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

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5
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105 views

Classifying map for a surface bundle

Let $E\longrightarrow X$ be a surface (with holes) bundle. The structure group is then $M_{g, s}$, the mapping class group of the fiber. It follows from the famous work of Penner that the classifying ...
1
vote
1answer
83 views

Spinor bundle of line bundle

Given any complex line bundle over a manifold $L\to M$, we know this admits a canonical Hermitian spinor bundle $S$. Suppose we know the first Chern class of the line bundle, i.e. $c_1(L)$ is known. ...
4
votes
0answers
105 views

A wrong example of a superconnection formula(Bismut)

In Bismut's paper, in Theorem 3.2, $$|\int_X\mu \left( Tr_s[\exp(-(\nabla^\xi+\sqrt{u}V)^2)]-\delta_Y(2\pi i)^{\dim_c N}Td^{-1}(-R^N)Tr_s[\exp(-(\nabla^\eta)^2)]\right)|\leq \frac{C}{\sqrt u}|\mu|_{C^...
6
votes
1answer
151 views

Reference for push-pull formula in cohomology

I would like a precise reference for the following fact. Assume that $$ \begin{array}{ccc} M\times_B N & \stackrel{f'}{\to} & N \newline \quad\downarrow g' & & \quad\downarrow g \...
11
votes
2answers
518 views

Algebraic Geometry for Topologists

As someone who is familiar with algebraic topology, say, at the level of Hatcher's book, and familiar with homological algebra and categories and applications in topology but has no idea what a ...
0
votes
0answers
23 views

Random Variables with Simply Connected Support [on hold]

Let $X$ be a random variable with simply connected support on real line. Define $Z_n=\sum_{i=1}^n X_i$ and $X_i\sim X$ for all $i$. Does $Z_n$ has simply connected support, for all $n\geq 1$?
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0answers
55 views

Where should I look for computing the intersection homology of projective varieties?

I'm learning about intersection cohomology topologically through MacPherson's "New York Times Article". This is a very nice guide which gives a nice idea on how to use these methods for low-...
2
votes
0answers
51 views

Twists of equivariant spectra

Let $A$ be a spectrum, defined by deloopings $A_n$ (n an integer). Then the identity $A = S^1\wedge A_1$ together with antipodal equivariant spectrum structure on $S^1$ gives genuine $\mathbb{Z}/2$-...
0
votes
0answers
58 views

Example of bundle-mapping over $S^4$ with singularity $S^2$

Could anyone give a non-trivial example of a bundle-mapping over $S^4$, i.e. find two complex rank 2 vector bundles $E_0,E_1$ over $S^4$ and a bundle mapping $$0\to E_0\overset{v}{\to}E_1\to0$$ such ...
3
votes
1answer
153 views

Rank 2 vector bundles over $\mathbb CP^2$

Is there any classification of the rank 2 complex vector bundles over $\mathbb CP^2$ up to diffeomorphism? Thank you.
4
votes
2answers
184 views

Rank 2 complex vector bundles over $S^4$

On $S^4$, we know that rank 2 complex vector bundles are classified by $\pi_3(U(2))=\mathbb Z$. Any element $g\in\pi_3(U(2))=\mathbb Z$ determines a complex vector bundle $E$ over $S^4$. Can we say ...
3
votes
1answer
126 views

Relation between conjugacy class, quotient isomorphism class, and signature of Fuchsian groups

Let $\Gamma\le SL(2,\mathbb{Z})$ be a finite index subgroup, not necessarily "congruence". Let $c_4,c_6$ be the number of conjugacy classes of elements of order 4 and 6 respectively, let $c_{-1}$ be ...
1
vote
0answers
111 views

Trivial cohomology for fibers implies isomorphism on cohomology

Let $f: Y \rightarrow X$ be a map of topological spaces such that for any $x \in X, f^{-1}(x)$ has trivial cohomology for some cohomology theory (in my case, cohomology with rational coefficients is ...
1
vote
1answer
184 views

What is the structure group of the Hopf fibration $S_1\rightarrow S_3 \stackrel{p}\rightarrow S_2$?

I am studying fiber bundles and have thoroughly reviewed the famous example of the Möbius strip. In that example, I learned how to discover that the structure group of the Möbius strip fiber bundle ...
5
votes
0answers
81 views

Constructible sheaves on general stratified spaces

I am not an expert in the field, so my question might be rather standard. Let $X$ be a compact metric space. Assume that $X=\cup_{i=1}^NS_i$ is a finite disjoint union of locally closed topological ...
-2
votes
1answer
116 views

Action of $\mathbb{Z}/3\mathbb{Z}$ on $P^{1}$ [closed]

I am reading from the book Topics in Galois theory by Serre. I have the following question , take $G=\mathbb{Z}/3\mathbb{Z}$. The group $G$ acts on $P^1$ by $$\sigma x\;=\;1/(1-x)$$ where $\sigma$ ...
1
vote
0answers
108 views

Vector bundles and equivariant vector spaces

It seems commonly accepted that most of the results of equivariant geometry for vector spaces yield analog result for vector bundles. In so far as I understand it, the reason for that is the ...
2
votes
0answers
112 views

Chern character (form) of a Gauss-Manin connection

Consider the trivial fibration $\mathbb{T}^2\to\mathbb{S}^1$, where $\mathbb{T}^2$ is the two-torus. Denote by $\mathbb{C}\to\mathbb{T}^2$ the trivial line bundle over $\mathbb{T}^2$, and equip it ...
3
votes
0answers
141 views

Manifolds and CW-complexes

Let us consider a category $C$ formed by topological spaces and continuous functions (or by smooth manifolds and smooth functions). We consider the morphism category $C_{2}$. An object of $C_{2}$ is a ...
3
votes
0answers
73 views

A model structure on semi-simplicial algebraic Kan complexes?

