The homology tag has no wiki summary.

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### Homology is computable because it is stable under suspension

I've heard it said that the reason why the homology groups of a space are a computable invariant is because they are a stable invariant in the sense that they are stable under suspension.
I'm ...

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### When do submanifolds lie in the same homology class? [closed]

Hello,
this may be a trivial question, but I am not very familiar with the topic.
Let (M,g) be a Riemannian Manifold. (In fact, we don't need the metric here.)
What exactly does it take for two ...

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### Projective dimension of zero module

Is there any consensus on what the projective dimension of the zero module should be? Here are three statements one commonly encounters in textbooks, sometimes with or without the condition $M\neq 0$:
...

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### Does a Dehn twist in the mapping class group of an cobordism give a BV-operator in string topology?

In her article Higher string topology operations, Godin in particular construct for each surface with $n$ incoming and $m \geq 1$ outgoing boundary circles an operation $H_\ast(BMod(S);det^{\otimes ...

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### What is the Hopf algebra structures in the homology of the based loop spaces of $E_7$ and $E_8$?

Since $\Omega X$ is a $H$-space, if it has homology of finite type, the homology acquires the structure of a Hopf algebra. Bott has shown that for $X=G$ a Lie group, in fact $H_*(\Omega X)$ is free ...

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### 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 ...

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### Homology dimension of the mapping class group of a surface with boundary

There is a result on the dimension bound for ${M_{g,n}}/S_n$, (the moduli space for Riemann surfaces of genus $g$ with $n$ marked points) that is
$H_{i}({M_{g,n}}/S_n)=0$, for $i\ge 6g-7+2n$ except ...

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### What does this naive attempt at $S^1$-equivariant homology describe?

After reading Cohen and Voronov's notes on string topology, one can find the following construction: Suppose we have a topological space $X$ with continuous action of $S^1$. This means we have a map ...

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### Computation of homology groups of $M_{g,n}$

First some definitions: $\bar{M_{g,n}}$ is Deligne-Mumford space, i.e., the moduli space of stable nodal complex projective curves of genus $g$ with $n$ marked points. It is a complex orbifold, ...

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### Proving homotopy invariance of cellular homology by constructing a chain homotopy

I'm trying to follow an argument in Lück's "Algebraische Topologie: Homologie und Mannigfaltigkeiten" (to which there apparently doesn't exist an english translation). The aim is to check homotopy ...

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### quasi-isomorphism

Is the distinction between quasi-isomorphism and `weak homotopy equivalence'
ONLY that the first means inducing an isomorphism in homology
and the second to an isomorphism of homotopy groups?

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### When are the homology and cohomology Hopf algebras of topological groups equal?

Suppose we have a topological group $G$, then the multiplication map $\mu$ and the diagonal map $\Delta$ provide the cohomology $H^\ast(G;R)$ (with Pontryagin coproduct and cup coproduct) and homology ...

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### The word “torsion” and its connection to geometry and homology

In an $R$-module $M$, an element $m \in M$ is said to be torsion if $am = 0$ for some $a \in R$ with $a \neq 0$.
Also, for a non-orientable (closed) surface such as the projective plane or the Klein ...

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### Singular homology of a graph.

By a graph I will understand an undirected graph without multiple edges or loops. By a morphism of graphs I will understand a map $f$ between the underlying sets of vertices, such that if $x$ and $y$ ...

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### Does this approach for the Poincare conjecture work?

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

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### When is there a deRham duality relation between the fundamental class and a top form.?

Hi, everyone:
I am reading a small expository paper on properties of CP2,
in which the intersection form is defined as an integral of
the wedge of two forms $w_1$, $w_2$, and these forms ...

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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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### Request: intermediate-level proof: every 2-homology class of a 4-manifold is generated by a surface.

Hi, everyone:
For the sake of context, I am a graduate student, and I have taken classes in
algebraic topology and differential geometry. Still, the 2 proofs I have found
are a little too terse for ...

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### Working with Intersection Forms in Homology. Computation.

Hi, everyone:
I am trying to work with the intersection form in 4-manifolds. Specifically,
I am working with $CP^2$ (complex projective 2-space.), whose form is given by $(1)$.
Now, I know how to ...

