arXiv research

Geometrically represents L-homology classes using normal maps.

We construct homology theories with coefficients in L-spectra on the category of ball complexes and we define products in this setting. We also obtain signatures of geometric situations in these homology groups and prove product formulae which we hope will clarify products used in the theory of the total surgery obstru…

This paper uses Steenrod homology to simplify surgery on generalized manifolds.

This paper presents an alternative approach to controlled surgery obstructions. The obstruction for a degree one normal map $(f,b): M^n \rightarrow X^n$ with control map $q: X^n \rightarrow B$ to complete controlled surgery is an element $σ^c (f, b) \in H_n (B, \mathbb{L})$, where $M^n, X^n$ are topological manifolds o…

Affirmative answer to a question about a map extending normal invariants.

For a closed topological $n$--manifold $K$ and a map $p:K\to B$ inducing an isomorphism $π_1(K)\toπ_1(B)$, there is a canonicaly defined morphism $b:H_{n+1}(B,K,\mathbb{L})\to \mathbb{S} (K)$, where $\mathbb{L}$ is the periodic simply-connected surgery spectrum and $\mathbb{S} (K)$ is the topological structure set. We …

Study vineyards linking TDA and knot theory, showing rich topological features.