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1
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0answers
46 views

computing homology of subvarieties of Euclidean spaces by persistent homology

Let $M$ be a submanifold of the Euclidean space $\mathbb{R}^n$. Let $G$ be a finite group acting on $M$ freely. I want to compute the homology (or even the cohomology ring) of $M/G$. Suppose the ...
-1
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0answers
17 views

Integral homology groups of the complexe projective n-plane [migrated]

I am reading "Morse theory" by Milnor and on page 27 we have proved that the homotopy type of CP^n is of a CW-complex of the form : a 0-cell attached to a 2-cell attached to a 4-cell ... attached to a ...
3
votes
0answers
93 views

Applications of cosheaf homology?

What are some applications of cosheaf homology within mathematics? Some ones I've heard of Sheaves (not cosheaves) are computing global sections and the Picard Group with a sheaf on projective space.
19
votes
3answers
572 views

Which paths in a graph are orthogonal to all cycles?

Start with some standard stuff. Suppose we have a directed graph $\Gamma$. I'll write $e : v \to w \,$ when $e$ is an edge going from the vertex $v$ to the vertex $w$. We get a vector space of 0-...
1
vote
1answer
206 views

Addition of two homology classes is zero in construction of Poincare Sphere

I ask here the question since it hasn't been answered in Math Stack Exchange. I am working through Greenberg and Harper, Lecture notes on Algebraic Topology, and I am having trouble with one ...
4
votes
1answer
133 views

“Small” simplicial complex with torsion trees

I am giving an expository talk soon about Duval-Klivans-Martin's paper Simplicial Matrix Tree Theorems, and I've been struggling to find a good example to do at the board. An important aspect of the ...
2
votes
0answers
71 views

why is $\cap \mu_B:H^k(\mathbb{R}^n,\mathbb{R}^n\setminus B;R)\to H_{n-k}(\mathbb{R}^n;R)$ an isomorphism? [closed]

I asked this http://math.stackexchange.com/q/1694046/309968 question already on MSE, but received no answer and I hope it's ok if I ask here for once. Let $R$ be commutative ring with $1_R$ Lemma: ...
6
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0answers
132 views

kernel of the mod $2$ Bockstein on the first cohomology group

Let $M$ be a path-connected finite $CW$-complex. Suppose the first integral homology group is $H_1(M;\mathbb{Z})= \mathbb{Z}_2^{\oplus r}\oplus A$ where $r\geq 1$ and $A$ is a finite abelian group of ...
4
votes
0answers
48 views

Obstructions to symplectically embedding compact manifolds of dimension $4$ or higher

It is known in Li's paper (http://arxiv.org/pdf/0812.4929v1.pdf) that in compact symplectic manifolds $(X^{2n},\omega)$ of dimension at least $2n\geq 4$, an immersed symplectic surface represents a $2$...
5
votes
1answer
209 views

a question about Bockstein spectral sequence

I find the following theorem for Bockstein spectral sequence at http://pages.vassar.edu/mccleary/files/2011/04/MC10.fin_.pdf, page 459: Question. for a fixed $k$, if $\beta$ does not hit $H_k(X;\...
1
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0answers
80 views

coefficient of homology of configuration spaces over real projective spaces

In the slides Characteristic Classes of Surface Bundles and Configuration Spaces, Miguel A. Xicot'encatl, page 38, what is the coefficient of the following homology? Could the coefficient be an ...
1
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0answers
24 views

Homology of the subcomplexes of the “diamond shaped” sphere under 1-norm in $R^n$ as a simplicial complex

The 1-norm on $\mathbb{R}^n$ is defined by $\|v\| = |v_1| + |v_2| + \cdots + |v_n|$ for a vector $v = (v_1, \ldots, v_n) \in \mathbb R^n$. The unit sphere $S^{n-1}_1$ under the 1-norm is a simplicial ...
10
votes
2answers
296 views

Rational homology sphere that is not Seifert manifold

I wonder if there is an example of rational homology sphere that is not a Seifert manifold. If there is, how can one construct such a rational homology sphere from a surgery of a knot in $S^3$?
10
votes
2answers
565 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}(...
0
votes
0answers
117 views

Homology of $S^n/G_x$

I tried to find the homology groups of the quotient of the unit sphere $S^{n-1}$ by an action of a finite subgroup $G$ of $SO(n)$. I'm especially concerned with $$H_i(S^{n-1}/G),\quad 1\leq i\leq n-2.$...
3
votes
0answers
179 views

What is the formula for the homology class represented by the diagonal?

Let $M$ be a compact oriented manifold and $\{\mu_{i}\}$ a basis for the homology $H_*(M, \mathbb{Z})$ (we are ignoring any torsion). Now consider the diagonal $\Delta_{M}$ inside $M\times M$. ...
3
votes
2answers
239 views

Is the “inverse” (i.e., the “cohomological”) numeration for singular (i.e., $H\mathbb{Z}$-)homology of spectra “acceptable”? [closed]

I have recently realized that in one of my (published) papers I have used the "inverse" numeration for the $H\mathbb{Z}$-homology of the objects of the stable homotopy category (so, if we consider ...
10
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0answers
183 views

Do homology classes have “special” representatives?

