Search Results

This may depend on your axioms, see S. Shelah "The consistency of Ext(G,Z)=Q", Israel J. Math. 39 (1981), no. 1-2, 74–82. There it is shown that it is consistent with the generalised continuum hypo …

It is zero, because $BDiff(S^1 \times S^1) \to BGL_2(\mathbb{Z})$ is split by the standard action of $GL_2(\mathbb{Z})$ on $S^1 \times S^1 = \mathbb{R}^2/\mathbb{Z}^2$. In other words $$BDiff(S^1 \tim …

It is the interesting one, with $Sq^1(x) = tx$ and the remainder determined by the axioms of Steenrod operations. This may be seen as the homotopy orbits is a model for BO(2), where one knows the Stee …

I think all elements are representable by honestly framed manifolds. Let $M$ be a closed $d$-manifold with a stable framing, and consider the obstructions to destabilising a stable framing. Asumng $M$ …

This follows immediately from a relative Serre spectral sequence. However, it is not the one usually called the "relative Serre spectral sequence", which has to do with a Serre fibration and its restr …

There is a ``Mayer--Vietoris" sequence $$\cdots \to \pi_2(Z, z) \to \pi_1(E, e) \to \pi_1(X, x) \times \pi_1(Y, y) \to \pi_1(Z,z) \to \pi_0(E) \to \cdots$$ that can be developed by fitting together th …

I seem to remember that a typical example is: let $X$ be the mapping telescope of countably many iterations of $\cdot 2 : S^2 \to S^2$. One calculates $H^3(X;\mathbb{Z})$, say with cellular cochains, …

Let $X=S^1 \vee S^2$ have $\pi_1(X) = \mathbb{Z} = \langle t \rangle$, and so $\pi_2(X) = \mathbb{Z}[t, t^{-1}]$, generated by the inclusion $\iota : S^2 \to X$. Attach a 3-cell to $X$ along $t\iota - …

The case $G=SO$ and $X=*$ is considered in Neumann, Walter D., Fibering over the circle within a bordism class. Math. Ann. 192 1971 191–192. where it is shown that a bordism class fibres over the c …

If $X$ is such a space (a CW complex, say), then it must be a suspension $\Sigma Y$ (as it is the suspension of $y :=\Omega X$). The James splitting gives $$\Sigma \Omega \Sigma Y \simeq \bigvee_{n=1} …

Let me rephrase the construction: you have a map $$\varphi : [0,1] \times M \to \mathbb{R}^n$$ which for each $t \in [0,1]$ is an embedding ($\varphi(t,x) = x+a(t)$ in your notation) and is transverse …

I think $[X] \frown \mathfrak{P}(m \oplus n) = 2\langle m \cdot n + n^3, [Y] \rangle$. The class $m \oplus n$ is better thought of as $m \otimes 1 + n \otimes x$ under the Kunneth decomposition, wher …

Not quite your question, but I'll say it anyway. The "fixed-points of actions on a $p$-acyclic space are $p$-acyclic" part of Smith theory easily extends to arbitrary $p$-groups. By induction on the …

The first identity $\pi_! \circ \pi^* = \vert G \vert \cdot \mathrm{Id}$ holds, and follows from knowing that $\pi_!$ is a $H^*(Y)$-module map via $\pi^*$, so $$\pi_!( \pi^*(x)) = \pi_!(1)\cdot x$$ an …

I think the following will do, but it is not pretty. Write an element of $\Delta^n$ as a tuple $0 \leq x_1 \leq \cdots \leq x_n \leq 1$. Identifying $[0,1]$ with $[-\infty, \infty]$, we may as well c …