Search Results

As to the cohomology, here is an answer, which is probably an "over-kill". Probably you can find an answer in much older literature like Borel's. Since $BT$ has torsion-free homology, COROLLARY 4.9 o …

As explained in comments by jdc $q$ induces an isomorphism. That leaves us with $j$. By taking a look at the cohomology of $RP^{2d-1}$ and its $(d-1)$-st skelton (which becomes $(2d-1)$-st skelton aft …

Here is an "answer" which may be or not be good enough for your purpose, but which is easy to prove. Let's start with Ravenel-Wilson-Yagita Theorem 1.20. Applied to Eilenberg-Maclane spaces, it impli …

Denote $G=\pi _1(X)$. Then we have a fibration $\tilde{X}\to X \to BG$, which leads to the Eilenberg-Moore spectral sequence $$ Tor ^{H^*(BG)}(H^*(X),H^*(pt))\Rightarrow H^*(\tilde{X})$$ provided tha …

$\newcommand\Sets{\mathrm{Sets}}\DeclareMathOperator\Nerve{Nerve}$Let $G$ be a finite group, $\mathcal{G}^0$ be the category of finite free $G$-sets and isomorphisms between them. Then $\mathcal{G}^0 …

The statement is false, here is a counterexample. First note that for a Lie Group $G$ and its closed subgroup $H$, we have a fibration $G/H\to BH\to BG$. $BG$ and $BH$ are not finite, but they are a …

As is commented by @Connor Malin, the action of the Steenrod algebra on $H^*B\mathbb{Z}/p$ is not faithful. Consider the case $p=2$. $Sq^3Sq^1$ acts trivially on $H^*(B \mathbb{Z}/2)$, since $Sq^{2n …

Here is a huge family of examples. Let $F:\mathcal{C}\to \mathcal{D}$ be, let's say, an additive functor between abelian categories. Let's say, $F$ is right or left exact. Its failure to be exact i …

This is an answer only to the question 1.2 (1.1 is already answered in a comment by Ulrich Pennig). Consider the adjoint representation of $SU(2)$ on its Lie algebra. Noting that the real dimension …

Write $kO$ for the connective $k$-theory, and $X$ for the connective delooping of $BTOP$. Then $H_*(BO;Q)$ and $H_*(BTOP,Q)$ are free commutative (in graded sense) generated by $\pi _*(kO,Q)\cong \p …

If it were the case, then $K(Z,n)$ would be a retract of $\underline{ku}_n$ where $\underline{ku}_n$ is the $n$-th infinite loop space associated to the spectrum $ku$. However, $ku$ is what is called …

Let $Y$ be a space, $V$ be a virtual bundle of dimension $0$ over $Y$ (this $V$ is your $-V$). Then $Thom(Y,2V)$ is almost never (except when $Y$ is contractible, or something like that) a smash prod …

I don't know how "systematic" the answer you are looking has to be, but for the quotients of the form $G_2/T$ and $G_2/P$ (P: Parabolic subgroup) you can find the results in Schubert presentation of t …

First of all, I think the concern of the OP is the fact that the homology of the infinite loop space doesn't agree with that of the spectra. So let's start with this. Let $Y$ be a spectrum, then it …

Different people use different notation on gradings, for example I would have called the bigrdading of $e_i$ $E_2^{1,2i}$. Supposing that this is not a typo, Quillen meant by $k$ in $E_s^{j,k}$ the …