Questions tagged [at.algebraic-topology]

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

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8
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2answers
864 views

Can we make rigorous the 'obvious' characterisation of singular homology?

It is a well known and often touted fact that the singular homology groups 'count the k- dimensional holes' in a space (see: How does singular homology H_n capture the number of n-dimensional "...
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2answers
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the hopf invariant of the hopf construction

I'm having some trouble with a problem about the Hopf construction, in the exercises for Ch. 4 of Mosher & Tangora. Given a map $g : S^{n-1} \times S^{n-1} \rightarrow S^{n-1}$, we get a map $h(...
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2answers
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tangent sphere bundle over sphere

are there some general description about tangent sphere bundle over sphere? (it is a special $S^{n-1}$bundle over $S^n$) say for n=1,it is trivial,$S^0\times S^1$,for n=2,it is $SO(3)\cong \mathbb{R}...
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1answer
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Functoriality of Poincaré duality and long exact sequences

Hi all, Today I was playing with the cohomology of manifolds and I noticed something intriguing. Although I might just have been caught out by a couple of enticing coincidences, it feels enough like ...
5
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1answer
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Is geometric realization of the total singular complex of a space homotopy equivalent to the space?

Let $X$ be a topological space and let $|Sing(X)|$ be the geometric realization of the total singular complex of $X$. Then $|Sing(X)|$ is a CW complex with one cell for each non-degenerate singular ...
13
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1answer
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An elementary proof that the degree of a map of spheres determines its homotopy type

I'm helping to teach an undergraduate algebraic geometry course (out of Hatcher's textbook). We have recently defined the degree of a map of spheres using homology, and the professor and I thought it ...
13
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6answers
4k views

Fundamental group of the line with the double origin.

In the simplest cases, the fundamental group serves as a measure of the number of 2-dimensional "holes" in a space. It is interesting to know whether they capture the following type of "hole". This ...
3
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3answers
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Homotopy Equivalence of Punctured Tori

I was told recently that if I take a 2-torus (genus 2) and remove 1 point, then this is homotopy equivalent to a torus with 3 points removed. This may be really easy but I don't see it. Thank you!
14
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2answers
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Is a conceptual explanation possible for why the space of 1-forms on a manifold captures all its geometry?

Let $M$-be a differentiable manifold. Then, suppose to capture the underlying geometry we apply the singular homology theory. In the singular co-chain, there is geometry in every dimension. We look at ...
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3answers
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Are there applications of algebraic geometry into algebraic topology?

It is described in many sources that algebraic topology had been a major source of innovation for algebraic geometry. It is said that the uses of cohomology, sheaves, spectral sequences etc. in ...
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1answer
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How to compute the (co)homology of a compact Riemann surface?

The situation is the following. A finite-index subgroup $\Gamma$ of $SL_2(\mathbb Z)$ acts on the upper-half plane $\mathcal H$. It has a fundamental domain, obtained by a union of translates of the ...
4
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0answers
300 views

(Equivariant) Sheaves of Equivariant Spectra?

This is a very naive question but 1)given a compact Lie group G, is there a good notion of a sheaf of equivariant spectra on a G-space X analogous to the model structure that Brown develops in his ...
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2answers
275 views

Can all induced maps be described categorically.?. (or at least as generally as possible)

Hi: I am new here. I went over the fAQ's, still, sorry if I break protocol. I am pretty confused about induced maps in different areas of algebraic topology; I do know how these induced maps are ...
9
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2answers
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Why the choice of the simplex for defining homology?

This question arose out of my attempts to understand another question. The most popular construction for the chain complex for defining singular homology uses the $n$-simplex. But it is also possible ...
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1answer
2k views

Thom isomorphism

Let$p:E\rightarrow B$ be an n dimensional vector bundle, R be a commutative ring. Assume that B is simplyconnected or char R=2. Then there is an element $U\in H^n(M(p);R)$ such that we have dual ...
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1answer
160 views

Does there exists a (possibly homological) characterization of the Jordan curve property in all dimensions?

