A branch of algebraic topology concerning the study of cocycles and coboundaries. It is in some sense a dual theory to homology theory. This tag can be further specialized by using it in conjunction with the tags group-cohomology, etale-cohomology, sheaf-cohomology, galois-cohomology, ...

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109 views

Questions about configuration spaces on the sphere and higher loop spaces

In the paper Configuration spaces on the sphere and higher loop spaces, Paolo Salvatore, Mathematische Zeitschrift November 2004, Volume 248, Issue 3, pp 527-540, I have some questions about ...
2
votes
0answers
70 views

A Künneth-Theorem version for relative singular cohomology

I'm not an expert in algebraic topology, but sometimes I need some results from this area, for example tools to determine singular cohomology groups of product spaces. The Künneth-Theorem which I ...
5
votes
0answers
126 views

Comparison of sheaves of modular forms

Let $\pi:E\to X$ the universal generalized elliptic curve over the compactified modular curve, with zero section $e: X\to E$. Now consider the following two sheaves on $X$: $e^*\Omega^1_{E/X}$ and ...
2
votes
0answers
33 views

Does anyone know if there is a generalization of symplectic Kodaira dimension beyond 4-manifolds?

I'm aware that in algebraic geometry, one has the Kodaira-Iitaka dimension, which generalizes the Kodaira dimension, but does anyone know if a correspondent generalization in the symplectic category ...
4
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0answers
123 views

Cohomology algebra generated by $n$ Steifel whitney classes and and $k$ dual classes subject only to $n+k$ defining relations? [closed]

Is the cohomology algebra $H^*(G_n(\mathbb{R}^{n+k}))$ over $\mathbb{Z}/2$ generated by the Steifel-Whitney classes $w_1, \dots, w_n$ of $\gamma^n$ and the dual classes $\overline{w}_1, \dots, ...
2
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0answers
23 views

Relation between symplectic blow-up of a compact manifold and fibre bundles over same manifold

The symplectic blow-up of a compact symplectic manifold $(X,ω)$ along a compact symplectically embedded submanifold $(M,σ)$ results in another compact manifold $(\tilde{X},\tilde{ω})$ given by ...
3
votes
1answer
246 views

Hypersurfaces without variable cohomology

Let $X$ be a smooth projective variety over $\mathbb C$ of dimension $n+1$. If $Y$ is a smooth very ample hypersurface, we know that except $H^n(X;\mathbb Q)\rightarrow H^n(Y;\mathbb Q)$, the ...
4
votes
1answer
108 views

Bott-Samelson construction of a perfect Morse function on G/T

An undergraduate student of mine is interested in writing a "senior" thesis on the topology of Lie groups. Let $G$ be a simply connected compact Lie group and $T$ a maximal torus. One way of ...
6
votes
2answers
240 views

Hodge map and the Cohomology Ring of a Riemannian Manifold

For a compact Riemannian manifold $M$, we know that the Hodge map $\ast$ and Laplacian $\Delta$ commute. From Hodge decomposition and its implied isomorphism between harmonic forms and cohomology ...
0
votes
1answer
59 views

Conjugation Cells [equivariant cohomology]

I'm studying conjugation spaces, I have found in many sources that a conjugation cell is a conjugation space (without a proof). The widest approach that I have found so far is this paper (example 3.5) ...
3
votes
0answers
98 views

Are there necessary and sufficient conditions for a chain complex $0 \to C_0 \to C_1 \to C_2 \to 0$ to be Poincare?

I am looking for necessary and/or sufficient conditions for the chain complex $0 \to C_0 \to C_1 \to C_2 \to 0$ over a principal ideal domain to be Poincare in the sense that $H_0 \cong H^2$, $H_1 ...
7
votes
1answer
266 views

Intuition for Picard-Lefschetz formula

I'm trying to develop some intuition for the (local) Picard-Lefschetz formula (which I'm encountering for the first time in Deligne's paper "La Conjecture de Weil, I"). To summarize the setup, we ...
4
votes
0answers
127 views

References for bilinear forms on chain complexes?

