Use this tag if your question involves some type of (co)homology, including (but not limited to) simplicial, singular or group (co)homology. Consider the tag (homological-algebra) for more abstract aspects of (co)homology theory.

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Cohomology in Differential Geometry

Below is a communicative diagram: $$\begin{array}[c]{ccc} ...
2
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1answer
23 views

Alexander Duality for Relative Homology

Is there a formulation for Alexander Duality for pairs of spaces $(A, B)$ such that $A\subset B\subset S^n$? I can't find a reference for this anywhere, but I think it is as follows, which I arrived ...
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34 views

(Co)homology of propositional logic

Sorry if this is a rather vague question, but it seemed like something that might be interesting. Let $P$ be a family of propositions, and let $\mathcal L(P)$ be the set of all compound propositions ...
2
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1answer
46 views

Kernel of induced map between singular chain groups

Let $p : \widetilde X \to X$ be a two-sheeted cover. This induces $p_\sharp : C_n(\widetilde X; \mathbb Z_2) \to C_n(X; \mathbb Z_2)$. I can show that $p_\sharp$ is surjective by noting that every ...
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0answers
19 views

Homology ring and cohomology ring

If I have that the cohomology ring of something has $0$ in all odd dimensions and $\mathbb{Z}$ in all even dimensions, and the the ring is isomorphic to $\mathbb{Z}[x_1, x_2, ..., x_i, ...]$, is it ...
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27 views

generalized cohomology

If I have a generalized cohomology theory $E$, then $E^n(X) = [\sum^{-n}X, E]$. I would like to know what $[\sum^{-n}*, E]$ looks like for $*$ a point. We can assume that $E$ is a nice $CW$ spectra ...
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1answer
27 views

Can a map from a p-simplex to the n-sphere be surjective?

If $p<n$, is this possible? I am confused about this. I am trying to prove that the i-th singular homotopy group of the n-sphere is a subset of the i-th homotopy group of $\mathbb{R}^n$ but I am ...
2
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1answer
37 views

Homology as Boundary of “Submanifold”

In the plane, imagine a horizonal figure eight, $\infty$. Let $\alpha$ be the curve which is convex from the leftmost point of the figure to its "middle", and concave from the middle until the ...
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31 views

Homology of $S^n - S^k\vee S^\ell$

Does anyone know a good trick to computing homology groups of the sphere minus the wedge of two spheres of possibly different dimension $S^n \setminus S^k\vee S^\ell$ ? Any particular $k$ and $\ell$ ...
3
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1answer
27 views

Wedge product descend to the cohomology

I found this statement in Raoul Bott "Differential Forms in Algebraic Topology": "Because the wedge product is an antiderivation, it descends to cohomology." Apparently this meant to be really obvious ...
2
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0answers
39 views

Pullback of a 1 form on the circle

Q: Let $M$ be a smooth compact manifold, and suppose there is a smooth map $F:M \rightarrow S^{1}$ whose derivative is non-zero at every point. Prove that the de Rham cohomology space $H^{1}(M)$ is ...
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0answers
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Finish the proof of Borsuk-Ulam theorem (Hatcher)

Hatcher at page 229 proposes to prove the Borsuk-Ulam theorem using the fact that any continuous map $f \colon\mathbb R P^n \to \mathbb RP^m$, $n > m$, induces the trivial map in cohomology with ...
2
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1answer
58 views

Original Papers on Singular Homology/Cohomology.

I am currently reading Singular Homology Theory and Cohomology on my own mainly from Hatcher's "Algebriac Topology" and "Topology and Geometry" by Bredon. Quite often it happens that it takes a lot of ...
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28 views

cohomology homomorphism between grassmannians induced by inclusion

Let $i:G_k(\mathbb{R}^n)\to G_k(\mathbb{R}^\infty)$ be inclusion of grassmannians. Then $H^*(G_k(\mathbb{R}^\infty);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k]$. $ ...
3
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1answer
23 views

Does every elliptic cohomology theory represent a complex-orientable $E_\infty$-ring spectra and vice-versa?

The last paragraph in Two-Vector Bundles and Forms of Elliptic Cohomology remarks that neither the spectrum $K(ku)$ nor tmf is complex orientable. In the case of $K(ku)$: "...the unit map for $K(ku)$ ...
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0answers
30 views

cohomology of finite dimensional grassmannians

What is the cohomology algebra of finite dimensional grassmannian $$ H^*(G_k(\mathbb{R}^N);\mathbb{Z}_2)? $$ $$ H^*(G_k(\mathbb{C}^N);\mathbb{Z}_p)? $$ $$ H^*(G_k(\mathbb{H}^N);\mathbb{Z}_p)? $$ I ...
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3answers
187 views

Is there a non-simply-connected space with trivial first homology group?