Recall that, according to Hodge, de-Rham cohomology classes of "nice enough" manifolds have "special" representatives - namely, harmonic forms. Now, how does one choose a "special" one among ...
4
votes
1answer
191 views

group completion theorem of homology as Hopf algebras

Let $M$ be a topological monoid with product $\mu$. Then $H_*(M)$ is a Hopf algebra with product $\mu_*$ and coproduct $\Delta_*$. The group-completion theorem by McDuff-Segal, 1976 gives that as a ...
3
votes
1answer
204 views

Constructing a homology class of degree $d(d-1)/2$ in $H_3(S^3)$

There is a nice construction of a class of degree $d^2$ in $H_3(S^3)$. Take a class $h$ of degree $d$ in $H_1(S^1)$, and then take its join with itself: $h*h$ is degree $d^2$ in $H_3(S^1*S^1)$, and $S^...
2
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0answers
78 views

Counting chain maps

I initially asked this question over at math stack exchange, you can find it here. I haven't really gotten any traction and I'm beginning to wonder if maybe its a harder question than I originally ...
2
votes
1answer
143 views

Configuration spaces of positive and negative particles

In the paper Mapping class group and function spaces: a survey, F. Cohen, M.A. Maldonado, page 3, line from bottom 1-3, it is given that for a $m$-manifold $M$, there is a map from the labelled ...
3
votes
1answer
128 views

group completion theorem by using homology fibrations

In the paper Homology fibrations and group completion theorem, McDuff-Segal (www.maths.ed.ac.uk/~aar/papers/mcdsegal.pdf), page 281: Let $M$ be a topological monoid such that $\pi_0M$ is generated by ...
5
votes
1answer
266 views

If a topological space has vanishing $n$th homology for every possible homology theory, does it have vanishing $n$th homotopy?

I don't have any strong preference as to whether or not the homology theories are required to be ordinary. Also, if this does not hold in general, does it hold for some nice category of spaces, like ...
-2
votes
1answer
95 views

stable splitting into a wedge sum [closed]

Suppose $X$ is a CW-complex such that there is a stable splitting of $X$ into wedge sum $$ \Sigma^t X\cong \bigvee _{k=1}^\infty Y_k. $$ (1). Does this imply $$ X\to \Sigma^tX\to \bigvee _{k=1}^\...
1
vote
1answer
177 views

Homology of manifold with action of group

Sorry for my ignorance in advance, this should be a very naive question and I would be happy for a reference. Let $G$ be an arbitrary group (not necessary finite) acting on two (connected) manifolds $...
9
votes
1answer
256 views

Dyer-Lashof operations and the homology of GL_n

For any ring R, $\bigsqcup_n {BGL}_n(R)$ is an $E_\infty$-space. Are there examples of rings where people have calculated $H_*(\bigsqcup_n {BGL}_n(R);\mathbb{Z}/2)$ and determined the Dyer-Lashof ...
1
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0answers
107 views

Homology and Burnside ring

If $G$ is a finite groupe, we denote $\mathcal{S}(G)$ the category of finite $G$-sets and $\mbox{I}(G)$ the set of isomorphism classes of it's objects. The Burnside ring of $G$, denoted by $\Omega(G)$,...
5
votes
1answer
235 views

Strange problem about triplets of differential forms

Suppose we have the following map: $$(\Omega^1(\mathbb{R}^n))^3\longrightarrow(\Omega^2(\mathbb{R}^n))^3$$ $$(\alpha,\beta,\gamma)\longmapsto(\mathrm{d}\alpha+\beta\wedge\gamma,\mathrm{d}\beta+\...
4
votes
1answer
530 views

A lower-dimensional algebraic topology problem between homology group and fundamental group

Let \begin{equation} A\stackrel{\alpha}{\longrightarrow}B\stackrel{\beta}{\longrightarrow}C\quad\quad (1) \end{equation} be a short sequence of (abelian or nonabelian) groups and homomorphisms. We say ...
3
votes
0answers
93 views

Homology of derivations of Differential Graded Lie algebra

Let $(L,d)$ be a Differential Graded Lie Algebra ($L=\bigoplus L_i$ and $d:L_i \to L_{i-1}$ satisfying the graded Leibniz rule). On the algebra $\mathrm{Der}L$ of derivations of $L$ define a grading ...
31
votes
2answers
1k views

Maps which induce the same homomorphism on homotopy and homology groups are homotopic

I am interested in the following question. Are maps which induce the same homomorphism on homotopy and homology groups homotopic? I am sure the answer is no, however I cannot imagine how to construct ...
1
vote
1answer
325 views

Example of torsion in orientable manifolds?

An orientable manifold can have torsion in its integer homology. But I believe by Poincare duality the manifold must be at least 4-dimensional -- isn't that right? Anyway are there simple examples ...
1
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0answers
124 views

Role of determinant of the matrix corresponding to $i$-th Homology group.

