3
votes
1answer
209 views

Compactly supported cohomology of homotopy equivalent manifolds

Are there examples of homotopy equivalent smooth, orientable manifolds $M$ and $N$ of the same dimension with non-isomorphic compactly supported cohomology rings?
5
votes
0answers
203 views

Are there any cool applications of the generalized Atiyah-Hirzebruch(-Serre) spectral sequence?

Both the Atiyah-Hirzebruch and the Serre spectral sequence can be constructed from a skeletal filtration of a CW-complex: We can construct the Atiyah-Hirzebruch spectral sequence by filtering $X$ by ...
7
votes
1answer
187 views

worked out examples in borel-moore homology

I'm trying to learn about BM homology. I've found a few references and they are all quite abstract and high-powered. I am actually familiar with derived categories and sheaves so I don't mind thinking ...
11
votes
2answers
415 views

Postnikov's algebraic reconstruction of cohomology from homotopy invariants

In his short paper (1951) and longer monograph (1955), Postnikov introduced what I believe are now called Postnikov systems or towers. It is my understanding that Postnikov systems have since then ...
4
votes
1answer
140 views

The Borel construction of equivariant cobordism

In $K$ theory, the Borel construction of equivariant cohomology is somehow not the right one. The $G$-equivaraint $K$ theory of a point should be the representation ring of $G$, but $K(BG)$ is this ...
9
votes
0answers
230 views

Cohomology and impossible figures

In connection with the MO question Occurrences of (co)homology in other disciplines and/or nature I recalled Roger Penrose's “On the cohomology of impossible figures": ...
1
vote
1answer
111 views

A formula for isotropy group $\pi_1(G_a)$

Let $G$ be a compact Lie group and $T$ be its maximal tours, and $a\in \mathfrak{g}^*$. and $G_a$ be the isotropy group of $G$ then $T\subset G_a$ and we know that $\pi_1(T)=\mathbb{Z}^n$. My ...
5
votes
2answers
202 views

computing second cohomology $H^2(O_a,\mathbb{Z})\cong \mathbb{Z}^n$ of a generic coadjoint orbit

Let $G$ be a compact , connected and simply connected Lie group and $a\in\mathfrak{g}^*$ (dual of Lie algebra of Lie group $G$). Then let $O_a$ be a generic coadjoint orbit then can we say ...
3
votes
1answer
175 views

Cohomologically minimal spaces

Let $X$ be a compact connected Hausdorff topological space. We say $X$ is a cohomologicaly minimal space, briefly CM space, if $X$ satisfies the following property: "For every proper subset ...
-1
votes
1answer
150 views

A closed manifold with a subset with the same ring cohomology

Is there an example of a closed manifold $M$ with a proper subset $A\subset M$ such the inclusion $i:A \to M$ gives a ring isomorphism $i^{*}$ between $\mathbb{Z}$-cohomologies? In this question ...
2
votes
0answers
77 views

The transfer map $H_*(BSO(3))\rightarrow H_*(BO(2))$: reference request

All cohomology and homology will be $Z/2$ coefficient. The restriction map $H^*(BSO(3))\rightarrow H^*(BO(2))$ is well-known to be the inclusion of the Dickson invariant $Z/2[w_2,w_3]$ into the ...
0
votes
0answers
137 views

Metalinear frame bundle on sphere or $\mathbb{C}P^n$

Let $M$ be a smooth manifold. A complex metalinear frame bundle $\tilde F(P)\to M$ of a rank $n$ complex vector bundle $P\to M$ is a principal $ML(n,\mathbb{C})$-bundle together with a covering map ...
22
votes
2answers
1k views

Why should I care about topological modular forms?

There seems to be a lot of recent activity concerning topological modular forms (TMF), which I gather is an extraordinary cohomology theory constructed from the classical theory of modular forms on ...
16
votes
1answer
602 views

For which varieties is the natural map from the Chow ring to integral cohomology an isomorphism?

My apologies if this question is too naive. Let $X$ be a smooth projective complex variety. There is a natural map $A^{\bullet}(X) \to H^{2\bullet}(X)$ of graded rings from the Chow ring of $X$ to ...
3
votes
0answers
68 views

Additive basis for the cohomology of real flag manifolds

By Borel's description the mod 2 cohomology algebra of the flag manifold is the polynomial algebra on the Stiefel-Whitney classes of canonical vector bundles modulo ideal generated by the dual ...
4
votes
0answers
215 views

extension of cohomology theories

In Rudyak's book it is proved (3.22 page 160) that every additive cohomology theory (in the sense of Eilenberg-Steenrod) defined on the category of finite dimensional pointed CW complexes can be ...
0
votes
1answer
326 views

Exact sequences of the cohomology induced by fiber bundle

I'm reading section 2.1 of Lawson's book, Spin Geometry. The book states the following fact. Let $X$ be a manifold and $E$ a vector bundle over it. Equip $E$ with a Riemannian structure. Let $P_O$ be ...
0
votes
0answers
69 views