More precisely, let $M$ be a subspace $\mathbb R^n$ with the following properties: $M$ is a topological manifold of dimension $n-1$. M is compact. Does there exist a homological characterization of ...
3
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2answers
661 views

Cohomology of complex projective spaces with coefficientes in a complex-orientable cohomology theory

Hello everyone, I'm having problems understanding a basic fact about complex-orientable cohomology theories: Let $E^{\ast}$ be a multiplicative cohomology theory and $x\in E^2({\mathbb C}\text{P}^{\...
24
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5answers
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(Co)homology of the Eilenberg-MacLane spaces K(G,n)

Let $(G, n)$ be a pair such that $n$ is a natural number, $G$ is a finite group which is abelian if $n \geq 1$. It is well-known that $\pi_n(K(G,n)) = G$ and $\pi_i (K(G,n)) = 0$ if $i \neq n$. Also ...
7
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1answer
644 views

Hopf fibration inside the retraction of R^4 minus line -> S^2?

This was inspired by this question. Let $Y = {\mathbb R}^4 \setminus$a coordinate line, which retracts to ${\mathbb R}^3 \setminus$a point, which retracts to $S^2$. What is an explicit immersion $S^...
32
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3answers
8k views

Why do the homology groups capture holes in a space better than the homotopy groups?

This is a follow-up to another question. A good interpretation of having an $n$-dimensional hole in a space $X$ is that some image of the sphere $\mathbb{S}^n$ in this space given by a mapping $f: \...
3
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1answer
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Quick ways to calculate cohomology of vector bundle/local system from transition functions?

Suppose I have a vector bundle (or local system, or something else given by transition functions) on a Riemann surface (or generally a (complex) manifold), and I want to compute its cohomology. The ...
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3answers
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Survey articles on homotopy groups of spheres

Are there general surveys or introductions to the homotopy groups of spheres? I'm interested especially in connections to low-dimensional geometry and topology.
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1answer
332 views

Homotopy colimits of cyclic spaces

Let $\Lambda$ denote Connes's cyclic category. It is an extension of the simplex category $\Delta$ (of nonempty finite linearly ordered sets) obtained by adding an automorphism of order $n+1$ to the ...
8
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2answers
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How does singular homology H_n capture the number of n-dimensional “holes” in a space?

This is a foundational doubt I have. How does singular homology H_n capture the number of n-dimensional holes in a space? We disregard the case of $H_0$ as it has the very satisfactory explanation ...
14
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2answers
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Ordinary cohomology of stacks

Let $\mathbf{X}$ be a stack over $Top$ (a lax sheaf of groupoids, or some such thing). If it admits a surjective representable map $F \to \mathbf{X}$ then one can form the iterated fibre product to ...
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1answer
991 views

Does there exist a classification of covering spaces in algebraic geometry?

This is a question based on the heuristics that most things in algebraic/differential topology has an analogue in algebraic geometry. The fundamental group classifies the covering spaces of a (...
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2answers
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Functoriality of fundamental group via deck transformations

Problem I'm trying to understand this with a view towards the etale fundamental group where we can't talk about loops. What I'm missing is how the fundamental group functor should work on morphisms, ...
3
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1answer
256 views

Cartesian-closed category of spaces with the Whitehead property?

I'm not sure if this is standard, but we'll call the property that every weak homotopy equivalence is an honest homotopy equivalence the Whitehead property (from Whitehead's theorem for CW complexes). ...
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2answers
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Čech cohomology of compact spaces via closed covers?

Let X be a compact space. Recall that its Čech cohomology $H^\bullet(X,\mathbb Z)$ is given by the colomit $\mathrm{colim}_U\big(H^*(C^\bullet(U;\mathbb Z),\delta)\big)$, where $U=(U_i)$ runs over ...
9
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2answers
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Thom's seminal cobordism paper in English?

In Quelques proprietes globales des varietes differentiables, Thom classifies unoriented manifolds up to cobordism. I've been struggling a bit to understand this paper, and while Stong's cobordism ...
5
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1answer
218 views

Nonuniqueness of maps of representing spaces

In Rudyak's On Thom Spectra, Orientability, and Cobordism, two variants of Brown's representability theorem are presented: given a natural transformation $f^*: E^* \to F^*$ of cohomology theories, ...
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4answers
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Simplicial complexes vs. geometric realization of abstract simplicial complexes

A finite abstract simplicial complex is a pair $D=(S,D)$ where $S$ is a finite set and $D$ is a non-empty subset of the power set of $S$ closed under the subset operation, e.g. $(\{a,b,c\},\{\emptyset,...
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2answers
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Does this approach for the Poincaré conjecture work?

Several months ago a paper was posted at http://arxiv.org/abs/1001.4164 called "Another way of answering Henri Poincaré's fundamental question." The author gave a talk on it today at my institution. ...
5
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2answers
730 views

Why is the intersection of complex submanifolds always positive.?

