Questions tagged [at.algebraic-topology]

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

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1answer
723 views

references for models of stable infinity categories

There's a fair amount of literature comparing different models for the homotopy theory of homotopy theories, or the homotopy theory of $(\infty,1)$-categories. Julie Bergner has a survey of this ...
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Meaning of orientation/orientability over rings other than the integers

This was asked as part of an earlier question. But since this part did not attract many answers, I am asking it separately. We consider the homology definition of an orientation for a manifold, as ...
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2answers
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Five lemma in HoTop* and arbitrary pointed model categories

Let $\textbf{HoTop}^*$ be the homotopy category of pointed topological spaces. In the following, the word "isomorphism" shall always mean isomorphism in $\textbf{HoTop}^*$, i.e. pointed homotopy ...
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4answers
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Classifying Space of a Group Extension

Consider a short exact sequence of Abelian groups -- I'm happy to assume they're finite as a toy example: $$ 0 \to H \to G \to G/H \to 0\ . $$ I want to understand the classifying space of $G$. Since ...
25
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3answers
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Reverse mathematics of (co)homology?

Background Exercise 2.1.16b in Hartshorne (homework!) asks you to prove that if $0 \rightarrow F \rightarrow G \rightarrow H \rightarrow 0$ is an exact sequence of sheaves, and F is flasque, then $0 \...
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Cobordisms of bundles?

Is there a notion of a cobordism which is compatible with bundle structure? That is, if I have bundles $E$ and $F$, is it the case that the manifold $W$ with $E$ and $F$ as boundary components, can ...
3
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1answer
130 views

Upper bound on the genus of a k-page graph

Is there an upper bound on the genus of a graph that has a book embedding on say k pages, or can the genus be arbitrarily large? If not a general bound is known, what happens for k=3?
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What is the Euler characteristic of a Hilbert scheme of points of a singular algebraic curve?

Let $X$ be a smooth surface of genus $g$ and $S^nX$ its n-symmetrical product (that is, the quotient of $X \times ... \times X$ by the symmetric group $S_n$). There is a well known, cool formula ...
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Burnside ring and zeroth G-equivariant stem for finite G

Let $G$ be a finite group. The theorem that the Burnside ring $A(G)$ is isomorphic to the zeroth stable stem $\pi^{G}_0(S)$ is usually said to originate from Segal. I search for a reference of a proof ...
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5answers
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two conjugate subgroups and one is a proper subset of the other? plus, a covering space interpretation.

Recently I've been reading J.P. May's A Concise Course in Algebraic Topology. In the section on the classification of covering groupoids, he mentions that sometimes a group G may have two conjugate ...
7
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1answer
374 views

Reference for equivalent definitions of the genus

Let $X$ be a (edit: nonsingular) projective complex algebraic curve. The genus of $X$ can be defined as the dimension of the space of holomorphic $1$-forms on $X$, which in turn can be defined either ...
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2answers
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Cohomology of Lie groups and Lie algebras

The length of this question has got a little bit out of hand. I apologize. Basically, this is a question about the relationship between the cohomology of Lie groups and Lie algebras, and maybe ...
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1answer
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complex structure on S^n

Using the chern character, it can be shown that there is no complex structure on $S^n$ if $n > 6$: See May's book: if $S^{2n}$ has a complex structure, let $\tau$ be the tangent bundle. $c_n(\tau) =...
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3answers
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Grassmannian bundle theorem

Let's consider a vector bundle $E$ of rank $n$ over a compact manifold $X$. Consider the associated Grassmannian bundle $G$ for some $k < n$, obtained by replacing each fiber $E_x$ by $Gr(k,E_x)$. ...
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2answers
608 views

Properties of the class of topological spaces possessing a CW-structure

Let ${\mathcal C}$ be the class of topological spaces which carry a CW-structure (note that I do not want to fix some particular CW-structure). Is it true that for a covering map $E\stackrel{f}{\to} ...
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Applications of the Brown Representability Theorem

Probably you can "google" this question, but I can't find anything relevant. The classical Brown Representability Theorem states: Denote $hCW_*$ the homotopy category of pointed CW-complexes. Let $F : ...
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unpointed brown representability theorem

The classical Brown Representability Theorem states: Denote $hCW_*$ the homotopy category of pointed CW-complexes. Let $F : hCW_* \to Set_*$ be a contravariant functor. Then $F$ is representable if ...
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3answers
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Cohomology of fibrations over the circle: how to compute the ring structure?