Hit problem and $\left( \mathbb{F}_2 \otimes_{A} P_6 \right)_{10}$

I try to determine the $\left(\mathbb{F}_2 \otimes_{A} P_6 \right)_{10}$ as a $\mathbb{F}_2$- vector space, in which $P_6$ is polynomial algebra $\mathbb{F}_2[x_1,x_2,\dots,x_6]$ and $A$ is Steenrod ...
1
vote
1answer
316 views

A generalization of cochain complex: quasi-cochain complex

It appears that we can generalize cochain complex to quasi-cochain complex, that still allow us to define cohomology. Definition: A quasi-cochain complex is a sequence of commutative monoids $M_n$ ...
3
votes
1answer
586 views

Pontryagin square on $Y\times S^1$ where $Y$ is three-dimensional

Let $Y$ be an oriented manifold of dimension three, and let $X=Y\times S^1$. We have $$H^2(X,\mathbb{Z}_2)=H^2(Y,\mathbb{Z}_2)\oplus H^1(Y,\mathbb{Z}_2).$$ Pick an element $m\oplus n\in ...
3
votes
0answers
96 views

consistent orientation for functorial pull-push in generalized cohomology

Given a correspondence $X_{\mathrm{in}} \stackrel{i_{\mathrm{in}}}{\leftarrow} X \stackrel{i_{\mathrm{out}}}{\to} X_{out}$ of suitable "spaces" of sorts, and given a (generalized) cohomology theory ...
7
votes
1answer
213 views

What's a good example/reference for cohomology classes on Springer fibers that aren't restricted from the flag variety

As usual, by Springer fiber, I mean the fixed points $X^u$ of a unipotent element $u$ of the group $G$ on the flag variety $X=G/B$. It's a lovely theorem that when $G=SL_n$, the induced map on ...
7
votes
1answer
257 views

On quadratic forms, Pontryagin Squares, and H^4(K(pi,2),U(1))

I am trying to get a concrete handle on the isomorphism $H^4(K(\pi_2,2),U(1)) \simeq \{$quadratic forms $\pi_2 \to U(1) \}$. This is explained in Eilenberg and Maclane's ...
3
votes
2answers
225 views

Equivariant cohomology and complex non-degenerate bilinear forms

Let $M = GL(n,\mathbb{C})$ be the set of non-degenerate bilinear forms on $\mathbb{C}^n$ (not necessarily symmetric). The general linear group $G = GL(n,\mathbb{C})$ acts on $M$ in the usual way ...
6
votes
2answers
343 views

Weights on equivariant cohomology?

Let $X$ be a quasi-projective variety over the complex numbers, equipped with an action of a linear algebraic group $G$. Is there a natural mixed Hodge structure on its equivariant cohomology? Is ...
3
votes
1answer
205 views

Reference request: construction of Steenrod operations for an odd p

Where in literature can one find a construction of Steenrod reduced powers (for an odd $p$) that (1) works for the singular cohomology of arbitrary topological spaces (or, more generally, for the ...
8
votes
1answer
297 views

“Cohomology at the infinity”: what does one call it

Suppose $X$ is a "good enough" Hausdorff topological space; we assume that $X$ is not compact. Now, for a natural number $k$ and an abelian group $G$, consider the group ...
7
votes
1answer
277 views

BRST cohomology definition

Is there written anywhere a full definition of BRST cohomology? All I have found so far is BRST cohomology in _______. As far as I can see, BRST cohomology is the ...
1
vote
0answers
466 views

Cech Cohomology of Sections of Holomorphic Bundle over Contractible Space

Is there anything that can be said in general about the Cech cohomology of the sheaf $\mathcal{F}$ of sections of a holomorphic bundle over a contractible space $X$? I know that such a bundle must ...
1
vote
1answer
189 views

cohomology of infinite product of EM spaces

Is it true that any map from an infinite product of Eilenberg-MacLane spaces in to an Eilenberg-MacLane space factor through, upto homotopy, a finite subproduct? Even if we take with field ...
1
vote
0answers
135 views

Cohomology with compact support and the nerve of a recouvrement

Let $X$ be a simplicial complexe and we assume it localy finite and finite dimensional. We suppose taht there exist a simplicial complexe $Y$ and a map assigning to each vertex $s\in Y$ a finite ...
7
votes
1answer
317 views

A formal group law over oriented bordism

My question is related to the following question by Mark Grant here on math overflow: Formal group law of unoriented cobordism There it is stated that $MO_*$ has a formal group law $F_0$, universal ...
0
votes
2answers
203 views

A question on composites of pushforward and pullback

Let a finite group $G$ acts on an orientable manifold $X$ freely. Denote $\pi:X\rightarrow Y=X/G$ be the quotient map. This covering map defines two maps between cohomology groups ...
1
vote
1answer
296 views

