A note on $H^1(Emb(M,N))$

A note on $H^* (\rm{Emb}_0 (M,N))$

Aim of this note is to gain cohomological information about the infinite-dimensional manifold $\rm{Emb}_0 (M,N)$ of asymptotically fixed embeddings of M into N from the topology of the target manifold N. This paper has been withdrawn by the author due a conceptual mistake.

定向同调和上同调操作

Directed homological and cohomological operations

本文提出了一种基于持久模块的定向上同调方法，探讨其性质及与同调操作的关联。

A note on rational homological stability for automorphisms of manifolds

By work of Berglund and Madsen, the rings of rational characteristic classes of fibrations and smooth block bundles with fibre $D^{2n}\sharp(S^n\times S^n)^{\sharp g}$, relative to the boundary, are for $2n\ge 6$ independent of $g$ in degrees $*\le (g-6)/2$. In this note, we explain how this range can be improved to $*\le g-2$ using cohomological vanishing results due to Borel and classical invariant theory. This implies that the analogous ring for smooth bundles is independent of $g$ in the same range, provided the degree is small compared to the dimension.

Obstruction theory on 8-manifolds

This note gives a uniform, self-contained, and fairly direct approach to a variety of obstruction-theoretic problems on 8-manifolds. We give necessary and sufficient cohomological critera for the existence of almost complex and almost quaternionic structures on the tangent bundle and for the reduction of the structure group to U(3) by the homomorphism U(3) --> O(8) given by the Lie algebra representation of PU(3).

Note on Totally Skew Embeddings of Quasitoric Manifolds over Cube

Totally skew embeddings are introduced by Ghomi and Tabachnikov. They are naturally related to classical problems in topology, such as the generalized vector field problem and the immersion problem for real projective spaces. In recent paper Topological obstructions to totally skew embeddings, {totaly skew} embeddings are studied by using the Stiefel-Whitney classes In the same paper it is conjectured that for every $n$-dimensional, compact smooth manifold $M^n$ $(n>1)$, $$N(M^n)\leq 4n-2α(n)+1,$$ where $N(M^n)$ is defined as the smallest dimension $N$ such that there exists a {\em totally ske…