I am looking for references that include general results and theorems for bilinear forms defined on chain complexes. That is, bilinear forms $\langle \cdot , \cdot \rangle_i : C_i \times C_i \to ...
6
votes
1answer
109 views

Rational cohomology of the Rosenfeld projective planes

The bioctonionic plane $(\mathbb{C} \otimes \mathbb{O})\mathbb{P}^2$, the quarteroctonionic plane $(\mathbb{H} \otimes \mathbb{O})\mathbb{P}^2$ and the octooctonionic plane $(\mathbb{O} \otimes ...
3
votes
1answer
126 views

cohomology module of unit tangent vector bundles over spheres

Let $S^m$ be the $m$-sphere and $\tau (S^m)$ the sphere bundle consisting of unit tangent vectors in the tangent bundle $TS^m$. Then we have a fibration $$ S^{m-1}\longrightarrow ...
2
votes
1answer
161 views

$n$-th cohomology of locally compact subsets in R^n

Where can I find a reference that for any locally compact (or just open) subset $U$ of $\mathbb{R}^n$, $H^n(U;\mathbb{Z})$ (the n-th Cech integral cohomology) is trivial?
3
votes
0answers
154 views

Cohomology and deformations of moduli of vector bundles

I believe that the following is well-known, but I cannot find a reference in the literature... Let $X$ be a smooth variety (in our case $X = M^s(r)$ coarse moduli space of stable rank $r$ vector ...
1
vote
0answers
105 views

cohomology ring of compact submanifolds of Euclidean spaces

Suppose we have a compact $m$-dimensional submanifold $M$ of $\mathbb{R}^N$ and we want to know the cohomology ring $H^*(M;\mathbb{Z})$. Let $\epsilon>0$ and a $m$-dimensional finite simplicial ...
1
vote
1answer
145 views

torsion part of the cohomology module of configuration spaces of manifolds

Let $M$ be a manifold and $C_n(M)$ the $n$-th unordered configuration space consisting of unordered $n$-tuples of distinct points in $M$. The mod $p$ homology module, $p$ prime, and the rational ...
3
votes
0answers
113 views

Equivariant Cohomology of flag varieties

Let $G$ be a simple simply connected algebraic group and $T$ be a maximal torus in $G$. Let $B$ be a Borel containing $T$ and $N(T)/T$ be the Weyl group. We have nice actions of $T$ and $W$ on the ...
2
votes
1answer
140 views

Stiefel-Whitney classes of tensor product $\xi^m \otimes \eta^n$, computation

Let $\xi^m$ and $\eta^n$ be vector bundles over a paracompact base space. Where can I find a reference to the Stiefel-Whitney classes of the tensor product $\xi^m \otimes \eta^n$ being computed as ...
5
votes
0answers
111 views

Testing the vanishing of cohomology fiberwise for a proper morphism from an Artin stack

Let $S$ be a Noetherian scheme, let $f\colon\mathscr{X} \rightarrow S$ be a proper morphism with $\mathscr{X}$ an algebraic stack, and let $\mathscr{F}$ be an $S$-flat coherent sheaf on $\mathscr{X}$. ...
0
votes
0answers
69 views

When can a 2-cocycle on a subgroup can be extended?