$\pi$-cohomology class — a variant of cohomology class

Let $X$ be a topological space with a triangulation. The triangulation defines a chain complex in $X$. Let $\mu_d$ be a cochain and $M^d$ be a chain. We use $< \mu_d, M^d > \in M$ to denote ...
10
votes
1answer
399 views

Results from Differential Cohomology

I've been working through some notes on differential cohomology for the past few months. I feel like I have a pretty decent grasp on the concepts and its construction, at least for differential ...
3
votes
1answer
196 views

Cohomology of Projective Classical Lie Groups

Let $G$ be a compact, connected, simply-connected Lie group with centre $Z(G)$, and consider the Lie group $G/Z(G)$. I believe that for $G$ a classical group, the Lie group $G/Z(G)$ is sometimes ...
3
votes
0answers
141 views

In the cohomology of Thom spectrum over LoopS^{2} and p-adic characteristic classes

Let $T$ denote the thom spectrum over $\Omega S^{2}$ defined by the map $1+3: \Omega S^{2} \to BG_{3}$ where $1 +3$ is a unit in $3$-adics. Here $G_{3}$ is the unit component of ...
0
votes
0answers
233 views

Cohomology of Grassmannian and equivariant cohomology

Let $G=U(n)$ and $EU(n)=V_{n}(\mathbb{C}^{\infty})$ the infinit Stiefel manifold. So i think that $BU(n)$ is the infinite Grassmannian $G_r(\mathbb{C}^{\infty})$. We have $H_{BU(n)}(pt)= ...
4
votes
2answers
140 views

Borel localization with Mayer-Vietoris sequence

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 \to H^k_G (U \cup V) \to H^k_{G}(U)\oplus ...
6
votes
0answers
204 views

Real schubert calculus

Studying real enumerative questions I noticed, that the even-even Schubert varieties of the real Grassmannian $Gr:=Gr_{2k}(2n,\mathbb R)$ behave analogously to the complex case. I call a partition ...
1
vote
1answer
224 views

What can we say about the map $\pi^{*}\pi_{*}$ for $\pi:X\rightarrow X/G$?

Let $X$ be a manifold or scheme with a finite group $G$ acting on it freely. Let $\pi:X\rightarrow X/G$ be the natural projection. We have $\pi_{\*}\pi^{\*}=|G|id$ on $H^*(X/G,\mathbb{Z})$. Can we say ...
5
votes
1answer
232 views

spectral sequence for cobordism without leaving smooth category

In Bott & Tu's marvelous book there is a derivation of the spectral sequence for a (smooth) fiber bundle for deRham cohomology done entirely in the realm of the smooth category. Unfortunately, as ...
0
votes
0answers
82 views

on variable and primitive cohomology of a hypersurface in a projective space

I have a smooth hypersurface D in $\mathbb{P}^n$: in many books about Hodge theory (as the ones of Voisin and Carlson) they take for granted that the primitive cohomology of D is equal the variable ...
7
votes
2answers
372 views

Injective maps on cohomology and Kahler manifolds

Compact Kahler manifolds have the property that surjective maps induce injections on cohomology with coefficents in $\mathbb{Q}$ (That is, if $X,Y$ compact Kahler, then a surjective map $\phi: X ...
6
votes
0answers
490 views

Is there any “deep” relation between the localization theorem of equivariant cohomology and the localization theorem of equivariant K-theory

First let's consider equivariant cohomology: if a compact Lie group $G$ acts on a compact manifold $M$. We have the equivariant cohomology $ H_G(M)$ defined as the cohomology of the cochain complex ...
12
votes
0answers
387 views

Steenrod algebra at a prime power

Let $n=p^k$ be a prime power. When $k=1$, the algebra of stable operations in mod $p$ cohomology is the Steenrod algebra $\mathcal{A}_p$. It has a nice description in terms of generators and ...
10
votes
1answer
342 views

Wanted: Odd-dimensional integral cohomology class whose square is nonzero

Does anyone know of a nice simple example of a space $X$ with an odd-dimensional integral cohomology class $a\in H^{2k+1}(X;\mathbb{Z})$ whose square is nonzero? I once thought that the ...
2
votes
1answer
201 views

Cohomology groups of quotient by finite group

I know there are already lots of questions about (co)homology groups of a quotient manifold, but please let me ask one more question. Let $G$ be a finite group acting on a manifold $M$ without fixed ...
6
votes
1answer
273 views

Torsion in $H^2(X,\mathbb{Z})$ induced by torsion in $H_1(X,\mathbb{Z})$

Let $X$ be a topological space. The universal coefficient theorem says that there is a short exact sequence $$ 0\rightarrow Ext(H_{k-1}(X,\mathbb{Z}),\mathbb{Z})\rightarrow ...
2
votes
1answer
350 views

equivariant cohomology

In this question we consider only cohomology with rational coefficients. All groups will be connected Lie groups. All group actions will be smooth. Let $M$ be a manifold. Let $G$ be a group acting on ...