Connected components of the space of proper gradient vector fields

We show that there exist two proper gradient vector fields on $$\mathbb {R}^n$$ R n which are homotopic in the category of proper maps but not homotopic in the category of proper gradient maps.

The Motivic Group H−1,−1BM

This chapter develops some more of the properties of the Borel–Moore homology groups 𝐻𝐵𝑀 −1,−1(𝑋). It shows that it is contravariant in 𝑋 for finite flat maps, and has a functorial pushforward for proper maps. If 𝑋 is smooth and proper (in characteristic 0), 𝐻𝐵𝑀 −1,−1(𝑋) agrees with 𝐻2𝒅+1,𝒅+1(𝑋, ℤ), and has a nice presentation, which this chapter explores in more depth. The main result in this chapter is the proposition that: if 𝑋 is a norm variety for ª and 𝑘 is 𝓁-special then the image of 𝐻𝐵𝑀 −1,−1(𝑋) → 𝑘× is the group of units 𝑏 such that ª ∪ 𝑏 vanishes in 𝐾𝑀 𝑛+1(𝑘)/𝓁. Again, this chapter also explores the historic trajectory of its equations.

Pull-Back of Metric Currents and Homological Boundedness of BLD-Elliptic Spaces

Using the duality of metric currents and polylipschitz forms, we show that a BLD-mapping f : X → Y between oriented cohomology manifolds X and Y induces a pull-back operator f* : Mk,loc(Y) → Mk,loc(X) between the spaces of metric k-currents of locally finite mass. For proper maps, the pull-back is a right-inverse (up to multiplicity) of the push-forward f* : Mk,loc(X) → Mk,loc(Y). As an application we obtain a non-smooth version of the cohomological boundedness theorem of Bonk and Heinonen for locally Lipschitz contractible cohomology n-manifolds X admitting a BLD-mapping ℝn → X.

HILBERT STRATIFOLDS AND A QUILLEN TYPE GEOMETRIC DESCRIPTION OF COHOMOLOGY FOR HILBERT MANIFOLDS

In this paper we use tools from differential topology to give a geometric description of cohomology for Hilbert manifolds. Our model is Quillen’s geometric description of cobordism groups for finite-dimensional smooth manifolds [Quillen, ‘Elementary proofs of some results of cobordism theory using steenrod operations’, Adv. Math., 7 (1971)]. Quillen stresses the fact that this construction allows the definition of Gysin maps for ‘oriented’ proper maps. For finite-dimensional manifolds one has a Gysin map in singular cohomology which is based on Poincaré duality, hence it is not clear how to extend it to infinite-dimensional manifolds. But perhaps one can overcome this difficulty by giving a Quillen type description of singular cohomology for Hilbert manifolds. This is what we do in this paper. Besides constructing a general Gysin map, one of our motivations was a geometric construction of equivariant cohomology, which even for a point is the cohomology of the infinite-dimensional space $BG$, which has a Hilbert manifold model. Besides that, we demonstrate the use of such a geometric description of cohomology by several other applications. We give a quick description of characteristic classes of a finite-dimensional vector bundle and apply it to a generalized Steenrod representation problem for Hilbert manifolds and define a notion of a degree of proper oriented Fredholm maps of index $0$.