4
votes
1answer
29 views

cohomology of classifying space of permutation groups

Let $\Sigma_k$ be the permutation group of order $k$. Let $r: \Sigma_k\to GL(k)$ be the regular representation by permuting the order of the standard basis of $\mathbb{R}^n$. Let $\rho: ...
5
votes
0answers
103 views

The action of Steenrod squares on the indeterminate in Bullett-Macdonald identities

The Bullet-Macdonald identity (c.f. On the Adem relations)is the following: $$P(s^2+st)P(t^2)=P(t^2+st)P(s^2).$$ where we have $P(s)=\Sigma _s s^iSq^i$. This is known to be equivalent to the Adem ...
4
votes
1answer
427 views

When does a cohomology class induce an isomorphism between homotopy groups?

A representative $\alpha$ of a cohomology class $[\alpha]\in H^n(X;\pi_n(X))$ is equivalent to a map from $X$ into an Eilenberg-Mac Lane space as $\alpha:X\to K(\pi_n(X),n)$. After applying a ...
24
votes
2answers
2k 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 ...
1
vote
1answer
277 views

A computation by the Shapiro Lemma

Hi: When I read the book "An introduction to Homological algebra" by Weibel, the page 206, line 9 says that "Shapiro's Lemma tell us that $H_q(S_n(X)\otimes_{Z}A)$ is zero if $q\neq 0$ and is ...
7
votes
1answer
332 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 ...
3
votes
1answer
216 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
155 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 ...
5
votes
1answer
246 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 ...
6
votes
0answers
182 views

When is the cohomology cross product square nonzero?

Let $X$ be a finite CW complex, and $A$ an abelian group. Given a nonzero class in singular cohomology $$x\in H^k(X;A),$$ when is its cross product square $$x\times x\in H^{2k}(X\times X;A\otimes A)$$ ...
5
votes
4answers
382 views

(Co)homological characterization of homotopy pullbacks

For a commutative square of spaces (of manifolds, or of simplicial sets): $$S=\left(\begin{array}{ccc} A & \to & B \newline \downarrow & & \downarrow \newline C & \to & ...
10
votes
3answers
940 views

Is there a good definition of (topological) K-Theory over arbitrary spaces?

Hi (this is my very first question here, so please don't hurt me...) for some time now i've been looking for a sufficiently aesthetical definition of (topological) K-theory of arbitrary spaces, yet ...
16
votes
1answer
588 views

How are these algebraic and geometric notions of homotopy of maps between manifolds related?

Let $M$ and $N$ be smooth manifolds, and $f,g: M \to N$ smooth maps. Denote by $(\Omega^\bullet M,\mathrm d_M)$ and $(\Omega^\bullet N, \mathrm d_N)$ the cdgas of de Rham forms in each manifold, and ...
10
votes
1answer
558 views

Which cohomology theories are real- and complex-orientable?

A complex-oriented cohomology theory $E^*$ is a multiplicative cohomology theory with a choice of Thom class $x\in\tilde{E}^2(\mathbb{C}P^\infty)$ for the universal complex line bundle (which can be ...
1
vote
1answer
660 views

Is there a connection between the theory of motives and homotopy theory?

I have read that motives were designed to be the common part of the many homology theories, a way of unifying them. But as I understand it: homotopy is closely related to homology, there is only 1 ...
27
votes
4answers
1k views

Integral cohomology (stable) operations

There have been a couple questions on MO, and elsewhere, that have made me curious about integral or rational cohomology operations. I feel pretty familiar with the classical Steenrod algebra and its ...
4
votes
1answer
639 views

Simplicial “universal extensions”, the hammock localization, and Ext

Let $M,B$ be $R$-modules, and suppose we're given an n-extension $E_1\to\dots\to E_n$ of $B$ by $M$, that is, an exact sequence $$0\to M\to E_1\to\dots\to E_n \to B\to 0.$$ A morphism of ...
15
votes
2answers
1k views

Is there a map of spectra implementing the Thom isomorphism?

A well known theorem in algebraic topology relates the (co)homology of the Thom space $X^\mu$ of a orientable vector bundle $\mu$ of dimension $n$ over a space $X$ to the (co)homology of $X$ itself: ...
9
votes
1answer
521 views

Representing cohomology of a sheaf à la Eilenberg-Maclane

Suppose that we are given a nice space $X$ and a sheaf of abelian groups $F$ on $X$. Fix an integer $n$. Then We have a contravariant functor from nice spaces over $X$ to abelian groups; Namely, to a ...
5
votes
1answer
1k views

quasi-isomorphism

Is the distinction between quasi-isomorphism and `weak homotopy equivalence' ONLY that the first means inducing an isomorphism in homology and the second to an isomorphism of homotopy groups?
7
votes
4answers
824 views

Difference between represented and singular cohomology?

Ordinary cohomology on CW complexes is determined by the coefficients. There are (more than) two nice ways to define cohomology for non-CW-complexes: either by singular cohomology or by defining ...
6
votes
2answers
827 views

Splitting of the Universal Coefficients sequence

The really beautiful way to prove the Universal Coefficients theorem, to my taste, is to use the fibration sequence $K(\mathbb{Z}, n) \to K(\mathbb{Z}, n) \to K(\mathbb{Z}/k, n)$ (I'm using ...
13
votes
1answer
725 views

Complex orientations on homotopy

I am wondering if there is a more "geometric" formulation of complex orientations for cohomology theories than just a computation of $E^*\mathbb{C}$P$^{\infty}$ or a statement about Thom classes. It ...
8
votes
2answers
678 views

Are the strata of Nakajima quiver varieties simply-connected? Do they have odd cohomology?

Nakajima defined a while back a nice family of varieties, called "quiver varieties" (sometimes with "Nakajima" appended to the front to avoid confusion with other varieties defined in terms of ...
6
votes
1answer
344 views

The (n+1)-st cohomology of K(Z/p,n).

I was looking through my notes for a homotopy theory course and found the following mysterious statement (K is of course the Eilenberg-Maclane space): $$H^{n+1}(K(\mathbb Z_p,n);\mathbb Z_p) \cong ...
7
votes
2answers
609 views

Rational Group Cohomology

This is a general question about group cohomology. I'm interested in the case when the coefficients are the rational numbers and hence I suppose when my groups are infinite. The question splits into ...
12
votes
7answers
923 views

How to get product on cohomology using the K(G, n)?

This came up in the question about Eilenberg-MacLane spaces. Given the definition of K(G, n), it's easy to prove that there is a map ...
12
votes
3answers
1k views

cohomology and Eilenberg-MacLane spaces

This question is related to this question from Dinakar, which I found interesting but don't yet have the background to understand at that level. Unless I'm mistaken, the rough statement is that ...
18
votes
5answers
2k views

Does the cohomology ring of a simply-connected space X determine the cohomology groups of ΩX?

One could try to apply the Eilenberg-Moore spectral sequence to the pullback diagram • → X ← •, obtaining a spectral sequence TorH•(X, R)(R, R) => H•(ΩX, R), but ...
10
votes
2answers
1k views

Finiteness conditions on simplicial sheaves/presheaves

Could someone give an overview, or just some examples, of "finiteness conditions" for simplicial sheaves/presheaves and/or simplicial schemes? Any answer or comment about this would be interesting, ...