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4
votes
2answers
198 views

cup product and Steenrod operations in Serre spectral sequence

Let $F\to E\to B$ be a fibration with $B$ simply-connected. Suppose all differentials in the cohomology Serre spectral sequence (corresponding to the above fibration) are zero maps. Then as a graded ...
2
votes
1answer
156 views

Schubert calculus and Pieri's formula

In the lecture notes Grassmannians: the first example of a moduli space. MIT Open Course Ware. page 7: Are there any formal publications (books/papers) where I can find the formula?
5
votes
1answer
169 views

group completion theorem of homology as Hopf algebras

Let $M$ be a topological monoid with product $\mu$. Then $H_*(M)$ is a Hopf algebra with product $\mu_*$ and coproduct $\Delta_*$. The group-completion theorem by McDuff-Segal, 1976 gives that as a ...
3
votes
0answers
58 views

Crossed homomorphisms between power series groups

Consider the group $\mathbb{C}[[z]]_1$ of the power series of the form $a_1 z + a_2 z^2 + \cdots$, with $a_1\neq 0$, under the operation of composition, and $\mathbb{C}[[z]]$ as a ...
5
votes
0answers
94 views

Steenrod operations on cohomology of grassmannians

Let $G_k(\mathbb{R}^n)$, $n\geq k$ and $G_k(\mathbb{R}^\infty)$ be the finite-dimensional and infinite-dimensional grassmannians respectively. Their cohomology rings are expressed in terms of ...
0
votes
1answer
188 views

When is the Thom class the Poincare dual of the zero section?

As the title suggests, when is the Thom class the Poincare dual of the zero section? For starters, it's true for the normal bundle of an immersion...
3
votes
1answer
110 views

Stokes-like Theorem for Dolbeault Operator

I have a simple question regarding complex geometry: is there an analog for the Stokes Theorem for the Dolbeault Operator $\bar{\partial}$? For instance, suppose that $M$ is a closed complex manifold ...
2
votes
0answers
49 views

cohomology ring of cross-section space of one-point compactification of tangent bundle

Let $M$ be an $m$-manifold whose cohomology is known. Let $TM$ be the tangent bundle of $M$ and $\xi$ be the fibre-wise one-point compactification of $TM$. Then $\xi$ is a $m$-sphere bundle over $M$. ...
0
votes
0answers
96 views

reference for groupoid cohomology

In nLab (groupoid cohomology) says: "Under the homotopy hypothesis theorem, plain (non-internal) groupoid cohomology is the same as the cohomology of homotopy 1-types." Are there references for ...
4
votes
0answers
149 views

Hochschild Cohomology of the Quantum Torus

I would like some advice on how to compute directly, or by a higher powered method the Hochschild Cohomology groups of the quantum torus using the stated complex I have found. I think there are ...
7
votes
1answer
417 views

Do modular forms show up in the cohomology of moduli spaces of unmarked curves?

Let $\overline{\mathcal M}_{g,n}$ be the compactified Deligne-Mumford moduli stack (although I don't think taking the coarse moduli space will make much of a difference here). If we decompose $g = 1 + ...
3
votes
1answer
242 views

What is the (co-)homology of $K(\mathbb{R}_\delta,n)$?

To elaborate on the question from the title, $\mathbb{R}_\delta$ is the additive group of real numbers (without any topology) and $K(\mathbb{R}_\delta,n)$ is an Eilenberg-MacLane space. I would like ...
1
vote
1answer
114 views

When the restriction of derived equivalence to a summand is a derived equivalence as well

I have a question about the equivalence of derived categories. Let $\mathcal{A} = \mathcal{A}'\oplus \mathcal{A}''$ and $\mathcal{B} = \mathcal{B}' \oplus \mathcal{B}''$ are direct sum of abelian ...
9
votes
1answer
226 views

Torsors trivializing over a fixed finite etale cover

Let $S$ be an integral regular scheme and let $T\to S$ be a finite etale morphism. Let $G$ be a smooth affine finite type group scheme over $S$. Is the set of $S$-isomorphism classes of $G$-torsors ...
0
votes
0answers
98 views

Chern classes, vanishing of smooth sections or vanishing of holomorphic?

I have seen both definitions and this is getting me more and more confused. Are Chern classes dual to the degeneracy cycles of smooth sections or holomorphic? They can't be the same thing, can they? ...
3
votes
0answers
66 views

cohomology ring of mapping spaces

In the lecture notes The homology of $\mathcal{C}_{n+1}$–spaces, n ≥ 0. F. Cohen, 1978, page 228-231, the cohomology ring $$ H^*(\text{Map}_*(S^n, S^n\wedge X);\mathbb{Z}_p) $$ is obtained for any ...
3
votes
0answers
80 views

Configuration spaces of positive and negative particles

In the paper Mapping class group and function spaces: a survey, F. Cohen, M.A. Maldonado, page 3, line from bottom 1-3, it is given that for a $m$-manifold $M$, there is a map from the labelled ...
1
vote
1answer
49 views