This question is based on a question when is the restriction $H^2(G,\mathbb{C}^*)\to H^2(K,\mathbb{C}^*)$ surjective? I am asking this as a new question as I already asked that user but got no ...
3
votes
1answer
96 views

Duistermaat-Heckman integral formula on compact manifold with boundary

Let a compact Lie group $G$ acts on a closed symplectic manifold $(M,\omega)$. If the action is Hamiltonian with $\mu$ the moment map, then the integral $$\int_M e^{i\mu (X)+\omega}$$ is equal to the ...
2
votes
2answers
140 views

Commutator 2-forms on Lie groups

Let $G$ be a compact Lie group and $\mathfrak g$ its Lie algebra. For any $f$ in the dual space $\mathfrak g^*$, we can define a skew-symmetric bi-linear form on $\mathfrak g$ by $(A,B)\mapsto ...
7
votes
1answer
396 views

A conjecture of Cheeger about intersection cohomology and $L^2$- cohomology

Let $X$ be a projective variety and let $D$ be a simple normal crossings divisor on $X$ Does $$IH^*(X;\mathbb C)\cong H_{(2)}^*(X\setminus D;\mathbb C)$$ hold true for each Kähler metric ...
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vote
0answers
150 views

Non-vanishing of a higher direct image

Let $f:X \to Y$ be a small birational morphism between threefolds. Assume $X$ has terminal singularities and the relative Picard of $f$ is $1$. Suppose the exceptional locus of $f$ is a curve $C$, and ...
0
votes
1answer
48 views

How to claculate the $T$-stable subgroup of second cohomology group

Let $G=\langle x,y,z,w \mid [y,x]=w^p=z, x^{p^2}=y^p=z^p=1 \rangle$ be a group, where $[u,v]=u^{-1}v^{-1}uv$, $p$ is a prime and the commutator which do not appear is 1. Let $N=\langle y,w \rangle ...
0
votes
0answers
279 views

Exterior product in relative cohomology

Let us consider two pair of spaces $(X, A)$ and $(Y, B)$. We set $(X, A) \wedge (Y, B) := (X \times Y, (X \times B) \cup (A \times Y))$. Given a cohomology theory $h^{\bullet}$, we can define a ...
39
votes
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594 views

Vector bundle $L$ admits connection if and only if degree of every direct summand of $L$ divisible by $\text{char}\,k$, intuition

Consider the following theorem of Atiyah. Let $X$ be a connected smooth projective curve over an algebraically closed field $k$. Then a vector bundle $L$ on $X$ admits a connection if and only if ...
1
vote
0answers
148 views

Bockstein cohomology

There is a notion of Bockstein homomorphism $\beta$. I am interested precisly in the case of sequence $$0 \rightarrow \mathbb{Z}/p \mathbb{Z} \rightarrow \mathbb{Z}/p^2 \mathbb{Z} \rightarrow ...
0
votes
0answers
86 views

When can the restriction map be zero or onto?

Let $G$ be a finite $p$ group of order $p^n$ and $N$ be a subgroup of $G$ of index $p$. Let $res:H^2(G,A) \rightarrow H^2(N,A)$ be the restriction map on cohomology, where $A$ is a trivial $G$ module. ...
3
votes
1answer
171 views

In which category does Illusies derived deRham complex live?

In "Complexe cotangent et déformations II", Illusie introduces the derived deRham complex as the pro-completion of the total complex associated to the double complex $$\Omega_{P_j^A(B)/A}^i$$ where ...
8
votes
1answer
400 views

Mathematics of Chiral Rings

Let $A$ be a graded vector space, and suppose that two commuting differentials $d_1$ and $d_2$ of degree +1 act on $A$, such that $A$ equipped with either is a chain complex. We now construct $C(A)$, ...
7
votes
2answers
433 views

Galois cohomologies of an elliptic curve

I asked this question at math stackexchange but did not get any answer and I was suggested to post the question here. I am studying basic theory of elliptic curves. I encountered Galois cohomology. ...
2
votes
1answer
125 views

A converse to Whitehead's Second Lemma (and more)

Let $k$ be an algebraically closed field of characteristic $0$ and let $\mathfrak{h}$ be a finite dimensional $k$-Lie algebra. I'm interested in knowing which (Lie algebraic) properties $\mathfrak{h}$ ...
14
votes
1answer
276 views