This question is inspired by Cohomology of fibrations over the circle Moreover, it can be considered a subquestion of the above, but somehow it seems to me that some of the more interesting points ...
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Mumford conjecture: Heuristic reasons? Generalizations? … Algebraic geometry approaches?

The Mumford conjecture states that for each integer $n$, we have: the map $\mathbb{Q}[x_1,x_2,\dots] \to H^\ast(M_g ; \mathbb{Q})$ sending $x_i$ to the kappa class $\kappa_i$, is an isomorphism in ...
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0answers
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Can one calculate Ext's between microlocalized perverse sheaves/D-modules using topology?

So, I know one really good technique for calculating Ext's between perverse sheaves/D-modules using topology: the convolution algebra formalism, worked out in great detail in the book of Chriss and ...
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2answers
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Homotopy Pushouts via Model Structure in Top

As far as I know, one way to take a homotopy colimit in a model category is to replace (up to acyclic fibration) all arrows in the diagram with cofibrations, and take the strict colimit of the ...
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3answers
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H-space structure on infinite projective spaces

Any Eilenberg-MacLane space $K(A,n)$ for abelian $A$ can be given the structure of an $H$-space by lifting the addition on $A$ to a continuous map $K(A\times A,n)=K(A,n)\times K(A,n)\to K(A,n)$. Does ...
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Are non-empty finite sets a Grothendieck test category?

A "test category" is a certain kind of small category $A$ which turns out to have the following property: the category $\widehat{A}$ of presheaves of sets on $A$ admits a model category structure, ...
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1answer
842 views

Why does the internal singular simplicial space realize to the same thing as the discrete singular simplicial set?

There are two version of the singular simplicial space of a topological space $X$, one discrete and one internal. At least if X is nice, both of them have homotopy equivalent geometric realizations (...
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Does homology detect chain homotopy equivalence?

Is the following true: If two chain complexes of free abelian groups have isomorphic homology modules then they are chain homotopy equivalent.
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Two kinds of orientability/orientation for a differentiable manifold

Let $M$ be a differentiable manifold of dimension $n$. First I give two definitions of Orientability. The first definition should coincide with what is given in most differential topology text books, ...
4
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1answer
864 views

relationship between borromean rings and hanging-a-picture-from-three-nails puzzle?

I recently heard the following puzzle: There are three nails in the wall, and you want to hang a picture by wrapping a wire attached to the picture around the nails so that if any one nail is removed ...
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3answers
984 views

What's the analogue of the Hilbert class field in the following analogy?

There's a wonderful analogy I've been trying to understand which asserts that field extensions are analogous to covering spaces, Galois groups are analogous to deck transformation groups, and ...
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5answers
837 views

Killing the torsion in homotopy

Origin This question was asked by John Baez in This Week's Finds in Mathematical Physics (Week 286). Therefore, please don't upvote this question (unless you really want to), but do upvote the ...
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4answers
494 views

Examples of the varying strengths of topological invariants

In my first algebraic topology class, I remember being told that the simplest reason for homology was to distinguish spaces. For example, if is X=circle and a Y= wedge of a circle and a 2-sphere then ...
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1answer
364 views

Killing Chern classes

Let $G$ be a compact connected Lie group and let $E\to B$ be a principal $G$-bundle. Suppose $a$ is a rational cohomology class of $E$ such that its pullback $b$ under an orbit inclusion map $G\to E$ ...
13
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1answer
804 views

Stable ∞-categories as spectral categories

Let C be a stable ∞-category in the sense of Lurie's DAG I. (In particular I do not assume that C has all colimits.) Then C does have all finite colimits, the suspension functor on C is an ...
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3answers
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categorical homotopy colimits

let $hTop_*$ denote the homotopy category of pointed spaces. I believe that it has no pushouts, in general. the reason is that you can't expect the involved homotopies to be compatible. can anyone ...
8
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2answers
364 views

Formulas for vector fields on Grassmannians?

