Search Results

A pretty direct argument was given by Frank Adams in the proof of 16.6 (page 336) in part III of his 1974 Chicago lectures (MR0402720). Thinking of the Atiyah--Hirzebruch spectral sequence for $K^*(X …

A possibly interesting analogue of the formula $H\mathbb{F}_{2*} H\mathbb{F}_2 = \otimes_{i\ge1} \mathbb{F}_2[\xi_i]$ is $H\mathbb{Z}_{(2)*} H\mathbb{Z}_{(2)} = \bigotimes^\mathbb{L}_{i\ge1} \mathbb{ …

The $A(2)$-module structure on $A(2)//A(1)$ does not extend to an $A$-module structure. In particular, there is no spectrum $X$ with $H^*(X; F_2) = A(2)//A(1)$ as an $A(2)$-module. Additively, $A(2) …

It is not generally true that each open subset of a CW complex admits the structure of a CW complex. Counterexamples were given in \bib{MR1157891}{article}{ author={Cauty, Robert}, title={Sur …

The $p$-completed algebraic $K$-theory of the algebraic closure of $\mathbb{Q}_p$, i.e., $K(\bar{\mathbb{Q}}_p; \mathbb{Z}_p)$, is equivalent to its second loop space, up to an issue about path compon …

Let $G = \mathbb{Z}/p \times \mathbb{Z}/p$ for $p$ odd. Then $H^3(BG; \mathbb{Z}) \cong \mathbb{Z}/p$ is not in the subring generated by Euler classes, since the non-trivial irreducible representatio …

An elementary answer to the first part of your question: Finite sets are more fundamental than their cardinalities. Consider the category of finite sets and bijective functions. Its geometric realiza …

You can try to define the determinant of an $n \times n$ matrix with entries in a bipermutative (or symmetric bimonoidal) category $R$ by an analogue of the usual signed sum of $n$-fold products. Howe …

With the aid of machine computations, you can readily determine the Adams differentials up to $t-s=30$ using the multiplicative structure, the relation between Steenrod operations in $\text{Ext}_A$ an …

For the relation $Sq(U) = \Phi(w)$, where $Sq$ is the total Steenrod squaring operation, $U$ is the Thom class, $\Phi$ is the Thom isomorphism and $w$ is the total Stiefel-Whitney class, I would cite …

Yes. For $i\ge1$ you can build $K(G,i)$ from the Moore space $M(G,i)$ by adding cells of dimension $\ge i+2$, so $H_i(K(G,i); Z) = G$ and $H_{i+1}(K(G,i); Z) = 0$. Hence $Ext(G, H) \cong H^{i+1}(K(G …

The $2$-completed stable $20$-stem $\pi_{20}(S)_2$ is cyclic of order $8$. Mimura and Toda (1963, Lemma 15.4) mr=157384 show the existence of a class $\bar\kappa_7 \in \pi_{27}(S^7)$ whose stable cla …

The first reference in this general area was: N. E. Steenrod, "Milgram's classifying space of a topological group", Topology 7 (1968) 349–368. Working in the category of compactly generated …

The convenient category CGH of compactly generated Hausdorff spaces has some poor colimits, since Hausdorffification may change the underlying point sets. The category CGWH of compactly generated we …

Some examples with one nonzero family of differentials: The classical Adams spectral sequence for $j/p$, the connective image-of-J spectrum reduced mod $p$, collapses at $E_3$, by Theorems 4.5 (at $p= …