Multiplicative cohomology theories and smash products

In his student guide on page 154, Adams gives a construction of products for cohomology using "pairings" of spectra (now known as maps from $E\wedge E\to E$). But then he says However, G. W. ...
0
votes
0answers
61 views

Homology of product of two groups [duplicate]

There is well known formula for the homology of product of two groups with coefficient in integers, that is $0 \rightarrow \oplus_{p+q=n}H_p(G,\mathbb{Z}) \otimes H_q(H,\mathbb{Z}) \rightarrow H_n(G ...
10
votes
1answer
310 views

T-equivariant cohomology of flag variety

Let $X=G/B$ , where $G=GL_n(\mathbb{C}^n)$ and $B$ be the upper triangular matrices. I am curoius about the structure of $H^*_T(G/B)$ which I consider as a $H_T^*(pt)$-module. If we just consider ...
1
vote
0answers
98 views

Schubert Calculus for the Full Flags

Almost all introductory texts on Schubert calculus discuss the Grassmannian case only. Does there exist a nice discussion of the full flag manifold case $SU(N)/T^{N-1}$? A low dimensional example ...
5
votes
1answer
218 views

Graded Hopf algebras and H-spaces

Let $k$ denote an algebraically closed field of characteristic $0$. Suppose $K=\bigoplus_{i\geq 0}K(i)$ is a Hopf $k$-algebra which admits a connected Hopf-grading (that is, a grading which is both ...
3
votes
1answer
187 views

geometric conditions on maps between manifolds inducing monomorphisms on cohomology

Let $M,N$ be manifolds whose dimensions may be different. Let $f: M\longrightarrow N$ be a smooth map. What geometric conditions on $f$ can we impose such that the induced homomorphism $$ f^*: ...
2
votes
1answer
106 views

positions of regular cubes in Euclidean space with all its vertices without distinction

Let $P$ be the standard regular cube, centered at the origin of $\mathbb{R}^3$ with all its vertices on the unit sphere. If the vertices are labelled by $1,2,3,4,5,6,7,8$, then the collection of ...
8
votes
1answer
200 views

references for Stiefel-Whitney class of Stiefel manifolds and Grassmannians

Let $M$ be a manifold. The total Stiefel-Whitney class of $M$ is defined to be the Stiefel-Whitney class of the tangent bundle $TM$ $$ w(M)=1+w_1(TM)+w_2(TM)+\cdots $$ I want to find references for $$ ...
20
votes
3answers
1k views

Why should curves be two-dimensional?

In Weil cohomology, a nice curve has cohomology up to degree 2, or more generally a nice $n$-dimensional variety has cohomology up to degree $2n$. I know that this was motivated at least in part by a ...
3
votes
2answers
269 views

mod p cohomology ring of alternating groups

Let $A_n$ be the alternating group of $\{1,2,\cdots,n\}$. (1). What is the cohomology ring $$ H^*(A_4;\mathbb{Z}/3) $$ and its Steenrod operation $P^i$'s? (2). Are there general results about the ...
3
votes
1answer
193 views

Geometric interpretation of the conjugation operation in the dual Steenrod algebra

As the dual mod 2 Steenrod algebra, $A$, is a Hopf algebra, it has the conjugation operation, $\chi:A\to A$. Milnor also gives a formula for this. I wonder if there is any source telling about a ...
10
votes
1answer
479 views

positions of a methane molecule with carbon atom at the origin

Let $\text{CH}_4$ be the molecule of Methane: The four hydrogen atoms form vertices of a regular tetrahedron with the carbon atom in the center of the regular tetrahedron. Here we regard all atoms ...
3
votes
0answers
162 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 $M\times M$. ...
8
votes
1answer
247 views

Stiefel-Whitney class of fibre bundles

Let $F,E,B$ be manifolds (we may assume them to be compact, without boundary if necessary) and $$F\to E\to B$$ be a fibre bundle. Suppose $$w(F), w(B),$$ the Stiefel-Whitney class of $F$ and $B$, are ...