Tagged Questions

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Simplicial homology of the skeleton of a simplex

Let $n$ and $k$ two natural numbers. We consider the (abstract) simplicial complex $K$ on $n$ vertices $v_1,\dots,v_n$ and such that a subset of $\{v_1,\dots,v_n\}$ is a face of $K$ if and only if it ...
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Cohomology Calculation

A couple of days ago I asked this Question on calculating hypercohomology I tried a similar example for $(\mathbb{C}^*)^2$, and I have a couple of questions. Here is my calculation: We have a ...
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Why Must the Degree of this Map be 0? [closed]

Let $f:S^3 \rightarrow S^1\times S^1\times S^1$ be a continuous map. Show that it's degree must be $0$. (Just a hint would be good)
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Cohomology Ring of Klein Bottle over $\mathbb{Z}_2$

I am trying to show that the cohomology ring of the Klein bottle with $\mathbb{Z}_2$ coefficients is $H^*(K,\mathbb{Z}_2) \cong \mathbb{Z}_2[x,y]/(x^3,y^2, x^2y)$. What I know: ...
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$C_g \simeq SX$ and $C_h \simeq SY$ [closed]

Hi need some help with this problem: Let $f : X \to Y$ . Then we can form the cofiber sequence $X \to Y \to C_f \to C_g \to C_h$ where $g: Y \to C_f$, $h: C_f \to C_g$, and $i: C_g \to C_h$. Show ...
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Compute the induced map on $\mathbb{CP}^n$

Let $d>0$ and $f:\mathbb{C}^{n+1}\rightarrow \mathbb{C}^{n+1}$ be given by $f(z_0,...,z_n)=(z_0^d,...,z_n^d)$. Let $F:\mathbb{CP}^n \rightarrow \mathbb{CP}^n$ be the induced map by $f$. Compute ...
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Relative cohomology of a vector space module non-zero vectors

I am trying to explicitly compute the relative cohomology groups $H^m(\mathbb R^n,\mathbb R^n_0;\mathbb Z)$, where $\mathbb R^n_0$ is all the non-zero vectors in $\mathbb R^n$. I think that the answer ...
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Is $H^0(S^0;G)\simeq G\oplus G$ or $G$?

In the article on topospaces for the (co)homology of spheres, it says $H^0(S^n,G)\simeq H^n(S^n,G)\simeq G$. Is this true when $n=0$? I think not, for if we view $S^0$ as the union of two ...
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Degree of an induced map on $\mathbb{CP}^n$

Let $r :\mathbb{C}^{n+1} \rightarrow \mathbb{C}^{n+1}$ be the map $r(z_0, z_1,\ldots, z_n)=(-z_0, z_1,\ldots, z_n)$. $r$ induces a map $\bar r : \mathbb{CP}^n \rightarrow \mathbb{CP}^n$. What is the ...
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Why is $H^2(\mathbb{R}P^2,\mathbb{Z})\simeq\mathbb{Z}_2$?

Why is the second cohomology group of $X=\mathbb{R}P^2$ with $\mathbb{Z}$-coefficients $\mathbb{Z}_2$? We can put the usual $\Delta$-structure on $X$ with two vertices, three $1$-simplices, say $a$, ...
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Wedge Sum of Two Spheres Homotopy Equivalent to a Compact Manifold?

