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.

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.

For Dewitt super groups $G$ modeled via an underlying finitely generated Grassmann algebra it is well known that when there exists a body group $BG$ compatible with the group operation on $G$, then, generically, the kernel $K$ of the body homomorphism is nilpotent. This is not true when the underlying Grassmann algebra is infinitely generated. We show that it is quasi-nilpotent in the sense that as a Banach Lie group its Lie algebra $\kappa$ has the property that for each $a\,\in \,\kappa ,\,\text{a}{{\text{d}}_{a}}$ has a zero spectrum. We also show that the exponential mapping from $\kappa$ to $K$ is surjective and that $K$ is a quotient manifold of the Banach space $\kappa$ via a lattice in $\kappa$.

We determine the extent to which the collection of Γ-Euler–Satake characteristics classify closed 2-orbifolds. In particular, we show that the closed, connected, effective, orientable 2-orbifolds are classified by the Γ-Euler–Satake characteristics corresponding to free or free abelian Γ and are not classified by those corresponding to any finite set of finitely generated discrete groups. These results demonstrate that the Γ-Euler–Satake characteristics corresponding to free abelian Γ constitute new invariants of orbifolds. Similarly, we show that such a classification is neither possible for non-orientable 2-orbifolds nor for non-effective 2-orbifolds using any collection of groups Γ.