Group completion of labelled configuration space on Euclidean spaces

In the lecture notes The Homology of $\mathcal{C}_{n+1}$-spaces, $n\geq 0$, F. Cohen, page 225 -226, it is obtained that there is a group completion on homology $$ \alpha_n: C(\mathbb{R}^n;X)\to ...
1
vote
0answers
54 views

cohomology ring of unordered configuration space on Euclidean spaces

Let $F(\mathbb{R}^n,k)/\Sigma_k$ be the $k$-th unordered configuration space on $\mathbb{R}^n$. In the lecture notes The Homology of $\mathcal{C}_{n+1}$-spaces, $n\geq 0$, F. Cohen, page 226, it ...
6
votes
3answers
472 views

Homological vs. cohomological dimension of a group/space

I have several related questions regarding homological vs. cohomological dimension of a space/group (this is not a duplicate of this). The standard definition of the cohomological dimension $cd(X)$ ...
0
votes
1answer
119 views

Geometric interpretation of Chern classes over flag manifolds

I would like to find that Chern classes of the tautological bundles over a flag manifold are dual to some cells in homology, analogously to what happens for the Grassmanian case. I have not been able ...
4
votes
1answer
233 views

Künneth formula for Bredon cohomology theory

Let $G$ be a finite group. Let $X$ and $Y$ be two $G$-CW complexes with known integer graded $G$-equivariant Bredon cohomology with constant coefficient systems. Is there any Künneth formula for this ...
1
vote
0answers
157 views

Functors similar to $H^i(\cdot)$

Suppose $T$ is a contravariant functor from the category of pointed topological spaces to the category of abelian groups, then we have homomorphisms $\alpha\colon T(X)\times T(Y)\to T(X\times Y)$ and ...
3
votes
1answer
117 views

group completion theorem by using homology fibrations

In the paper Homology fibrations and group completion theorem, McDuff-Segal (www.maths.ed.ac.uk/~aar/papers/mcdsegal.pdf), page 281: Let $M$ be a topological monoid such that $\pi_0M$ is generated by ...
0
votes
1answer
291 views

On a claim of Zagier on extending a map to cocycle

Zagier, in his paper 'Some Surprising Consequences of the Cohomology of SL$_2(\bf{ Z})$' (link, p. 6), studies the action of $\Gamma=PSL_2(\bf Z)$ on a vector space $V$, denoting the action by $v\ |\ ...
2
votes
2answers
446 views

What is the $\mathbb Z/2$-cohomology of $\mathrm B^n(\mathbb Z/2)$?

I would like to know the cohomology groups $\mathrm H^\bullet(\mathrm B^n(\mathbb Z/2);\mathbb Z/2)$. I assume that this is a standard computation, but I'm not sure where to look up the answer (and, ...
2
votes
0answers
353 views

Differential and pre-differential of Jacobi identity

Let M be a manifold. To what extent all Lie algebra structures with tensorial property on $\chi^{\infty}(M)$ are studied? That is a Lie algebra structure for which $[X,fY]=f[X,Y]$. (For ...
1
vote
0answers
103 views

Generalization of De Rham cohomology for spinor fields

Is there a generalization of De Rham cohomology for spinors fields? I can see that one can construct p form fields out of spinor field by contraction of the type $\bar{\psi} \gamma^{a_1} ...
1
vote
1answer
107 views

Schubert Polynomials for Complex Projective Space

The Borel picture of the cohomology ring of a flag variety gives a description as a coset space, without identifying any representatives for the classes. Lascoux and Schützenberger gave specific ...
2
votes
2answers
269 views

Is Eilenberg-Maclane $\wedge$ Moore space the spectrum of the cohomology theory $H^*(\ ,G)$?

In the web page http://www.encyclopediaofmath.org/index.php/Moore_space it can be found the following statement: If $K(\mathbb Z,n)$ is the Eilenberg–MacLane space of the group of integers ...
0
votes
1answer
159 views

Cohomology of Homogeneous Complex Manifolds

Let $M$ be compact $G$-homogeneous manifold, equipped with the equivariant complex structure, when $G$ is a semi-simple algebraic group. The obvious example is every flag manifold. In that case, all ...
-2
votes
1answer
74 views

stable splitting into a wedge sum [closed]

Suppose $X$ is a CW-complex such that there is a stable splitting of $X$ into wedge sum $$ \Sigma^t X\cong \bigvee _{k=1}^\infty Y_k. $$ (1). Does this imply $$ X\to \Sigma^tX\to \bigvee ...
1
vote
1answer
206 views

What is the cokernel of the map $H^2\big(\pi_1(X), \mathbb Z\big) \longrightarrow H^2(X,\mathbb Z).$

For a manifold $X$ (for simplicity, assumed to be compact), let $\pi_1(X)$ be the fundamental group of $X$. What is the cokernel of the map $$H^2\big(\pi_1(X), \mathbb Z\big) \longrightarrow ...
2
votes
0answers
95 views

Generating free groups by small subgroups and an element

Let $F$ be a free group of countably infinite rank, and let $L \leq F$ be a finite index subgroup. For some prime $p$ and $k \in \mathbb{N}$, let $\sigma \colon L \to ...
3
votes
2answers
232 views

Upper bound on Betti numbers of an intersection of hypersurfaces (or quadrics)

I have a problem I have been stuck with since several weeks now, and yet I believe it should be easy to specialists. Let $k$ be an algebraically closed field, $m$ and $n$ two integers. Let ...
6
votes
4answers
698 views

Is there a (satisfying) proof that cellular cohomology is isomorphic to simplicial cohomology that doesn't use relative cohomlogy?

