# Tagged Questions

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### Torsion in homology or fundamental group of subsets of Euclidean 3-space

Here's a problem I've found entertaining. Is it possible to find a subset of 3-dimensional Euclidean space such that its homology groups (integer coefficients) or one of its fundamental groups ...
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### What is (co)homology, and how does a beginner gain intuition about it?

This question comes along with a lot of associated sub-questions, most of which would probably be answered by a sufficiently good introductory text. So a perfectly acceptable answer to this question ...
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### Spaces with same homotopy and homology groups that are not homotopy equivalent?

A common caution about Whitehead's theorem is that you need the map between the spaces; it's easy to give examples of spaces with isomorphic homotopy groups that are not homotopy equivalent. (See Are ...
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### Smooth classifying spaces?

Take G to be a group. I care about discrete groups, but the answer in general would be welcome too. There are the various ways to construct the classifying space of G, bar construction, cellular ...
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### Teaching homology via everyday examples

What stories, puzzles, games, paradoxes, toys, etc from everyday life are better understood after learning homology theory? To be more precise, I am teaching a short course on homology, from ...
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### Why torsion is important in (co)homology ?

I've once been told that "torsion in homology and cohomology is regarded by topologists as a very deep and important phenomenon". I presume an analogous statement could be said in the context of ...
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### Does homology have a coproduct?

Standard algebraic topology defines the cup product which defines a ring structure on the cohomology of a topological space. This ring structure arises because cohomology is a contravariant functor ...
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For any group G we can form the lower central series of normal subgroups by taking $G_0 = G$, $G_1 = [G,G]$, $G_{i+1} = [G,G_i]$. We can check this gives a normal chain $G_0 > G_1 > ... > ... 3answers 2k views ### Homology theory constructed in a homotopy-invariant way Singular homology sends homotopic morphisms on equal morphisms and weakly equivalent spaces on isomorphic objects. So singular homology is in fact defined on the homotopy category of topological ... 2answers 968 views ### Are the homology and cohomology Serre spectral sequences dual to each other? If we use homology and cohomology over a field$k$, if a space has homology and cohomology groups of finite type in each degree, then$H_\ast(X;k)$is dual to$H^\ast(X;k)$using the universal ... 2answers 672 views ### Differentials in the Lyndon-Hochschild spectral sequence The Lyndon-Hochschild(-Serre) spectral sequence applies to group extensions in a manner analogous to the Serre-Leray spectral sequence applied to a fibration. Does anyone know of a good description (... 1answer 919 views ### On the wikipedia entry for Borel-Moore homology The wikipedia page on Borel-Moore homology claims to give several definitions of it, all of which are supposed to coincide for those spaces$X$which are homotopy equivalent to a finite CW complex and ... 2answers 572 views ### A homology theory which satisfies Milnor's additivity axiom but not the direct limit axiom? Let us agree on the following: a "homology theory" means a functor$h_*$from the category of pointed CW complexes to the category of graded abelian groups, together with natural isomorphisms$h_{*+1}(...
Let $X$ be a smooth projective variety. If I've understood correctly, the Weil conjectures imply that it is possible to compute the Betti numbers of $X(\mathbb{C})$ by computing the local zeta ...