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

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1
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1answer
80 views

Mapping Class Group (MCG) of connected sum of 3-torus and $S^2\times S^1$

I know that the MCG (isotopy-classes of orientation preserving homeomorphisms) of 3-torus $(S^1\times S^1 \times S^1)$ is $SL(3,Z)$, since it is an Eilenberg–MacLane space, giving $ MCG(T^3)=Out(\pi_1(...
0
votes
0answers
35 views

Systole of a flat surface

Is the systole (length of the shortest saddle connection) of a flat surface $(X,\omega)$ ($X$ is a Riemann surface and $\omega$ an abelian differential on it with zeros in the points $\Sigma=\{p_1,\...
7
votes
1answer
750 views

semi flat connections

Let $L\to V$ be complex line bundle and $F_{t}:V\to V$, $t\in [0,1]$, be a loop of diffeomorphisms, $F_0=F_1=$ identity. For every $x\in V$, we get a loop $\gamma_x(t)=\{F_t(x)\}$ whose class in $\...
20
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4answers
2k views

Model structure on Simplicial Sets without using topological spaces

The category of simplicial sets has a standard model structure, where the weak equivalences are those maps whose geometric realization is a weak homotopy equivalence, the cofibrations are ...
2
votes
1answer
54 views

On the notion of conelike stratified (cs-) space

The notion of cs-stratification of a topological space is apparently due to Siebenmann, see also the paper by N. Habegger and L. Saper in the paper "Intersection cohomology of cs-spaces and Zeeman's ...
4
votes
1answer
96 views

Importance of $E_n$-algebras over ring structures on $\pi_*(E)$

Hopefully this question is not too vague to be closed. I am looking for examples of when a construction/theorem that involves $E$-(co)homology or even simply the ring $E_*$ requires an understanding ...
5
votes
1answer
199 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
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1answer
163 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 ...
21
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0answers
648 views

Applications of arithmetic topology to number theory

There is a well-known analogy between 3-manifolds and number fields, with knots corresponding to prime ideals. Are there any results in number theory that have been proven using topology through this ...
4
votes
2answers
138 views

rho invariant of manifolds

[I thought that I had already posted this question, but I couldn't find it in a search, so I apologize if I'm posting twice.] Let $G$ be a finite group. Then the rational oriented bordism ring $\...
23
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8answers
3k views

Why should I prefer bundles to (surjective) submersions?

I hope this question isn't too open-ended for MO --- it's not my favorite type of question, but I do think there could be a good answer. I will happily CW the question if commenters want, but I also ...
1
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1answer
114 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 ...
6
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2answers
232 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 $...
0
votes
1answer
390 views

Can monodromy be described by the same matrix for chosen generators in case of the same singularity type?

Let $X$ be a surface in $\mathbb{P}^3$. We have a fibration $f: X \longrightarrow \mathbb{P}^1$, and $f^{-1}(s_1)$ and $f^{-1}(s_2)$ have the same singularity type. Let $\gamma_1$ and $\gamma_2$ be ...
0
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0answers
93 views

Join of $G$-CW-Complexes

I want to understand the CW-structure on the join of $G$- CW complexes for my master's thesis. Let $G$ be a discrete group and $X$ and $Y$ $G$-CW-complexes. Furtheremore, let $X*Y$ denote the join $$[...
8
votes
2answers
261 views

How to compute $[CP^2, G/PL]$?

Let $E^4$ be the two stage Postnikov space appearing in the homotopy type of the classifying space $G/PL$. One of its properties is that it only has two nontrivial homotopy groups $\pi_2(E)=Z/2Z$ and $...
6
votes
1answer
150 views

Group bundles for topological spaces without universal cover

I‘m currently writing my Bachelor Thesis on (Co-)Homology with local coefficients. Let me first describe the situation: There are two approaches in defining Homology with local coefficients of a ...
6
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0answers
134 views

Where can I find basic “computations” of equivariant stable homotopy groups?

I am new to this subject; so please correct me if I will say something wrong or if you don't like my notation. In particular, I don't know whether it is reasonable to consider an infinite group $G$ (...
5
votes
1answer
243 views

Question on period map, Gauss-Manin connection and complex coordinates of $\mathcal{H}^1(k)$

Let $\mathcal{L}_g$ be the space of abelian differentials on Riemann surfaces of genus $g\ge 2$ and $\mathcal{TH}_g:=\mathcal{L}_g/Diff_0^+(S_g)$ be the Teichmuller space of abelian differentials on ...
9
votes
1answer
207 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
723 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 ...
24
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0answers
875 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 ...
3
votes
1answer
179 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.
3
votes
1answer
131 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 ...
0
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0answers
23 views

Random Variables with Simply Connected Support [closed]

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 $n\geq 1$?
2
votes
0answers
107 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-...
5
votes
2answers
220 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 ...
44
votes
10answers
9k views

Nice proof of the Jordan curve theorem?

As a student, I was taught that the Jordan curve theorem is a great example of an intuitively clear statement which has no simple proof. What is the simplest known proof today? Is there an intuitive ...
2
votes
0answers
60 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 $\mathbb{Z}/2$-...
1
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0answers
124 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 ...
40
votes
12answers
14k views

Algebraic Topology Beyond the Basics:Any Texts Bridging The Gap?

Peter May said famously that algebraic topology is a subject poorly served by its textbooks. Sadly, I have to agree. Although we have a freightcar full of excellent first-year algebraic topology texts ...
9
votes
2answers
359 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 H^n(BG;\mathbb{R}...
4
votes
1answer
387 views

“Polygons and gravitons” and Kodaira's theorem

I'm trying to understand the paper by Hitchin called: ''Polygons and gravitons". I'm stuck at page 470. At this point, he does some computations and obtains the conformal structure of the real ...
1
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1answer
188 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
84 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
123 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 $\sigma$ ...
2
votes
0answers
104 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 ...
2
votes
0answers
113 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
147 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 ...
8
votes
1answer
639 views

What's the detailed proof of “the composition of planar tangles is well-defined”?

In the planar algebra theory (see here or there section 2), a planar tangle is an isotopy class; then to define the composition of two tangles, we need to choose a representative in each classes. See ...
3
votes
0answers
76 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 ...
79
votes
7answers
8k views

Homotopy groups of Lie groups

Several times I've heard the claim that any Lie group $G$ has trivial second fundamental group $\pi_2(G)$, but I have never actually come across a proof of this fact. Is there a nice argument, ...
3
votes
0answers
93 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
189 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
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0answers
103 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 ...
7
votes
3answers
678 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
208 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
166 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 $N:=V(f_{2},f_{3})$ ...
2
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0answers
47 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 ...
0
votes
1answer
320 views

p-local space vs p-completion

I am having some trouble understanding the difference between the $p$-completion and a $p$-local space. If $X$ a simply connected space has all higher homotopy groups finitely generated, then the $p$-...