Questions tagged [homology]

Homology is a general way of associating a sequence of algebraic objects such as abelian groups or modules to other mathematical objects such as topological spaces.

212 questions
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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 ...
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“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 ...
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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 https://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: ...
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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 ...
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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 $2$...
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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 $... 1answer 479 views Dyer-Lashof operations and the homology of GL_n For any ring R,$\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)$and determined the Dyer-Lashof ... 0answers 124 views Homology and Burnside ring If$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)$,... 1answer 258 views Strange problem about triplets of differential forms Suppose we have the following map: $$(\Omega^1(\mathbb{R}^n))^3\longrightarrow(\Omega^2(\mathbb{R}^n))^3$$ $$(\alpha,\beta,\gamma)\longmapsto(\mathrm{d}\alpha+\beta\wedge\gamma,\mathrm{d}\beta+\... 1answer 548 views A lower-dimensional algebraic topology problem between homology group and fundamental group Let $$A\stackrel{\alpha}{\longrightarrow}B\stackrel{\beta}{\longrightarrow}C\quad\quad (1)$$ be a short sequence of (abelian or nonabelian) groups and homomorphisms. We say ... 0answers 118 views Homology of derivations of Differential Graded Lie algebra Let (L,d) be a Differential Graded Lie Algebra (L=\bigoplus L_i and d:L_i \to L_{i-1} satisfying the graded Leibniz rule). On the algebra \mathrm{Der}L of derivations of L define a grading ... 2answers 2k views Maps which induce the same homomorphism on homotopy and homology groups are homotopic I am interested in the following question. Are maps which induce the same homomorphism on homotopy and homology groups homotopic? I am sure the answer is no, however I cannot imagine how to construct ... 1answer 790 views Example of torsion in orientable manifolds? An orientable manifold can have torsion in its integer homology. But I believe by Poincare duality the manifold must be at least 4-dimensional -- isn't that right? Anyway are there simple examples ... 0answers 230 views Role of determinant of the matrix corresponding to i-th Homology group. I was thinking about the proof of the Lefschetz's Fixed point theorem and the ingeniuty of the Hopf's Trace formula, i.e. associating the trace of the matrix for deciding about the fixed points. Now ... 0answers 90 views Homology and Exterior Square Let G be a finite group and G \wedge G denote the exterior square of G. It is well known that the second integral homology H_2(G,\mathbb{Z}) is the kernel of homomorphism x \wedge y \mapsto [... 5answers 919 views Take contraction wrt a vector field twice and define kernel mod image. Does that give anything interesting? First, we make the following observation: let X: M \rightarrow TM be a vector field on a smooth manifold. Taking the contraction with respect to X twice gives zero, i.e.$$ i_X \circ i_{X} =0.$$... 0answers 170 views Tubular neighborhoods in the proof of the Morse homology theorem I have a question regarding the proof of the Morse homology theorem given by D. Salamon in "Morse theory, the Conley index and Floer homology". The full text can be found here: http://www.mtm.ufsc.br/... 2answers 227 views The cohomology groups of \Omega U(n) Let \Omega U(n) be the loop space of U(n). Is it true that the cohomology groups H^*(\Omega U(n); \mathbb{Z}) are torsion-free? How can one calculate these groups? 1answer 395 views Software for computing Thurston's unit ball Is there any software which can be used for computing Thurston's unit ball(for second homology of 3-manifolds) of link complements? In particular can I do that with SnapPy? PS: even a table for ... 1answer 497 views Naturality of a Kunneth formula for cohomology Let X,Y be CW complexes. By Kunneth formula, we have a group isomorphim$$ H^n(X\times Y;G) \cong \oplus_{p+q=n} H^p(X;H^q(Y;G))$$Is there a natural map realizing this isomorphism? 2answers 191 views Interpretation of H_1(A_\mathbb{C}^{top},\mathbb{Q}) In the paper "Sato-Tate Distributions and Galois Endomorphism Modules in Genus 2" (arxiv: http://arxiv.org/abs/1110.6638), the authors use the singular homology H_1(A_\mathbb{C}^{top},\mathbb{Q}) (... 0answers 525 views Homology of Lie groups Let G be a Lie group and G^{\delta} the underlying group (with discrete topology). Obviously, we have a continuous map of groups i:G^{\delta}\rightarrow G which induces a map between classifying ... 1answer 396 views Second betti number of compact analytic spaces Let V be a proper singular complex algebraic variety, possibly nonprojective (dim(V)=n>0). I would like to know: 1) if its second Betti number is non zero, 2) same question but now V is a ... 9answers 3k views Why localize spaces with respect to homology? A basic construction in algebraic topology is the localization of spaces or spectra with respect to a homology theory: one formally inverts the E-homology isomorphisms, reflecting each space into ... 0answers 249 views Divisibility in homology/homotopy I have a simply-connected CW-complex F of finite-type, and I know that the imprimitivity of its particular integral homology is divisible by an odd prime p; that is,$$ \forall n,\exists \delta, \... 0answers 101 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 ... 0answers 477 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 :) 4answers 359 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. ... 1answer 209 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$G$-... 1answer 275 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$R$-... 1answer 566 views worked out examples in borel-moore homology I'm trying to learn about BM homology. I've found a few references and they are all quite abstract and high-powered. I am actually familiar with derived categories and sheaves so I don't mind thinking ... 1answer 187 views 0-homologous surface bounds Given a map$f : S \to M^4$from a compact closed not necessarily connected oriented surface to a compact oriented 4-manifold, such that$f_*([S])$is zero in$H_2(M)$, is there a compact oriented 3-... 1answer 380 views Are totally degenerate chains null-homologous? Let$X$be a CW complex. Suppose$\gamma\in C_n(X)$is a cycle which is a sum of maps$\sigma:\Delta^n\to X$which factor as$\Delta^n\to\mathbb R^{n-1}\to X$. Does it follow that$\gamma\$ is null-...
Usually we define singular homology via the complex of singular chains: $$C_\ast(X)=\bigoplus_{n\geq 0}\mathbb Z\langle\sigma:\Delta^n\to X\rangle$$ where the right hand side denotes the free abelian ...