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In this paper, we compute ${\rm Ext}_{A}^{s}(\widetilde{H}^*(B\mathbb{Z}/p),\mathbb{F}_p)$ for $s\leq 1$. Using this result, we investigate the behavior of $\varphi_3^{\mathbb{F}_p}$ and $\varphi_s^{\widetilde{H}^*(B\mathbb{Z}/p)}\ (s\leq1)$ for an odd prime $p$., Comment: 36 pages

The normalized singular chains of a path connected pointed space $X$ may be considered as a connected $E_{\infty}$-coalgebra $\mathbf{C}_*(X)$ with the property that the $0^{\text{th}}$ homology of its cobar construction, which is naturally a cocommutative bialgebra, has an antipode, i.e. it is a cocommutative Hopf algebra. We prove that a continuous map of path connected pointed spaces $f: X\to Y$ is a weak homotopy equivalence if and only if $\mathbf{C}_*(f): \mathbf{C}_*(X)\to \mathbf{C}_*(Y)$ is an $\mathbf{\Omega}$-quasi-isomorphism, i.e. a quasi-isomorphism of dg algebras after applying the cobar functor $\mathbf{\Omega}$ to the underlying dg coassociative coalgebras. The proof is based on combining a classical theorem of Whitehead together with the observation that the fundamental group functor and the data of a local system over a space may be described functorially from the algebraic structure of the singular chains., Comment: This version contains minor changes based on a referee report mostly concerning presentation and typos. We have also added new acknowledgements

The (pointed) coarse shape category Sh * ( Sh ⋆ * ), having (pointed) topological spaces as objects and having the (pointed) shape category as a subcategory, was recently constructed. Its isomorphisms classify (pointed) topological spaces strictly coarser than the (pointed) shape type classification. In this paper we introduce a new algebraic coarse shape invariant which is an invariant of shape and homotopy, as well. For every pointed space ( X , ⋆ ) and for every k ∈ N 0 , the coarse shape group π ˇ k * ( X , ⋆ ) , having the standard shape group π ˇ k ( X , ⋆ ) for its subgroup, is defined. Furthermore, a functor π ˇ k * : Sh ⋆ * → Grp is constructed. The coarse shape and shape groups already differ on the class of polyhedra. An explicit formula for computing coarse shape groups of polyhedra is given. The coarse shape groups give us more information than the shape groups. Generally, π ˇ k ( X , ⋆ ) = 0 does not imply π ˇ k * ( X , ⋆ ) = 0 (e.g. for solenoids), but from pro- π k ( X , ⋆ ) = 0 follows π ˇ k * ( X , ⋆ ) = 0 . Moreover, for pointed metric compacta ( X , ⋆ ) , the n -shape connectedness is characterized by π ˇ k * ( X , ⋆ ) = 0 , for every k ⩽ n .

In this note, we determine the action of the first differential in the Atiyah–Hirzebruch spectral sequence for the nth Morava K-theory of the p-adic construction Dp(ℓ)(X)=Cℓ(p)∧ΣpX∧p of a pointed space X for n⩾1. Our formula gives us a way to compute the first differentials in the Atiyah–Hirzebruch spectral sequence for the Morava K-theory of iterated loop spaces.Analogous formulas are also proved for the Steenrod operations.