Questions tagged [at.algebraic-topology]

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

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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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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 ...
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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 ...
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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 ...
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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 ...
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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!
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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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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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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 ...
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(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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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 ...
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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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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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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 ...
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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. ...
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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 ...
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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 $n<5$...
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 ...