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

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9
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1answer
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
0answers
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
1answer
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
2answers
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
2answers
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
1answer
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
2answers
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
0answers
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
0answers
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
0answers
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
1answer
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
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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
2answers
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
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
15answers
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
1answer
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
2answers
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
1answer
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
1answer
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
2answers
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
1answer
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
0answers
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
1answer
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
2answers
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
0answers
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 ...