I was thinking about the proof of the Lefschetz's Fixed point theorem and the ingeniuty of the Hopf's Trace formula, i.e. associating the trace of the matrix for deciding about the fixed points. Now ...
1
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0answers
78 views

Homology and Exterior Square

Let $G$ be a finite group and $G \wedge G$ denote the exterior square of $G$. It is well known that the second integral homology $H_2(G,\mathbb{Z})$ is the kernel of homomorphism $x \wedge y \mapsto [...
14
votes
5answers
702 views

Take contraction wrt a vector field twice and define kernel mod image. Does that give anything interesting?

First, we make the following observation: let $X: M \rightarrow TM $ be a vector field on a smooth manifold. Taking the contraction with respect to $X$ twice gives zero, i.e. $$ i_X \circ i_{X} =0.$$...
2
votes
0answers
121 views

Tubular neighborhoods in the proof of the Morse homology theorem

I have a question regarding the proof of the Morse homology theorem given by D. Salamon in "Morse theory, the Conley index and Floer homology". The full text can be found here: http://www.mtm.ufsc.br/...
1
vote
2answers
201 views

The cohomology groups of $\Omega U(n)$

Let $\Omega U(n)$ be the loop space of $U(n)$. Is it true that the cohomology groups $H^*(\Omega U(n); \mathbb{Z})$ are torsion-free? How can one calculate these groups?
8
votes
1answer
314 views

Software for computing Thurston's unit ball

Is there any software which can be used for computing Thurston's unit ball(for second homology of 3-manifolds) of link complements? In particular can I do that with SnapPy? PS: even a table for ...
8
votes
1answer
393 views

Naturality of a Kunneth formula for cohomology

Let $X,Y$ be CW complexes. By Kunneth formula, we have a group isomorphim $$ H^n(X\times Y;G) \cong \oplus_{p+q=n} H^p(X;H^q(Y;G))$$ Is there a natural map realizing this isomorphism?
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2answers
186 views

Interpretation of $H_1(A_\mathbb{C}^{top},\mathbb{Q})$

In the paper "Sato-Tate Distributions and Galois Endomorphism Modules in Genus 2" (arxiv: http://arxiv.org/abs/1110.6638), the authors use the singular homology $H_1(A_\mathbb{C}^{top},\mathbb{Q})$ ($...
8
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0answers
430 views

Homology of Lie groups

Let $G$ be a Lie group and $G^{\delta}$ the underlying group (with discrete topology). Obviously, we have a continuous map of groups $i:G^{\delta}\rightarrow G$ which induces a map between classifying ...
3
votes
1answer
284 views

Second betti number of compact analytic spaces

Let $V$ be a proper singular complex algebraic variety, possibly nonprojective ($dim(V)=n>0$). I would like to know: 1) if its second Betti number is non zero, 2) same question but now $V$ is a ...
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9answers
2k views

Why localize spaces with respect to homology?

A basic construction in algebraic topology is the localization of spaces or spectra with respect to a homology theory: one formally inverts the $E$-homology isomorphisms, reflecting each space into ...
0
votes
0answers
230 views

Divisibility in homology/homotopy

I have a simply-connected CW-complex $F$ of finite-type, and I know that the imprimitivity of its particular integral homology is divisible by an odd prime $p$; that is, $$ \forall n,\exists \delta, \...
1
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0answers
84 views

some intuition about the degree of a map

Consider a map $$ f: \Sigma \to X/\sigma,$$ where $\Sigma=\Sigma_g/\Omega$ is a quotient of a Riemann surface by an antiholomorphic involution, $\sigma:X\to X$ is an antiholomorphic involution of some ...
2
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0answers
353 views

Cech homology (!) of the Warsaw Circle

Can anyone can give me a reference to the fact that first Cech homology (not cohomology!) group of the Warsaw Circle is $\mathbb{R}$? Thank you in advance :)
1
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4answers
323 views

Homology of infinite intersection

If $X_1\supseteq X_2\supseteq \ldots$ is a sequence of "nice" compact spaces, I would like to know whether the natural map from $H_*(\cap X_i)$ to the inverse limit $\lim \, H_*(X_i)$ is surjective. ...
1
vote
1answer
173 views

What are finite groups $H$ such that $H^n(H,\mathbb{Q/Z}) \cong H_n(G,\mathbb{Z})$?

Let $G$ be a finite group and $G^{\prime}$ be its commutator subgroup. Let $\mathbb{Z}$ and $\mathbb{Q}$ denote the integers and rationals. $\mathbb{Z}$ and $\mathbb{Q/Z}$ treated as trivial $G$-...
1
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1answer
180 views

Universal coefficient theorem for local ring

Let $R$ be a commutative local artin $k$-algebra,where $k$ is a field with characteristic $0$.I wonder whether universal coefficient theorem holds in this case.Namely,if $C$ is a chain of flat $R$-...