The homotopy classification of four-dimensional toric orbifolds

Let X be a 4-dimensional toric orbifold. If $H^{3}(X)$ has a non-trivial odd primary torsion, then we show that X is homotopy equivalent to the wedge of a Moore space and a CW-complex. As a corollary, given two 4-dimensional toric orbifolds having no 2-torsion in the cohomology, we prove that they have the same homotopy type if and only their integral cohomology rings are isomorphic.

On the homotopy classification of proper Fredholm maps into a Hilbert space

We classify the homotopy classes of proper Fredholm maps from an infinite-dimensional Hilbert manifold into its model space in terms of a suitable version of framed cobordism. Our construction is an alternative approach to the classification introduced by Elworthy and Tromba in 1970 and does not make use of further structures on the ambient manifold, such as Fredholm structures. In the special case of index zero, we obtain a complete classification involving the Caccioppoli–Smale mod 2 degree and the absolute value of the oriented degree.

𝜔+-Type Index of GL+(2)-Paths

In this paper, we establish an {\omega^{+}}-type index theory for paths in the general linear group {\mathrm{GL}^{+}(2)}. This is done by the complete homotopy classification for such paths. We also compare this index theory with the ω index theory for paths in the symplectic group {\mathrm{Sp}(2)} and obtain a generalization of Bott formula for iterated paths in {\mathrm{GL}^{+}(2)}. As applications, the minimal periodic solution problem and the linear stability of general differential systems are studied.

Homotopy classification of morphisms of differential graded algebras

We present homotopy classification of morphisms of differential graded algebras {f:A\to A^{\prime}} in terms of induced {A_{\infty}} -algebra morphisms of corresponding minimal models {\{f_{i}\}\kern-1.0pt:\kern-1.0pt(H(A),\{m_{i}\})\kern-1.0pt\to\kern-1.0pt(H(A% ^{\prime}),\{m^{\prime}_{i}\})} .

On Subcritically Stein Fillable 5-manifolds

We make some elementary observations concerning subcritically Stein fillable contact structures on 5-manifolds. Specifically, we determine the diffeomorphism type of such contact manifolds in the case where the fundamental group is finite cyclic, and we show that on the 5-sphere, the standard contact structure is the unique subcritically ?llable one. More generally, it is shown that subcritically fillable contact structures on simply connected 5-manifolds are determined by their underlying almost contact structure. Along the way, we discuss the homotopy classification of almost contact structures.

On geometric formality of rationally elliptic manifolds in dimensions 6 and 7

We discuss the question of geometric formality for rationally elliptic manifolds of dimension 6 and 7. We prove that a geometrically formal six-dimensional biquotient with b2 = 3 has the real cohomology of a symmetric space. We also show that a rationally hyperbolic six-dimensional manifold with b2 ? 2 and b3 = 0 can not be geometrically formal. As it follows from their real homotopy classification, the seven-dimensional geometrically formal rationally elliptic manifolds have the real cohomology of symmetric spaces as well.