Tagged Questions
0
votes
0answers
28 views
K and KO spectra
In Switzer's algebraic topology book, ch 11 page 216, he defines the K and KO spectra. He then goes on to say: "Since they are $\Omega$-spectra, we have
$\tilde{KO}^0(X) \cong [X,x_0;\mathbb{Z} ...
1
vote
1answer
56 views
filtration on the (co)homology of a space from the filtration of a space
Fix $n\!\in\!\mathbb{N}$. Let $X$ be a topological space and $X_0\subseteq X_1\subseteq X_2\subseteq \ldots$ subspaces of $X$. Let $\iota_k:X_k\rightarrow X$ be the inclusion. Let
...
3
votes
1answer
43 views
Sufficient condition for a direct limit of abelian groups to be infinitely generated
I have the following setup. The CW-complexes $\Gamma_n$ are equipped with maps $\gamma_n\colon\Gamma_{n+1}\rightarrow\Gamma_{n}$ and it is known that the rank of their first cohomology groups is ...
1
vote
0answers
43 views
Cohomology and 1-forms with compact support
I'm, having troubles with the following
Let $U$ be a bounded open set in $\mathbb{R}^{2}$ such that $\mathbb{R}^{2}\setminus U$ has $n+1$ connected components. Prove that $\dim(H_c^{1}(U))=n$.
I ...
6
votes
0answers
78 views
When does a cohomology theory have a ring structure?
I've looked around and I can't quite seem to find an answer to this question. When does a cohomology theory admit a non trivial product structure? I was trying to compute a cohomology ring from a CW ...
5
votes
0answers
51 views
Injective Resolutions in $\mathfrak{Ab}(X)$
Using right derived functors of the global sections functor, I'd like to calculate the first cohomology group of the constant sheaf $\mathbf{Z}$ on $S^1$ with its usual topology, ...
7
votes
2answers
123 views
What's the point of spectra?
I'm familiar with the definition of a spectrum, the one due to Adams, however, I'm not really sure why someone would want to define such a thing. I know they allow one to generalize homology and ...
2
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0answers
39 views
Simplicial cup product on torus
I'm trying to compute the simplicial cup product on the torus (using $\Delta$-complexes) but running into a problem: each way I draw the fundamental polygon I get different answers! When I draw it as ...
2
votes
1answer
48 views
Simplicial cohomology of $ \Bbb{R}\text{P}^2$
I've managed to confuse myself on a simple cohomology calculation. I'm working with the usual $\Delta$-complex on $X = \mathbf{R}\mathbf{P}^2$ and I've computed the complex as ...
3
votes
0answers
82 views
Simple exercise in cohomology
I know this is a simple exercise but I am stuck unfortunately.
Question:
Use de Rham cohomology to prove that the sphere $S^2$ is not diffeomorphic to the torus $T$. You may assume that ...
3
votes
1answer
85 views
Algebraic Topology Double Complexes
I am going through Bott and Tu and trying to do Exercise 9.13 which says
When a homomorphism $f: K \rightarrow K'$ of double complexes induces $H_d$-isomorphism, it also induces $H_D$-isomorphism.
...
2
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0answers
57 views
Prove Poincare duality theorem with Morse theory.
First let us consider a smooth n-manifold. And find a Morse function f. Now let's consider -f. A singular point of f with index k is a singular point of -f with index n-k. Thus we have a canonical ...
2
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0answers
42 views
$f^\ast (a \smile b) = f^\ast(a) \smile f^\ast(b)$ using simplicial chains to define cochains
Let $f \colon X \to Y$ be a continuous map between topological spaces $X$ and $Y$, $f_\ast$ be the induced homomorphism of singular chains $C_k^s(X;G)$, $C_k^s(Y;G)$ and $f^\ast$ be the induced ...
0
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0answers
43 views
De Rham Cohomology of Product of Manifold with an Open Interval
Let $X$ be a submanifold of $\mathbb{R}.$ Prove that $H^{k}_{DR} (X) = H^{k}_{DR} (X\times (0,1)).$ I know that we should consider maps $\iota_a: X\to X\times (0,1)$ by $\iota_a(x) = (x,a)$ for ...
1
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1answer
93 views
A table of homology and cohomology groups
Does anyone know where I can find a table of the homology and cohomology groups, with different coefficients, of standard spaces - $S^1\times S^1$, Klein bottle, projective space, etc.?
1
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0answers
44 views
Cohomology of a chain complex
I know that one can define a chain complex for a CW complex X by taking the chain groups $C_n(X)$ as the free group generated by the $n$-cells, $C_n(X;\mathbb{Z}) = \mathbb{Z}\langle ...
5
votes
1answer
95 views
When is a map essential in Čech cohomology?
I read a nice survey of parts of game theory, Foundations of Strategic Equilibrium, by Hillas and Kohlberg. Something where I stumble is the discussion of Mertens stability. There is a definition that ...