Is there a path connected topological space such that its fundamental group is non-trivial, but its first homology group is trivial? Since the first homology group of a space is the abelianization of ...
2
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1answer
38 views

Simplicial Homology: The definition of cycles

I'm trying to convince myself beyond a doubt that $n$-cycles should be defined as elements of $\ker \partial _n$. My intuition is along the lines of "a cycle is a boundary of some chain (not ...
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55 views

Exactness of a certain sequence

Let $M$ be a connected manifold of dimension $m$ and let $\beta\in H^{1}(S^{1})$ such that $\int_{S^{1}}\beta = 1$. Let $\pi_{1}:M\times S^{1}\rightarrow M$ and $\pi_{2}: M\times S^{1}\rightarrow ...
0
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1answer
18 views

homology of suspension

Let $\Sigma$ be suspension. For any CW-complex, or topological space, does the reduced homology satisfy $$ \tilde H_*(\Sigma^k X)=s^k\tilde H_*(X)? $$ Here $s^k H$ is a copy of $H$ such that an ...
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16 views

Configuration space of product spaces

Let $M,N$ be manifolds. Let $F(M,n)$, $F(N,n)$ be ordered configuration spaces of order $n$. Let $F(M,n)/\Sigma_n$, $F(N,n)/\Sigma_n$ be the unordered configuration spaces of order $n$, for ...
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Why are period integrals naïve periods?

Apologies for the long question. I recall the definition of a (naïve) period according to Kontsevitch and Zagier [KS]: A (naïve) period is a complex number whose real and imaginary parts are ...
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1answer
26 views

Exterior derivative cohomology

Let $\Omega^k (U)$ denote the set of differential $k$-forms on an open subset $U\subseteq \mathbb{R}^n$. For each $k\in \mathbb{N}$ the exterior derivative $d_k=d : \Omega^{k-1} (U) \rightarrow ...
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23 views

maps between suspension of complex projective spaces and special unitary groups

How to do the following question? I get totally lost... this question is given by the professor in our final exam paper.
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1answer
100 views

About simple connectedness

Two topological spaces $X$ and $Y$ are homotopic if there exists continuous $f: X \to Y$ and $g: Y\to X$ such that $f\circ g$ is homotopic to $Id_Y$ and $g\circ f $ homotopic to $Id_X$ (regardless any ...
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homotopy invariance for singular homology for maps of pairs

Let R be a ring, $(X,A), (Y,B)$ pairs of topological spaces and $f,g:(X,A)\to (Y,B)$ continuous maps of pairs such that there exists $H:X\times I\to Y$ homotopy with $H(A\times I)\subseteq B$, ...
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3answers
69 views

Homology and Homotopy in the Plane

Suppose we're living in the plane minus (possibly infinitely many) isolated points, which I'll call poles. Intuitively, the following two statements seem reasonable: Loops in the plane are homotopic ...
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1answer
30 views

simplicial homology in Hatcher book.

I was studying simplicial homology in Hatcher's Algebraic topology book.In one paragraph book says following: Some obvious general questions arise: Are the groups $H_n(X)$ independent of the choice of ...
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1answer
33 views

cohomology monomorphism between grassmannians and product of projective spaces

Let $S^1\times\cdots \times S^1( n\text{ times })=\prod_n U(1)\to U(n)$ be the inclusion. This induces a map between classifying spaces $$ \prod_nBS^1\to BU(n).$$ i.e., $$ ...
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0answers
36 views

De Rham cohomology of $\mathbb{R}^2 \setminus \{k~\text{points}\}$

This question is motivated by Exercise 1.7 from Differential Forms in Algebraic Topology by Bott & Tu. The original question in the text concerns the de Rham cohomology of $\mathbb{R}^2$ with ...
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35 views

Help understanding the proof of a theorem about Cohomology of Vector Bundles

I am trying to understand a paper called Betti tables of graded modules and cohomology of vector bundles, but i am stuck in Proposition 6.8 which states: Let $\mathcal{E}$ be a vector bundle on ...
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all differentials collapse of the Serre spectral sequence

Let fibration $$ SO(n)\to SO(n+1)\to S^n, $$ consider the Serre spectral sequence of cohomology $(E^{*,*}_k,d_k)$, $k\geq 2$, $E^{p,q}_2=H^p(S^n;\mathbb{Z}_2)\otimes H^q(SO(n);\mathbb{Z}_2)$. How ...
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2answers
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The first Chern class of projective line $\mathbb{CP}^1$

I am studying the Chern class using by some textbooks and lecture notes. One day, I found an example of the first Chern class of $\mathbb{CP}^1$. Let $\xi$ be a tautological line bundle of ...
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1answer
35 views

homomorphism between cohomology induced by the multiplication of an H-space

Define the product on $\mathbb{C}P^\infty$ in the following way: \begin{eqnarray*} \phi:\mathbb{C}P\overset{\Delta}\longrightarrow(\mathbb{C}P^\infty)^k\overset{\mu}\longrightarrow ...
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0answers
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fibration sequence of projective spaces

