k views

### Why should I care about topological modular forms?

There seems to be a lot of recent activity concerning topological modular forms (TMF), which I gather is an extraordinary cohomology theory constructed from the classical theory of modular forms on ...

**0**

votes

**1**answer

161 views

### The Jordan-Brouwer Separation Theorem

Theorem $S^{n-1}$ disconnects $S^n$ into two open connected components, which have $S^{n-1}$ as frontier.
In $R^3$, if we replace sphere of standard torus with genus $g\geq1$, we may have "The ...

**1**

vote

**1**answer

156 views

### Explicit computation of the action of a Dehn twist on the fundamental group of a surface

Let $S$ be a compact orientable surface of genus $g$. Now let $p\in S$ and $\gamma$ a closed simple curve on $S$ disjoint from $p$. It is not very difficult to compute the action of a Dehn twist along ...

**6**

votes

**2**answers

481 views

### All mapping space between CW complexes is a CW complex?

Let $\mathrm{Map}(X,Y)$ denote the (unbased) cellular mapping space from $X$ to $Y$.
If $X$ and $Y$ are finite CW complexes, is $\mathrm{Map}(X,Y)$ a CW complex?
Can we know the cell structure of ...

**7**

votes

**1**answer

194 views

### Correspondence between operads and monads requires tensor distribute over coproduct?

In checking the details of the correspondence between operads over a symmetric monoidal category and monads on some associated endofunctor of the category, I cannot make the obvious proof work without ...

**6**

votes

**0**answers

119 views

### Induction map in equivariant K-theory

Let X be a space with $Z/2$ action. There is a map from $K(X)$ to the equivariant K-group $K_{Z_{2}}(X)$, which is called "the induction map". (It is a standard operation in equivariant stable ...

**1**

vote

**0**answers

100 views

### Is there a characterization of the terminal infinity topos?

In ETCS, the terminal topos $\mathcal{Set}$ is characterized in the internal language as a well-pointed topos with a natural numbers object that satisfies the axiom of choice.
Is there a ...

**13**

votes

**1**answer

595 views

### Periodic Orbit property

A topological space $X$ satisfies "Periodic orbit property", briefly POP, if for every continuous map
$f:X \to X$, there exist a natural number $n$ and a point $x_{0}\in X$ such that ...

**1**

vote

**0**answers

210 views

### Tensor product of d.g-algebras

I'd like to prove that the tensor product functor $- \otimes Y$, where $Y$ is a d.g-algebra over a field of characteristic 0, preserves finite products of d.g-algebras. This statement is in a paper by ...

**2**

votes

**1**answer

280 views

### Homotopy groups of filtered homotopy limits

Let $X$ be the homotopy limit of a filtered system of simplicial sets $X_i$. When are the morphisms $\pi_j(X)\to \varprojlim \pi_j(X_i)$ surjective for all $j\ge 0$? This seems to be no problem when ...

**3**

votes

**0**answers

115 views

### What are the Čech-local equivalences of (simplicial pre)sheaves?

Let $\mathcal{C}$ be small category and let $J$ be a Grothendieck topology on $\mathcal{C}$. The Čech model structure on $[\mathcal{C}^\mathrm{op}, \mathbf{sSet}]$ is defined to be the left Bousfield ...

**1**

vote

**1**answer

128 views

### Naturality of commutative model for cell attachment in Rational Homotopy Theory

If $X$ is a simply connected space and $\alpha\in\pi_n(X)$ and $Y=X\cup_{\alpha} e^{n+1}$.
and $(\wedge V,d)$ be the minimal model for $X$,then $(\wedge V\oplus \mathbb{Q}
u,d)$ is a commutative model ...

**3**

votes

**1**answer

237 views

### Explicit examples presheaves associated to higher direct images which fail to be sheaves

So I would like to have a few simple examples where the presheaf associated to higher direct image of sheaf fails to be sheaf. So I'm looking for two (natural and simple) topological spaces $X$ and ...

**7**

votes

**2**answers

243 views

### Integral versus real (universal) characteristic classes

I'm pretty confused about the precise relation of the integral and the real cohomology of the classifying space $BG$ of a compact Lie group $G$. The natural map $H^n(BG;\mathbb{Z})\to ...

**8**

votes

**0**answers

242 views

### Lie algebras vs. graph complexes

A ribbon graph is a graph in which every vertex has valence at least three and is equipped with a cyclic ordering of its adjacent half edges. The ribbon graph complex $\mathcal{G}_*$ is the chain ...