Let $X=S^2$v $S^2$ (wedge sum). The homology groups are $H_0(X,\mathbb{Z})= \mathbb{Z}$, $H_1(X,\mathbb{Z})= 0$, and $H_2(X,\mathbb{Z})= \mathbb{Z} \oplus\mathbb{Z}$. I can see that $X$ is not ...
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So I have shown that the natural projection $\pi: \mathbb{CP^n}\rightarrow \mathbb{CP^n/CP^k}$ induces a monomorphism $\pi^*:H^*(\mathbb{CP^n/CP^k},\mathbb Z)\rightarrow H^*(\mathbb{CP^n},\mathbb Z) ... 0answers 41 views When is a graded ring the cohomology ring of a CW-complex? Let$A^*$be a graded-commutative ring with$A^n = 0$for sufficiently large$n$and each$A^n$finitely generated. When does there exist a finite CW-complex$X$with$H^*(X) \cong A$as graded rings? ... 1answer 27 views $H^1(X) = [X,\mathbb{T}]$? This is a stupid question, but here goes. I have a compact Hausdorff space$X$, and I am talking about$[X,\mathbb{T}]$, the group of homotopy classes of maps$X \to \mathbb{T}$, where$\mathbb{T}$... 0answers 52 views De Rahm Cohomology of Complex Grassmannian Since the complex Grassmannian$G_k(\mathbb{C}^n)\cong SU(n)/S(U(k)\times U(n-k))$is connected and simply connected, the first two de Rahm cohomology groups are given by $$... 0answers 18 views Comparison between Eilenberg-Steenrod excision and Brown representability excisive One of the Eilenberg-Steenrod axioms for unreduced cohomology is excision, which states that H^n(X,A)\cong H^n(X\setminus U,A\setminus U), for good subspaces such as when \overset{\circ}U\subseteq ... 0answers 37 views Minimum regularity Of Stoke's theorem to hold in smooth manifold. Stokes’ Theorem on Manifolds is often express as follows: Given a differential m-form \omega whose support is the C^{\infty} m-dimensional compact manifold {\cal{M}} with boundary ... 1answer 64 views Decomposition of cohomology group on S^{n} If we have decomposition of cohomology group on S^{n} it looks like H^{n}(S^{n})=H^{n}(S^{n})_{+}\oplus H^{n}(S^{n})_{-}, where H^{n}(S^{n})_{\pm} cohomology of invariant or anti-invariant n ... 1answer 185 views Relative de Rham Cohomology is Homotopy Invariant Suppose f:N\rightarrow M is a smooth map between two manifolds. Relative de Rham cohomology is defined through the complex \Omega^{q}(f)=\Omega^{q}(M)\oplus\Omega^{q-1}(N) with ... 1answer 66 views De Rham cohomology for \mathbb{R^2} De Rham cohomology groups for \mathbb{R^2}. H^{0}_{dR}(\mathbb{R}^{2})=\mathbb{R} since Z^{0}(\mathbb{R}^{2}) is the one dimensional space of locally constant functions on \mathbb{R}^{2} and ... 1answer 43 views Prove that H^{2}(S^{2})\neq 0 Prove that H^{2}(S^{2})\neq 0 Suppose \omega is an area form of S^{2}. An arbitrary two form on S^{2} is closed as if f(x,y)dx\wedge dy\in\Omega^{2}(S^{2}) then d(f(x,y)dx\wedge dy)=0. I ... 0answers 40 views Intuition of higher push-forward constant sheaves. Let us consider the higher phsh-forward sheaves R^if_*\mathbb{R} of a map f:X\rightarrow Y between two compact manifolds. We assume that the fibers has a constant dimension, say n. I think ... 0answers 48 views In the Universal Coefficient Theorem, how does the cohomology generator relate to the homology generators? Consider homology and cohomology of some space X where the homology groups are finitely generated. Consider tor(H^i(X)), the torsion part of H^i(X). How do the generators of tor(H^i(X)) ... 0answers 56 views There does not exist a map S^2\times S^2\to \mathbb{CP}^2 with odd degree. The following is a problem from a topology qualifying exam I am studying for: Show there does not exist a map S^2\times S^2\to \mathbb{CP}^2 with odd degree. I think I am doing something wrong, ... 0answers 44 views The vanishing (?) cohomology of the Milnor fiber Setup. Say we have a germ of a holomorphic function f:(\mathbb C^{n+1},0)\to (\mathbb C,0) with a critical point at the origin. There is an \epsilon>0 small enough so that f becomes a ... 0answers 35 views Cohomology of S^2\times S^2/\mathbb{Z}_2 The product of two spheres admits a diagonal \mathbb{Z}_2-action, (x,y)\mapsto (-x,-y). I'm trying to compute the integral singular cohomology ring of the orbit space X of this action. X is ... 1answer 60 views Examples of de Rham cohomology being easier to compute that singular cohomology De Rham's theorem states that for any smooth manifold M the singular cohomology and de Rham cohomology of M are isomorphic. Are there any examples of manifolds for which it is easier to compute ... 1answer 31 views Why is the generator of H^0(S^n,\mathbb{Z}) the identity of H^*(S^n,\mathbb{Z})? I know the cohomology ring H^*(S^n,\mathbb{Z})\simeq\mathbb{Z}[X]/(X^2), but the computation of this uses that fact that a generator of H^0(S^n,\mathbb{Z}) is the identity for the cohomology ring ... 0answers 33 views Cohomology-Homology bilinear form of Seifert surfaces Let C_\ast be any chain complex of R-modules. Then for any k\in\mathbb{Z} we obtain a R-bilinear map$$\langle-,-\rangle:H^k\!C_\ast\times H_kC_\ast\longrightarrow R, ... 