Question~1: How to construct a fibration sequence $$ S^3\to S^2 \to \mathbb{C}P^\infty\to \mathbb{H}P^\infty ? $$ Does $$S^3\simeq \Omega \mathbb{H}P^\infty ? $$ (Since $\mathbb{C}P^\infty\simeq ...
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1answer
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$H_2(M)$ is free abelian for any simply connected $4$-manifold

In Naber's book "Topology, Geometry and Gauge Fields. Foundations", it is stated that for each $4$-manifold $M$ which is smooth, closed, connected and simply connected we have $H_0(M) = H_4(M)= ...
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1answer
26 views

Examples spectral sequence

I have to make a talk about spectral sequences, so I'd like to present some concrete examples of computation, after the general definition. I'd like to present three examples of spectral sequences: ...
5
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1answer
151 views

Euler characteristic, genus and cohomology: a deep connection?

For a smooth projective curve $V$ over the complex numbers, the algebraic genus, defined as the dimension of the linear system $L(\omega)$, where $\omega$ is the canonical divisor, coincides with the ...
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34 views

non-abelian Galois cohomology

Let $1 \to A \to B \to C \to 1$ be a short exact sequence of (not necessarily abelian) $G$-modules. Passing to non-abelian cohomology, we have the exact sequence of pointed sets $$ 1 \to A^G \to B^G ...
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1answer
32 views

What is $C_n(X)$?

This article says the following: Let $X$ be a triangulated space and let $C_n(X)$ be a real vector space with $n$-simplices $[x_0,x_1,x_2,\dots,x_n]$. Each different combination of $x_i's$ forms a ...
0
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1answer
46 views

Restricting the DeRham cohomology class of a submanifold to a coordinate neighborhood.

Suppose $M$ is an $n$-manifold and $A$ a $k$-dimensional submanifold, both compact and oriented. Let the deRham cohomology class of $A$ be denoted $[\phi_A]$. The class is defined by ...
2
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2answers
61 views

Cohomology groups of coherent sheaves for very small and very big twists.

Let $\mathcal{F}$ be nonzero coherent sheaf over the projective space $\mathbb{P}_k^n$. The Serre vanishing Theorem says that $h^i \mathcal{F}(d)=0$ for $i>0$ and $d\gg 0$. I am wondering if it is ...
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2answers
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Factor sets and group extensions (Homological algebra- Hilton and Stammbach VI.10.1)

Show that an extension $$A\xrightarrow{i} E\xrightarrow{p} G$$ may be described by a factor set, as follows. Let $s:G\rightarrow E$ be a secion so that $ps=1_G$. Every elmenet of $E$ is of the form ...
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1answer
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Applying the functor $H_*$ to the inclusion sequence $A\rightarrow B\rightarrow C$

Does applying the functor $H_*$ to the sequence of inclusions $A\rightarrow B\rightarrow C$ induce a map $\phi_3: H_*(B)\rightarrow H_*(C )$, such that if $\phi_1:H_*(A)\rightarrow H_*(B)$, and ...
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1answer
36 views

How does one actually take the dual of a chain complex?

I know the following about the chain complex used for computing the homology groups of the torus $S^1 \times S^1$: The complex is $0 \to^{\delta_3} \mathbb{Z}[U] \oplus \mathbb{Z}[L] \to^{\delta_2} ...
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0answers
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Question about Tate resolution and cohomology groups of nonzero coherent sheaves.

Let $\mathcal{F}$ be a nonzero coherent sheaf on the projective space $\mathbb{P}_{k}^m$. I am trying to show that for every integer $d$ there is $j$ for which $h^j\mathcal{F}(d-j) \neq 0$. My ...
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0answers
37 views

$H^1(X, G_1 \times G_2)$ in terms of simpler first cohomologies

Let $X$ be a variety over a field $k$ of characteristic 0 (not necessarily with $k = \bar{k}$). Let $G_1, G_2$ be linear algebraic groups over $k$. My question is: Can I write $H^1(X, G_1 \times ...
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1answer
51 views

Does an odd degree map on $S^n$ descend to an odd degree maps on $\mathbb{R}P^n$?

Suppose there is a map $f:S^n\to S^n$ that induces non-trivial on $\mathbb{Z}/2$ homology group homomorphisms, further suppose $f$ descends to $f':\mathbb{R}P^n\to\mathbb{R}P^n$. Does it then follows ...
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$H^q(\mathfrak{g},K;V)$ is equal to $Ext_{\left(\mathfrak{g},K\right)}^q\left(\mathbb{C},V\right)$?

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$. Let $K$ be a closed subgroup of $G$ with corresponding Lie subalgebra $\mathfrak{k}$. Let $V$ be a $\left(\mathfrak{g},K\right)$-module. Then, I ...
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votes
1answer
21 views

1-forms as a direct sum

Let us define the spaces $C_0, C_1$ and $C_2$ of differential $0,1,2$ forms respectively on the sphere $S^2.$ Is it true that $C_1$ is the direct sum of $d(C_0) \oplus \delta^* (C_2)$? I think this ...