By an algebraic semi-simplicial kan complex I mean a semi-simplicial set (i.e. a presheaf on the category of finite ordered sets and injective order preserving maps), which is a Kan complex (in the ...
3
votes
0answers
90 views

Galois categories and the connected components functor

In stacks 0BMQ, a Galois category is defined to be a functor $F:\mathsf C\longrightarrow \mathsf{FinSet}$ such that $\mathsf C$ is finitely bicomplete, every object ...
3
votes
2answers
185 views

cohomology of configuration space of punctured variety

Given a smooth projective variety $X$ of dimension $l$, we denote with $F(X,n)$ the configuration space of points $$ F(X,n):=\{(x_{1}, \dots, x_{n})\in X^{n}\: : \: x_{i}\neq x_{j}\text{ for each }i,j ...
1
vote
0answers
101 views

cohomology ring of homogenous manifold

Let $[d_1^{t_1}, \dotsc, d_s^{t_s}]$ be a partition of a positive integer $n$, i.e., $\sum d_r t_r = n$. I want to know the de Rahm cohomology ring of the following types of homogenous spaces : $$ G/H ...
2
votes
0answers
103 views

Quaternionic projective bundle in complex Grassmann bundle

"What is the fundamental class of the projective bundle of lines of a quaternionic bundle in the Grassmann bundle of 2-planes of the underlying complex bundle?" In Quaternionic projective space in ...
4
votes
1answer
205 views

Quaternionic projective space in complex Grassmannian

I would like to consider the quaternionic projective space $\mathbb{PH}^{n-1}\subset\mathbb{G}_2(\mathbb{C}^{2n})$ as a subvariety of the Grassmannian of complex 2-planes. For a real vector $e\in\...
2
votes
1answer
162 views

Intersection of two curves is not Cohen Macaulay

Let be $R=\mathbb{C} \lbrace x,y,z \rbrace$ the formal series ring and let $f_{1},f_{2},f_{3} \in R$ be nonzero elements of $R$. (a) Consider the varieties $M:=V(f_{1},f_{2})$ and $N:=V(f_{2},f_{3})$ ...
0
votes
1answer
186 views

Open problems where Haskell meets Category theory or Hopf algebras [closed]

I couldn't find any idea to obtain a problem where Haskell programming language meets Category Theory, Algebraic Topology or Hopf algebras for an original and interesting problem. Also, I wonder ...
1
vote
0answers
61 views

Homotopy invariant deletions of open faces of simplicial complexes

Given a finite simplicial complex (as a topological space) $\Delta$ and a face $\tau$, suppose we delete the interior of $\tau$ (a point if $\tau$ is a vertex, otherwise homeomorphic to an open ball ...
7
votes
3answers
666 views

Is $S^1\vee S^1$ an Eilenberg-Mac Lane Space to a Homotopy Purist?

I believe that $S^1\vee S^1$ is the Eilenberg-Mac Lane space $K(\mathbb{Z}\ast\mathbb{Z},1)$. One can prove this by constructing its universal cover and observing that it is contractible. My question ...
4
votes
1answer
127 views

Seeking very regular $\mathbb Q$-acyclic complexes

This question was raised from a project with Nati Linial and Yuval Peled We are seeking a $3$-dimensional simplicial complex $K$ on $12$ vertices with the following properties a) $K$ has a complete $...
2
votes
0answers
46 views

Multiplicativity of the analytic index (or of kernel bundle)

What I want to ask is the multiplicativity of the analytic index of a family of Dirac operators. In the single operator case the analytic index of elliptic operator is multiplicative. This is proved ...
4
votes
0answers
218 views

Baum Connes Conjecture [closed]

I have recently decided on a topic for my master thesis. I want to compare the Baum Connes conjecture as it is formulated in topology to the conjecture as it is formulated in functional analysis. I ...
6
votes
3answers
328 views

Lifting symmetries to the universal cover

If $X$ is a connected topological space with universal cover $p: \tilde{X} \to X$, I believe any homeomorphism $f : X \to X$ can be 'lifted' to a homeomorphism $\tilde{f} : \tilde{X} \to \tilde{X}$. ...
3
votes
1answer
168 views

$\mathbb Z_2$-homotopy type of a $k$-connected, $(k+1)$-dimensional simplicial complex with a free involution

If $K$ is a finite, $k$-connected, $(k+1)$-dimensional simplicial complex then, by the theorems of Hurewicz and Whitehead, $|K|$ is homotopy equivalent to a point or to a wedge of $(k+1)$-dimensional ...
7
votes
1answer
306 views

Fundamental group of the space of maps into a classifying space

Let $P : E \to X$ be a principal $G$-bundle, where $G$ is a connected topological group. $P$ is classified by a map $f: X \to BG$. The group of gauge transformations $\mathcal{G}$ of $P$ is defined to ...
3
votes
0answers
120 views

Can ring spectra be thought of as some sort of operad in $Top$?