0answers 12 views How to Resolve Extension Issues in Equivariant (Co)Homology Computations I am computing equivariant homology, which is just the usual homology of the Borel construction. I have reached a few extension issues. I have used the Gysin sequence, Leray-Serre spectral sequence, ... 0answers 82 views Eilenberg-Moore Spectral Sequence for Homology with Coefficients in the Integers I am trying to learn about the Eilenberg-Moore spectral sequence to compute homology and cohomology. I have been using Hatcher's book on spectral sequences and also McCleary's "A User's Guide to ... 1answer 111 views Universal coefficient theorem with ring coefficients The universal coefficient theorem for cohomology reads: $$0 \to Ext(H_{n-1}(C), R) \to H^n(C;R) \to Hom(H_n(C), R) \to 0,$$ where$C$is a chain complex of free abelian groups and$R$is a ring. It ... 1answer 33 views Is there any simple example that$lim^1$terms appear? limit of cohomology does not behave well in the sense that there will be$lim^1$term. Is there any simple example that$lim^1$terms appear? Thanks! 3answers 61 views Cohomology groups of real projective space My question concerns the cohomology groups$H^k(RP^n,\mathbb{Z}_2)$. We know that$H_k(RP^n,\mathbb{Z}_2) = \mathbb{Z}_2$if$0 \leq k \leq n$and is trivial otherwise. I looked up the solution and it ... 1answer 34 views $\tilde{H}^i(\sum X) \cong \tilde{H}^{i-1}(X)$I need help with this problem Let$\sum X =C^+X \cup_X C^-X$be the union of two cones on$X$with a common base. Show that$\tilde{H}^i(\sum X) \cong \tilde{H}^{i-1}(X)$. Dose this give an ... 2answers 47 views Topological dimension and derham cohomological dimension If G is a compact complex manifold then does the topological dimension bound the deRham cohomological dimension below? By derham cohomological dimension, I mean the largest extended natrual number ... 1answer 69 views Equality of rank for homology and cohomology groups via the universal coefficient theorem I'm having trouble understanding a passage from the proof of Corollary 3.37 in Hatcher's Algebraic Topology, namely the fact that the universal coefficient theorem implies$$... 1answer 42 views Existence of a suitable cover for$S^{2}$and a given sheaf I am trying to find a Leray covering for the 2-sphere with respect to the sheaf$\mathcal{F}=\mathbb{Z}$. I am also assuming that a contractible open covering satisfies$H^{i}(U,\mathbb{Z})$for all ... 0answers 30 views Cohomology of Circle from unreduced Eilenberg-Steenrod Axioms I would like to compute the cohomology groups of$S^1$straight from the unreduced Eilenberg-Steenrod axioms. My motivation is to be able to calculate the cohomology group of spheres in any dimension ... 1answer 68 views equivariant cohomology in case of free actions (basic question) Suppose$X$is a topological space and$G$is a topological group, and$G$acts on$X$. Here is my question: If$G$acts freely on$X$, then what are the maps showing$(X \times EG)/G$is homotopy ... 1answer 76 views Question about Relative Cohomology I need help with the following question please: Suppose that a space$X \subseteq Y $retracts onto some subspace$A \subseteq X $. When do I have$H^\ast ( Y,X) \cong H^\ast (Y,A)$? Thanks. 1answer 110 views Solving an exercise in Milnor-stasheff's “characteristic classes” I am trying to solve the following exercise (which is an exercise in Milnor-Stasheff's book). It basically says the following: Let$ M =S^n $be the$n$-sphere and let$TM$be its tangent ... 1answer 83 views De Rham cohomology of$T^*\mathbb{CP}^n$I am a bit rusty on my de Rham cohomology, and I'm hoping that someone here could help me. I want to find the cohomology of$T^*\mathbb{CP}^n$(seen as a real manifold). Now, this should be equal to ... 1answer 44 views Can we compare cohomology rings with different coefficients? I have an example sheet that asks me to compute the cohomology rings for two spaces, say X and Y, with coefficients in$\mathbb{Z}$and$\mathbb{Z}_d$respectively. It then asks whether X and Y are ... 1answer 53 views $S^{1}$-bundles over$\mathbb{RP}^2$How many$S^1$-bundles over$\mathbb{RP}^2$do exist? Is it true that there exist only two bundles - trivial and not? 1answer 94 views Cohomological Whitehead theorem Let$X$and$Y$be CW complexes (resp. Kan complexes) and let$f : X \to Y$be a continuous map (resp. morphism of simplicial sets). The following seems to be a folklore result: Theorem. The ... 1answer 79 views If M is a non-orientable closed connected 3 manifold prove H1(M) is an infinite group. This is an example from a question sheet (non-assessed) of a university class. If M is a non-orientable, closed, connected 3 manifold, prove$H_1(M;\mathbb{Z})$is an infinite group. I know that since ... 1answer 91 views Cohomology to compute number of holes? Can one use cohomology to compute the number of holes in a space$E$, where$E=R\times R$,$R$is a Riemann surface of genus$g$, - i.e., is$\dim(H^n(E))$, and by Künneth's formula,$H^{n}(E) \cong ...
Let $M$ be a closed 2-dimensional manifold (a surface). Assume that I have a more or less explicit expression for a Čech 2-cocyle $h_{ijk} \in H^2(M, G)$. I want to know the expression of $h_{ijk}$ as ...