The author studies the problem how a map $L:M\to ℝ$ on an $n$-dimensional manifold $M$ can induce canonically a map $A_M\left(L\right):T^{*}{T}^{\left(r\right)}M\to ℝ$ for $r$ a fixed natural number. He proves the following result: “Let $A:T^{(0,0)}\to T^{(0,0)}\left(T^{*}{T}^{\left(r\right)}\right)$ be a natural operator for $n$-manifolds. If $n\ge 3$ then there exists a uniquely determined smooth map $H:{ℝ}^{S\left(r\right)}\times ℝ\to ℝ$ such that $A=A^{\left(H\right)}$.”The conclusion is that all natural functions on $T^{*}{T}^{\left(r\right)}$ for $n$-manifolds $(n\ge 3)$ are of the form $\{H\circ ({\lambda }_M^{\langle 0,1\rangle },\cdots ,{\lambda }_M^{\langle r,0\rangle })\}$, where $H\in C^{\infty }\left({ℝ}^r\right)$ is a function of $r$ variables.

The author proves that for a manifold $M$ of dimension greater than 2 the sets of all natural operators $TM\to (T_k^{r*}M,T_{\ell }^{q*}M)$ and $TM\to T{T}_k^{r*}M$, respectively, are free finitely generated $C^{\infty }\left({\left({ℝ}^k\right)}^r\right)$-modules. The space $T_k^{r*}M=J^r{(M,{ℝ}^k)}_0$, this is, jets with target 0 of maps from $M$ to ${ℝ}^k$, is called the space of all $(k,r)$-covelocities on $M$. Examples of such operators are shown and the bases of the modules are explicitly constructed. The definitions and methods are those of the book of I. Kolář, P. W. Michor and J. Slovák [Natural operations in differential geometry,...

In this nice paper the author proves that all natural symplectic forms on the tangent bundle of a pseudo-Riemannian manifold are pull-backs of the canonical symplectic form on the cotangent bundle with respect to some diffeomorphisms which are naturally induced by the metric.

[For the entire collection see Zbl 0699.00032.] Natural transformations of the Weil functor $T^A$ of A-velocities [I. Kolař, Commentat. Math. Univ. Carol. 27, 723-729 (1986; Zbl 0603.58001)] into an arbitrary bundle functor F are characterized. In the case where F is a linear bundle functor, the author deduces that the dimension of the vector space of all natural transformations of $T^A$ into F is finite and is less than or equal to $dim\left(F_0{ℛ}^k\right)$. The spaces of all natural transformations of Weil functors into linear...

For each graph G, we define a chain complex of graded modules over the ring of polynomials whose graded Euler characteristic is equal to the chromatic polynomial of G. Furthermore, we define a chain complex of doubly-graded modules whose (doubly) graded Euler characteristic is equal to the dichromatic polynomial of G. Both constructions use Koszul complexes, and are similar to the new Khovanov-Rozansky categorifications of the HOMFLYPT polynomial. We also give a simplified definition of this triply-graded...

The Mumford conjecture predicts the ring of rational characteristic classes for surface bundles with oriented connected fibers of large genus. The first proof in [11] relied on a number of well known but difficult theorems in differential topology. Most of these difficult ingredients have been eliminated in the years since then. This can be seen particularly in [7] which has a second proof of the Mumford conjecture, and in the work of Galatius [5] which is concerned mainly with a “graph” analogue...