15
votes
1answer
236 views
Twisted Cech cohomology
Let $X$ be a CW-complex with contractible universal cover $\tilde{X}$ and fundamental group $\pi = \pi_1X$. Twisted (co)homology is found by lifting the cell structure on $X$ to a $\pi$-invariant ...
2
votes
1answer
65 views
Cohomology $SO(3)$
We have that De Rham cohomology of $SO(3) \simeq \mathbb{R}P^{3}$ is $\mathbb{R}$ in degree $0$ and $3$ and $0$ in degree $1$ and $2$. But I saw that $H^{*}(SO(3)) \simeq \mathbb{Z}_{2} $ in degree 2. ...
0
votes
1answer
30 views
Cellular cohomology complex projective spaces
I have to calculate the cohomology of complex projective spaces $\mathbb{C}P^{n}$ using cellular cohomology (I know that we have a CW decomposition of $\mathbb{C}P^{n}$ in $n+1$ cells of even ...
0
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0answers
43 views
Cohomology of Stiefel manifold
How can I compute $H^{*}(V_{k}(\mathbb{C}^{n}))$? Where I denote with $V_{k}(\mathbb{C}^{n})$ the Stiefel manifold of $k$-frame in $\mathbb{C}^{n}$.
0
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0answers
35 views
Chern classes fiber bundle projective space
I calculated the cohomology of $\mathbb{C}P^{n}$ using the spectral sequence associate to the fibration $S^{1} \hookrightarrow S^{2n+1} \rightarrow \mathbb{C}P^{n}$. How can I find the first Chern ...
1
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0answers
38 views
Equivariant localization and integration equivariant forms
I have two problems:
Let it $\Omega^{*}_{G}:=(\mathbb{C}[\mathfrak{g}]\otimes\Omega^{*}(M))^{G}$ be the complex of equivariant differential forms on a differential manifold $M$ (in which acts a Lie ...
2
votes
1answer
126 views
Visualized definition of cohomology
I cannot imagine how cohomology is related to graph theory, actually I read solid definition from wiki, and to be honest, I cannot understand it.
e.g I know what is homology (in simple term), group ...
2
votes
1answer
92 views
Cohomology of Grassmannian
Let $G_r$ the infinite complex Grassmannian manifold. We know that $H^{*}(G_r)=\mathbb{C}[x_{1}, \cdots, x_{n}]$ where $x_i$ are the Chern classes of tautological bundle. But $H^{*}(G_r)$ is also ...
1
vote
0answers
37 views
Equivariant Mayer Vietoris and Borel localization
We have this theorem:
Let $U$, $V$ two open sets of manifold $M$, ($U \cup V = M$). If they are $G$-stable the induced sequence in cohomology
$$ \cdots \rightarrow H^{k}_{G}(U \cup V) \rightarrow ...
0
votes
0answers
31 views
Equivariant differential
http://www.math.sunysb.edu/~myoung/localization.pdf
How is $\iota$ defined on page $4$, line $3$?
11
votes
2answers
136 views
Different ways of representing a second cohomology class
There are probably many ways of talking about a second (integral) cohomology class of a smooth, closed, orientable manifold $M$ of dimension $n$. Here are a few, with $\alpha\in H^2(M,\mathbb{Z})$:
...
7
votes
1answer
86 views
If $H_n(X;\mathbb{Z})$ are all f. g. free abelian, then $H^*(X;\mathbb{Z}) \otimes \mathbb{Z}_p \cong H^*(X; \mathbb{Z}_p)$?
An exercise in Hatcher's book asks to prove that whenever $X$ is a space with the homology groups $H_n(X; \mathbb{Z})$ finitely generated free abelian for each $n \geq 0$, then $H^*(X; \mathbb{Z}) ...
1
vote
1answer
90 views
confusion about cup product in cohomology ring
I have some confusion that i would like to clarify. The product in cohomology is not the cup product $\smile$ but it is another product $*$ that is constructed from cup product. Indeed wrinte ...
2
votes
2answers
222 views
Brouwer's fixed point theorem for free?
I think I found a proof of Brouwer's fixed point theorem which is much simpler than any of the proofs in my books.
One part is standard: Suppose there is an $f:D^n \rightarrow D^n$ with no fixed ...
0
votes
1answer
55 views
What are elements in $H^1(\Gamma, \mathbb{C})$?
Let $\Gamma$ be a curve. What are explicit elements in $H^1(\Gamma, \mathbb{C})$? For example, let $\Gamma$ be the plane curve $y=x^2$. What is $H^1(\Gamma, \mathbb{C})$? Thank you very much.
5
votes
1answer
114 views
Are maps inducing the same cohomology homomorphisms homotopic?
It is not hard to show that given $f,g: X \rightarrow Y$, with $f$ and $g$ homotopic the induced homomorphisms $f^*, g^* : H^* (Y, \mathbb{Z}) \rightarrow H^* (X, \mathbb{Z})$ are the same.
Is the ...
