In this note, we give an alternate proof of the Farrell–Jones isomorphism conjecture for the affine Artin groups of type $\widetilde B_n$.

For every $k \geq 2$ and $n \geq 2$, we construct n pairwise homotopically inequivalent simply connected, closed $4k$-dimensional manifolds, all of which are stably diffeomorphic to one another. Each of these manifolds has hyperbolic intersection form and is stably parallelisable. In dimension four, we exhibit an analogous phenomenon for spin$^{c}$ structures on $S^2 \times S^2$. For $m\geq 1$, we also provide similar $(4m-1)$-connected $8m$-dimensional examples, where the number of homotopy types in a stable diffeomorphism class is related to the order of the image of the stable J-homomorphism $\pi _{4m-1}(SO) \to \pi ^s_{4m-1}$.

The trace of the $n$-framed surgery on a knot in $S^{3}$ is a 4-manifold homotopy equivalent to the 2-sphere. We characterise when a generator of the second homotopy group of such a manifold can be realised by a locally flat embedded $2$-sphere whose complement has abelian fundamental group. Our characterisation is in terms of classical and computable $3$-dimensional knot invariants. For each $n$, this provides conditions that imply a knot is topologically $n$-shake slice, directly analogous to the result of Freedman and Quinn that a knot with trivial Alexander polynomial is topologically slice.

Let $X^{n}$ be an oriented closed generalized $n$-manifold, $n\ge 5$. In our recent paper (Proc. Edinb. Math. Soc. (2) 63 (2020), no. 2, 597–607), we have constructed a map $t:\mathcal {N}(X^{n}) \to H^{st}_{n} ( X^{n}; \mathbb{L}^{+})$ which extends the normal invariant map for the case when $X^{n}$ is a topological $n$-manifold. Here, $\mathcal {N}(X^{n})$ denotes the set of all normal bordism classes of degree one normal maps $(f,\,b): M^{n} \to X^{n},$ and $H^{st}_{*} ( X^{n}; \mathbb{E})$ denotes the Steenrod homology of the spectrum $\mathbb{E}$. An important non-trivial question arose whether the map $t$ is bijective (note that this holds in the case when $X^{n}$ is a topological $n$-manifold). It is the purpose of this paper to prove that the answer to this question is affirmative.

We work in the smooth category. Let $N$ be a closed connected orientable 4-manifold with torsion free $H_1$, where $H_q := H_q(N; {\mathbb Z} )$. Our main result is a readily calculable classification of embeddings$N \to {\mathbb R}^7$up to isotopy, with an indeterminacy. Such a classification was only known before for $H_1=0$ by our earlier work from 2008. Our classification is complete when $H_2=0$ or when the signature of $N$ is divisible neither by 64 nor by 9.

The group of knots $S^4\to {\mathbb R}^7$ acts on the set of embeddings $N\to {\mathbb R}^7$ up to isotopy by embedded connected sum. In Part I we classified the quotient of this action. The main novelty of this paper is the description of this action for $H_1 \ne 0$, with an indeterminacy.

Besides the invariants of Part I, detecting the action of knots involves a refinement of the Kreck invariant from our work of 2008.

For $N=S^1\times S^3$ we give a geometrically defined 1–1 correspondence between the set of isotopy classes of embeddings and a certain explicitly defined quotient of the set ${\mathbb Z} \oplus {\mathbb Z} \oplus {\mathbb Z} _{12}$.

We observe an inductive structure in a large class of Artin groups of finite real, complex and affine types and exploit this information to deduce the Farrell–Jones isomorphism conjecture for these groups.

The aim of this paper is to show the importance of the Steenrod construction of homology theories for the disassembly process in surgery on a generalized n-manifold Xn, in order to produce an element of generalized homology theory, which is basic for calculations. In particular, we show how to construct an element of the nth Steenrod homology group $H^{st}_{n} (X^{n}, \mathbb {L}^+)$, where 𝕃+ is the connected covering spectrum of the periodic surgery spectrum 𝕃, avoiding the use of the geometric splitting procedure, the use of which is standard in surgery on topological manifolds.

We prove that the Hatcher–Quinn and Wall invariants of a self-transverse immersion f: Nn ↬ M2n coincide. That is, we construct an isomorphism between their target groups, which carries one onto the other. We also employ methods of normal bordism theory to investigate the Hatcher–Quinn invariant of an immersion f: Nn ↬ M2n−1.

We show that closed $\widetilde{\mathbb{S}\mathbb{L}}\,\times \,{{\mathbb{E}}^{n}}$-manifolds are topologically rigid if $n\,\ge \,2$, and are rigid up to $s$-cobordism, if $n\,=\,1$.

We study the topological 4-dimensional surgery problem for a closed connected orientable topological 4-manifold $X$ with vanishing second homotopy and ${{\pi }_{1}}\left( X \right)\,\cong \,A\,*\,F\left( r \right)$ , where $A$ has one end and $F\left( r \right)$ is the free group of rank $r\,\ge \,1$. Our result is related to a theorem of Krushkal and Lee, and depends on the validity of the Novikov conjecture for such fundamental groups.