That singular and de Rham cohomologies of a smooth manifold are isomorphic has two proofs that I know of. The classical one uses Stokes' theorem to give the isomorphism explicitly. The second proof ...
1
vote
1answer
126 views

Unordered configuration space of $\mathbb{R}P^1$

In the paper GEOMETRY OF TRUNCATED SYMMETRIC PRODUCTS AND REAL ROOTS OF REAL POLYNOMIALS, JACOB MOSTOVOY, Bull. London Math. Soc. (1998) 30 (2): 159-165, Theorem 2. (b): ...
1
vote
1answer
196 views

Multiplicative structure in the cohomological Leray-Serre spectral sequence - please elucidate a proof

Let $\pi \colon X \to B$ be a fibration with $B$ a path-connected CW complex. Write $B^p$ for the $p$-th skeleton of $B$ and set: $X_p = \pi^{-1}(B^p)$, $F_p^m = \ker [H^m(X) \to H^m(X_{p-1})]$, ...
3
votes
1answer
179 views

When is the cohomology of a space Ext(k,k)?

If $X$ is aspherical, we know that $H^*(X,k) = \text{Ext}_R(k,k)$, with $R = k\pi_1$. For non-aspherical spaces, do we ever have $H^*(X,k) = \text{Ext}_R(k,k)$ for some ring $R$? Obviously we need ...
3
votes
0answers
93 views

Action of automorphisms on cohomology with supports

Let $x$ be the closed point of an $n$-dimensional local scheme $X$, essentially smooth over a field $k$. Let $M$ be a sheaf on the category of smooth $k$-varieties (in either Zariski or Nisnevich ...
3
votes
1answer
120 views

LES for relative cohomology via sheaves

I was unable to find a suitable answer for the following question: Once one learns that singular cohomology is the same as cohomology with coefficients in locally constant sheaf, it is natural to try ...
18
votes
1answer
577 views

What are explicit obstructions to realizability of formal group laws as complex-oriented ring spectra?

Recall that a complex-oriented spectrum is a ring spectrum E with a map $MU \to E$. Analogously, a ring with a (1-d commutative) formal group law is (represented by) a ring $R$ with a map $L \to R$ ...
9
votes
1answer
318 views

Identifying a Hopf algebra cohomology theory

Here is a cohomology theory for a Hopf algebra, which I am sure has appeared elsewhere. I met it in the van Est spectral sequence for Hopf algebras. Apologies for my being stupid here, but it would be ...
8
votes
2answers
449 views

Interpretation of the monomorphism $H^2(\pi_1(X),\mathbb{Z}) \rightarrow H^2(X,\mathbb{Z})$

Let $X$ be a nice topological space and denote by $\pi_1(X)$ its fundamental group. It is well-known that there is a well-defined map $$ 0 \rightarrow H^2(\pi_1(X),A) \rightarrow H^2(X,A),$$ where ...
5
votes
0answers
129 views

Are all these K3 surfaces supersingular?

Consider all the smooth K3 surfaces given by $X^4+W^2X^2+XW^3 = f(Y,Z,W)$ or $X^4+XW^3 = g(Y,Z,W)$ over $\mathbb F_{2}$ with $f$ or $g$ homogenous of degree 4. There are a lot of choices for $f$ and ...
7
votes
1answer
292 views

What is an example of a formal group law that is Landweber-exact but not flat?

Quick Background: The $p$-series of $F$ (where $F$ is a formal group law over a graded ring $R$) will be of the form $[p](x) = px + v_1x^{p^1} + ... + v_nx^{p^n} + ...$ ; $(F, R)$ is Landweber-exact ...
0
votes
0answers
61 views

Cohomology operators inducing local basis of $1-$forms

Suppose that $\partial$ is a non-trivial ($\partial \neq 0$) cohomology operator on an $m-$dimensional manifold $M$ (that is: $\partial:\Omega(M)\to\Omega(M)$ is a degree $1$ derivation such that ...
2
votes
1answer
285 views

Integration currents VS Poincaré Dual

Let $M$ be a complex manifold of dimension $n$ and $S \subset M$ a closed complex submanifold of complex codimension $r$. Let $[S] \in H_{2r}(S)$ be the fundamental class of $S$. We have the ...
2
votes
1answer
177 views

When is finding an explicit inverse of an isomorphism not possible

My question is about Shapiro's lemma. Consider the isomorphism $\phi: H^n(G, Hom_{ZH}(ZG, A))\cong H^n(H,A)$ of shapiro's lemma. I would like to describe this via cochains. So the obvious map is ...
2
votes
0answers
90 views

Deformations of associative algebras and Hochschild cohomology

I am studying the deformation theory of associative algebras (and Poisson algebras) and came across a question for which I cannot find an answer: Let $(A,\mu)$ be a commutative associative algebra ...