3
votes
0answers
110 views
cohomology isomorphism
Let $X$ be a finite dimensional CW complex and $A$ be a closed subset in $X$ and $N$ a regular neighborhood of $A$ that deformation retracts onto it. why do we have for each $i$,
$$H^{i}(X-A;\mathbb ...
4
votes
1answer
115 views
Highest DeRahm Cohomology
Let $X$ be a $C^\infty$ manifold, compact oriented and connected of dimension $n$. How do you prove that the integration map $$\int_X: \omega \mapsto \int_X \omega $$ from $H^n_{DR}(X)$ to ...
5
votes
1answer
117 views
Follow up on intersection forms
For which topological spaces $X$ can I define an intersection form $b(\cdot, \cdot)$?
I know at least one example: If $X$ is a closed orientable $2n$-manifold then one can define an intersection ...
5
votes
1answer
89 views
Cohomology group of free quotient.
Let $G$ be a finite group acting on a manifold $M$ without fixed point. The standard Leray-Cartan-Serre spectral sequence argument shows that
$$
H^k(M,\mathbb{Q})^G\cong H^k(M/G,\mathbb{Q}).
$$
This ...
4
votes
0answers
104 views
Cohomology of fiber bundle with a section
Let $f:E\rightarrow B$ be a $C^{\infty}$-fiber bundle. Assume that there is a section $s:B\rightarrow E$ of this bundle. One easy consequence of the existence of section is that map
$$
...
3
votes
1answer
136 views
Poincaré duality
Let $X$ be a a compact oriented manifold of dimension $n$. Assume that its (co)homologies have no torsion. Then Poincaré duality says that
$$
H^{k}(X,\mathbb{Z})\cong H_{n-k}(X,\mathbb{Z})
$$
holds ...
7
votes
1answer
173 views
Confusion on Cech cohomology
From Harvard math qualification exam, 1990.
Let $X$ be a smooth manifold with an open cover $N<\infty$ sets $\{B_{n}\}^{N}_{1}$ which are contractible. Assume that $$\pi_{0}(B_{n}\cap B_{m})\le ...
5
votes
1answer
698 views
Intersection form on quotient manifold
I have a simple algebraic topology question. Let $M$ and $N$ be 2-dimensional oriented manifolds (say $H^{2}(M,\mathbb{Z})\cong \mathbb{Z}\alpha_{M}$ and $H^{2}(M,\mathbb{Z})\cong ...
1
vote
0answers
66 views
Rational cohomology of quotient by group action
Let $X$ be a topological space with a continuous action by a finite group $G$. Hopefully under some assumptions on $X$ one can identify the rational cohomology of $X/G$ with the $G$-invariants of the ...
34
votes
2answers
810 views
Algebraic Topology Challenge: Homology of an Infinite Wedge of Spheres
So the following comes to me from an old algebraic topology final that got the best of me. I wasn't able to prove it due to a lack of technical confidence, and my topology has only deteriorated since ...
2
votes
0answers
263 views
Homology and cohomology: why does Poincaré duality fail for domains with boundary?
Poincaré duality says that for a compact, orientable manifold without boundary the $k$th and $(n-k)$th homology groups are isomorphic.
For domains with boundary, it's easy to construct examples where ...
7
votes
2answers
133 views
Geometric interpretation of injective/projective resolutions?
I understand the geometric interpretation of derived functors, as well as their usefulness in giving a simple, purely algebraic description of cohomology.
I also understand how resolutions are used ...
2
votes
0answers
123 views
On a canonical homomorphism in cohomology
When $K$ is a simplicial complex, the dual complex $C^*(K)$ to the chain complex $C_*(K)$ has a concrete interpretation: an element in $C^n(K)$ is given by assigning an integer to every oriented ...
5
votes
2answers
387 views
Intersection Pairing and Poincaré Duality
Let $M$ be an $n$-dimensional compact and oriented manifold. Then one can define the intersection pairing $H_k(M,\mathbb Z) \times H_{n-k}(M,\mathbb Z) \to \mathbb Z$. One possible formulation of the ...
1
vote
2answers
209 views
A Problem Reading Like “Snake Lemma” meets Cohomology
When studying homology, I have been told that there is a result called the Snake Lemma that via induction affords us with a long exact sequence of homology groups. I am wondering if the ...
2
votes
1answer
133 views
Approaching a Cohomology Computation
If $U \subseteq \mathbb{R}^2$ is the complement of $d > 0$ points in the plane, and $H^k$ denotes the $k$-th cohomology group, how should I verify that $H^k (U)$ equals $\mathbb{R}$, ...
11
votes
1answer
331 views
Why isn't $H^*(\mathbb{R}P^\infty,\mathbb{F}_2)\cong \mathbb{F}_2[[x]]$?
We just computed in class a few days ago that $$H^*(\mathbb{R}P^n,\mathbb{F}_2)\cong\mathbb{F}_2[x]/(x^{n+1}),$$ and it was mentioned that $H^*(\mathbb{R}P^\infty,\mathbb{F}_2)\cong \mathbb{F}_2[x]$, ...