The Wikipedia article on (real) Grassmannians gives a simple argument that the Euler characteristic satisfies a recurrence relation $$\chi G_{n,r} = \chi G_{n-1,r-1} + (-1)^r \chi G_{n-1,r}$$. This ...
12
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2answers
792 views

Infinity de Rham quasi-isomorphism

This question is similar to Do chains and cochains know the same thing about the manifold? in the sence that both deal with a natural "comparison" quasi-isomorphism that does not preserve the ring ...
2
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1answer
593 views

Any reason why K_23(Z) has order 65520?

I'm rereading my notes and they mention that $K_{23}(\mathbb Z) = \mathbb Z/(65520)$ This looks like a good point to stop and ask whether there is any explanation for this $K$-group of integers (23 ...
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5answers
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Equivalence of ordered and unordered cech cohomology.

Given a topological space X and a finite cover X = $\cup X_i$, one can define Cech cohomology of a sheaf of abelian groups F with respect to the cover $\{X_i\}$ in two different ways: (Ordered): ...
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4answers
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Algorithm or theory of diagram chasing

One of the standard parts of homological algebra is "diagram chasing", or equivalent arguments with universal properties in abelian categories. Is there a rigorous theory of diagram chasing, and ...
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4answers
551 views

Realizing complexes with bases as cellular complexes

This is a question a friend of mine asked me some time ago. I suspect the answer is "no" but can't prove it. Every free complex of abelian groups is isomorphic to the reduced cellular complex of some ...
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4answers
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Characteristic classes in generalized cohomology theories?

Hello, 'ordinary' Stiefel-Whitney classes are elements of the singular cohomology ring and are constructed using the Thom isomorphism and Steenrod squares. So I think they should exist for any (...
3
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1answer
266 views

disagreement between two definitions of the singular boundary map

Hi everyone, I have a little problem with the definition of singular boundary map in singular homology theory. It appears to be some disagreement between two authors. The first one is Hatcher in his '...
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4answers
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Equivariant singular cohomology

One can define the $G$-equivariant cohomology of a space $X$ as being the ordinary singular cohomology of $X \times_G EG$ --- I think this is due to Borel? (See e.g. section 2 of these notes) ...
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3answers
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How does one find vanishing algebraic cycles?

I have a question, related to what I asked before. Let's consider a smooth hyperplane section $X$ of a smooth projective variety $Y$ over $\mathbb C$. According to Weak Lefschetz theorem, cohomology ...
17
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3answers
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What are the fibrant objects in the injective model structure?

If C is a small category, we can consider the category of simplicial presheaves on C. This is a model category in two natural ways which are compatible with the usual model structure on simplicial ...
18
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1answer
789 views

Do chains and cochains know the same thing about the manifold?

This question was inspired by Poincaré quasi-isomorphism Let $M$ be a closed oriented $n$-manifold. The cap product with the fundamental class of $M$ induces an isomorphism $H^i(M,\mathbf{Z})\to ...
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2answers
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Poincaré quasi-isomorphism

Suppose we have a simplicial combinatorial manifold (just a triangulated manifold) and its Poincaré dual cell complex. Corresponding homology simplicial and homology cell complexes are quasi-...
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Why do finite homotopy groups imply finite homology groups?

Why does a space with finite homotopy groups [for every n] have finite homology groups? How can I proof this [not only for connected spaces with trivial fundamental group]? The converse is false. $\...
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2answers
492 views

(Co-) Homology associated to Waldhausen K-Theory

Waldhausen K-Theory takes as input a Waldhausen category C and produces a spectrum K(C). I would like to know what is known about generalized (co-) homology theories that can be realized by this ...
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846 views

Geometry of the multilagrangian Grassmannian

Let's introduce the following variety $MG(3,6)$, which is a "multisymplectic" analog of a Lagrangian Grassmannian $LG(3,6)$. Consider a 3-form $\omega = dx1 \wedge dx2 \wedge dx^3 - dx4 \wedge dx5 \...
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2answers
449 views

Branched coverings over orbifolds with reflector lines

It is well known that if $F\to B$ is a $n$-finite branched covering over an orbifold with cone-points then the orbifold Euler's characteristics are related via $\chi(F)=n(\chi(B)-\sum_i^r\frac{a_i-1}{...