It is a result of May's work on operads that the homotopy category (or $\infty$-category, if you prefer) of connective spectra is equivalent to a full subcategory of the category of representations of ...
3
votes
1answer
222 views

Equivalent definition of a homotopy of functions

It is well known that given $X,Y$ arbitrarily topological spaces, $I$ the unit interval, and continuous functions $f, g : X \rightarrow Y,$ a homotopy between the functions is a continuous function $H ...
1
vote
1answer
228 views

Homotopy type of an oriented, closed, simply connected manifold

It is well known that every closed, oriented, simply-connected four-manifold $M$ is homotopy equivalent to a CW-complex consisting on a 0-cell, a wedge of two spheres and a 4-cell. I was wondering ...
3
votes
1answer
120 views

explicitly embedding a simplicial $d$-complex into $\mathbb{R}^{2d+1}$, or algorithms for doing so

A classical result in topology for which I can't find a reference for is that a simplicial complex $K$ of dimension $d$ with $n$ vertices can be linearly embedded into $\mathbb{R}^{k}$ when $k=2d+1$. ...
2
votes
1answer
208 views

Manifolds whose diffeomorphism group has the homotopy type of a manifold itself

I have a very stupid question. Let $M$ be a closed smooth manifold. In particular cases the homotopy type of the diffeomorphism group $Diff(M)$ can be very pathological. For example, in the case of $M=...
4
votes
1answer
108 views

Topological invariance of Stiefel-Whitney classes for open smooth manifolds

It is well known that Stiefel-Whitney classes are homotopy invariant for closed smooth manifolds. But in the case of open manifolds even $w_1$ is not a homotopy invariant (take just open cylinder and ...
3
votes
1answer
114 views

Kunneth formula for de Rham cohomology twisted by flat vector bundle

It is well known that for two manifolds $M$ and $N$ (let's say they are compact), the Kunneth formula says that $H(M\times N)=H(M)\otimes H(N)$, where $H$ denotes de Rham cohomology with complex ...
6
votes
1answer
219 views

Infinite families in stable homotopy groups

I will be very grateful for any advise or reference on the following. 1- How much is known about infinite families in ${_2\pi_*^s}$, the $2$-component of the stable homotopy ring? 2- How much is ...
4
votes
0answers
152 views

'Noether normalization' for finite group schemes

Throughout let $p$ be a prime, and let $k$ be a field of characteristic $p$. Let $G$ be a compact Lie group. Such a $G$ can always be embedded as a closed subgroup of $SU(n)$ for some $n$. This ...
8
votes
1answer
328 views

Computational complexity of computing simplicial homology

Is there any literature regarding the fastest known algorithm to compute the homology groups of a simplicial complex (on n vertices)? What about computing the fundamental group? The context is to tell ...
0
votes
1answer
95 views

Inverse Galois Problem ; Galois group of some branched cover of $P^{1}$ defined over $\mathbb{Q}$

I was trying to read a paper on Inverse Galois problem . I understands what the inverse Galois problem is. It asks if every finite group is the Galois group of some extension of the rationals. The ...
8
votes
0answers
165 views

What is known about maps between loop spaces of Spheres? - Reference request

What is know in general about the maps $\Omega^rS^n\rightarrow\Omega^sS^m$ between loop spaces of Spheres, or, perhaps to phrase it better, the groups $[\Omega^rS^n,\Omega^sS^m]$ for various values ...
8
votes
1answer
291 views

Is the natural isomorphism $|FX_\bullet| \cong F|X_\bullet|$ lax symmetric monoidal?

Let $\mathcal{V_1}$ and $\mathcal{V_2}$ be cocomplete symmetric monoidal categories, each endowed with a cosimplicial object $\Delta^\bullet=\Delta^\bullet_{\mathcal{V}_i}:\Delta \to \mathcal{V}_i$. ...
7
votes
0answers
91 views

Homology of inverse limits over inverse systems more complicated than towers

Most textbooks discussions of homology of inverse limits of chain complexes consider only “towers,” i.e. inverse systems indexed by the natural numbers. I’d like to find a reference that explains what ...
1
vote
1answer
246 views

$G_1 \rtimes G_2 \cong G_3 \implies BG_1 \times BG_2 \simeq BG_3$? $G_1 \times G_2 \cong G_3 \implies BG_1 \times BG_2 \simeq BG_3$? [closed]

As summarized in the title, suppose there is an isomorphism between $G_1 \times G_2$ and $G_3$, is it always true that $BG_1 \times BG_2$ is homotopy equivalent to $BG_3$? If it is not always true ...