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### What would be the ramifications of homotopy theory being as easy as homology theory?

Greg Muller, in a post called Rational Homotopy Theory on the blog "The Everything Seminar" wrote
"I tend to think of homotopy theory a little bit like ‘The One That Got Away’ from mathematics as a ...

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### Are fundamental groups of aspherical manifolds Hopfian?

A group $G$ is Hopfian if every epimorphism $G\to G$ is an isomorphism. A smooth manifold is aspherical if its universal cover is contractible. Are all fundamental groups of aspherical closed smooth ...

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### Given a ramified cover of a Riemann surface, is there a good choice of basis for H_1 of the source?

Suppose we are given a map $f: X \to Y$ between two Riemann Surfaces, with branch points $p_1,p_2,\dots,p_n$ and known multiplicities at these points. Assuming we have a basis of $H_1(Y, ...

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### Homology of Surfaces with Holes

The classification theorem for surfaces says that the complete set of homeomorphism classes of surfaces is
{ $S_g : g \geq 0$ } $ \cup$ { $N_k : k \geq 1$ },
where $S_g$ is a sphere with $g$ ...

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### “Homotopy-first” courses in algebraic topology

A first course in algebraic topology, at least the ones I'm familiar with, generally gets students to a point where they can calculate homology right away. Building the theory behind it is generally ...

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### Homology with Coefficients

We can define the (first) homology of a surface $S$ by working with graphs embedded in $S$. That is, we take any (oriented) graph which is 2-cell embedded in $S$, and take cycles modulo boundaries in ...

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### De Rham homology

Suppose M is an arbitrary smooth manifold and D is its bundle of 1-densities.
On the category of finite-dimensional vector bundles over M and linear differential operators between them
there is a ...

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### homology with compact supports

In one of the exercises in McDuff and Salamon, they mention homology with compact supports. I know how to define *co*homology with compact supports, but I can't picture the homology version. How do ...

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### Intersection form in twisted homology (homology with local coefficients)

The answer to this question should be obvious, but I can't seem to figure it out. Suppose we have a surface $F$, and a representation $\rho : \pi_1(F)\to SU(n)$. We can define the homology with local ...

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### What can be said about the homotopy groups of a CW-complex in terms of its (co)homology?

One example is the Hurewicz theorem which tells us that (e.g) a CW-cx with only one 0-cell has a nontrivial fundamental group if H_1 is nontrivial. What other examples are there? (The CW-complexes I ...

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### Is it always possible to compute the Betti numbers of a nice space with a well-chosen Lefschetz zeta function?

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 ...

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### CW-structures and Morse functions: a reference request

The following is probably well known, but I wasn't able to locate a reference in the literature.
Let $f$ be a Morse function on a smooth compact manifold $M$ without boundary and let $\rho$ be a ...

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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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### 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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### 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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### 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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### 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 ...

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### Cohomology of associative algebras

Let $A$ be an associative algebra over a commutative ring $k$. I've read statements saying that Hochschild (co)homology is the "right" notion of (co)homology for associative algebras. When $A$ is ...

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### What is a cohomology theory (seriously)?

This question has bugged me for a long time. Is there a unifying concept behind everything that is called a "cohomology theory"?
I know that there exist generalized cohomology theories, Weil ...

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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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### Are there two non-diffeomorphic smooth manifolds with the same homology groups?

I know that there definitely are two topological spaces with the same homology groups, but which are not homeomorphic. For example, one could take T^2 and S^{1}\vee S^{1}\vee S^{2} (or maybe ...

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### For which spaces is homology (or cohomology) determined by the Eilenberg-Steenrod axioms

This is a spinoff of Can anyone give me a good example of two interestingly different ordinary cohomology theories? . By an ordinary homology theory, I mean a functor on topological spaces which ...

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### Can one calculate the (co)homology of the loopspace of a Lie group from its Lie algebra?

Compact connected simply-connected Lie groups have so much structure that you can calculate their cohomology from their Lie algebras using Lie algebra cohomology (certain Ext-groups) and similarly ...

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### Why is homology not (co)representable?

This is in the same vein as my previous question on the representability of the cohomology ring. Why are the homology groups not corepresentable in the homotopy category of spaces?

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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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### 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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### 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 ...

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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 ...