Hi, everyone: I was finally able to show that all complex manifolds are orientable, by generalizing to many variables the fact that , for a single complex variable, the Jacobian matrix is ...
4
votes
1answer
702 views

Pontrjagin square (and possible typo in Mosher & Tangora?)

There's an exercise at the end of Ch. 2 of Mosher & Tangora's "Cohomology Operations and Applications in Homotopy Theory", which says: Suppose the cocycle $u\in C^{2p}(X;Z)$ satisfies $\delta u=...
8
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2answers
511 views

Cofinal inclusions of Waldhausen categories

Let $\mathcal{C}$ be a Waldhausen category. Suppose that $\mathcal{B}$ is a subcategory of $\mathcal{C}$, and that $\mathcal{B}$ is closed under extensions. If $\mathcal{B}$ is strictly cofinal in $\...
8
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2answers
639 views

Reference request for relative bordism coinciding with homology in low dimensions

It's a standard fact that, for finite CW complexes, the relative (edit: oriented) bordism group $\Omega_n(X,A)$ coincides with the homology $[H_\ast(X,A;\Omega_\ast(pt))]_n\simeq H_n(X,A)$ for $n<5$...
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2answers
637 views

What is an example of a non-regular, totally path-disconnected Hausdorff space?

I need this for a counterexample: the multiplication in the fundamental group $\pi_1(\Sigma X_+)$, when it is equipped with the topology inherited from $\Omega \Sigma X_+$, fails to be continuous for ...
3
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2answers
623 views

CW structure on spaces obtained by attaching cells wildly

Is there necessarily a CW structure on a space build out of cells without demanding them to be attached in "right" order? More precisely, let $X$ be a topological space such that the map $\emptyset\...
40
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7answers
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Non-examples of model structures, that fail for subtle/surprising reasons?

An often-cited principle of good mathematical exposition is that a definition should always come with a few examples and a few non-examples to help the learner get an intuition for where the concept's ...
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3answers
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Connected components of space of maps between two manifolds

Question: What are the connected components of the familiar spaces of functions between two (let's say compact and smooth, for simplicity) manifolds $M$ and $N$? Specifically, I'm thinking of the ...
32
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4answers
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Mathematically mature way to think about Mayer–Vietoris

This question is short but to the point: what is the "right" abstract framework where Mayer-Vietoris is just a trivial consequence?
6
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1answer
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Maslov index of a pullback bundle

This question has bugged me as I read McDuff-Salamon's book on pseudoholomorphic curves. I'll use their terminology. Let $\Sigma$ be a compact surface possibly with boundary, $M$ an almost-complex ...
6
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1answer
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When does a “representable functor” into a category other than Set preserve limits?

This might be a dumb question. If $C$ is an ordinary category, then for any $c \in C$ the covariant representable functor $\text{Hom}(c, -) : C \to \text{Set}$ preserves limits. However, it can ...
6
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2answers
570 views

Graphs of maps between manifolds as cycles and intersection theory

I'm guessing that the answer to this question is well-known, but I'm struggling to find anything to help me. Let $X,Y$ be compact manifolds of dimension $n,m$ respectively. Let $f:X \to Y$ be a ...
0
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1answer
440 views

Natural embedding GL_n(C) -> C^{n^2} \ {0} induces zero on cohomology

The question looks like an exercise in elementary algebraic topology, but I didn't manage to solve it. I am considering this question because it is a toy example in a problem I'm thinking about. Let'...
16
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2answers
694 views

The kernel of the map from the handlebody group to Outer automorphisms of a free group

Let $K$ be a compact oriented 3-dimensional handlebody of genus $g$. The group $H_g$ of isotopy classes of diffeomorphisms of $K$ is called the handlebody group. (It embeds as a subgroup of the ...
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2answers
2k views

Group action, Fixed point set and Orbit Space

I want to know to what extent is the group action determined by its fixed point data and orbit data, i.e. if $G$ acts on $M$ in two ways with the same fixed point set and orbit space, on what ...
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3answers
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How to demonstrate $SO(3)$ is not simply connected?

A quote from Wikipedia's article on the Rotation group: Consider the solid ball in $\mathbb{R}^3$ of radius $\pi$ [...]. Given the above, for every point in this ball there is a rotation, ...
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3answers
601 views

Is every (finite) group action on R^n by diffeomorphisms conjugate to a linear action?

I want to know if every smooth (finite)group action on $\mathbb{R}^n$ is conjugate to some linear action.Thank you!