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Register a new userNew proofs of theorems of Kathryn Mann
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Authors
Shigenori Matsumoto
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We give a shorter proof of the following theorem of Kathryn Mann cite{M}: the identity component of the group of the compactly supported $C^r$ diffeomorphisms of $R^n$ cannot admit a nontrivial $C^p$-action on $S^1$, provided $ngeq2$, $r eq n+1$ and $pgeq2$. We also give a new proof of another theorem of Mann: any nontrivial endomorphism of the group of the orientation preserving $C^r$ diffeomorphisms of the circle is the conjugation by a $C^r$ diffeomorphism, if $rgeq3$.
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Let $Pi_g$ be the surface group of genus $g$ ($ggeq2$), and denote by $RR_{Pi_g}$ the space of the homomorphisms from $Pi_g$ into the group of the orientation preserving homeomorphisms of $S^1$. Let $2g-2=kl$ for some positive integers $k$ and $l$. Then the subset of $RR_{Pi_g}$ formed by those $varphi$ which are semiconjugate to $k$-fold lifts of some homomorphisms and which have Euler number $eu(varphi)=l$ is shown to be clopen. This leads to a new proof of the main result of Kathryn Mann cite{Mann} from a completely different approach.
The chromatic polynomial and its generalization, the chromatic symmetric function, are two important graph invariants. Celebrated theorems of Birkhoff, Whitney, and Stanley show how both objects can be expressed in three different ways: as sums over all spanning subgraphs, as sums over spanning subgraphs with no broken circuits, and in terms of acyclic orientations with compatible colorings. We establish all six of these expressions bijectively. In fact, we do this with only two bijections, as the proofs in the symmetric function setting are obtained using the same bijections as in the polynomial case and the bijection for broken circuits is just a restriction of the one for all spanning subgraphs.
We indicate two short proofs of the Goresky-MacPherson topological invariance of intersection homology. One proof is very short but requires the Goresky-MacPherson support and cosupport axioms; the other is slightly longer but does not require these axioms and so is adaptable to more general perversities.
A low-dimensional version of our main result is the following `converse of the Conway-Gordon-Sachs Theorem on intrinsic linking of the graph $K_6$ in 3-space: For any integer $z$ there are 6 points $1,2,3,4,5,6$ in 3-space, of which every two $i,j$ are joint by a polygonal line $ij$, the interior of one polygonal line is disjoint with any other polygonal line, the linking coefficient of any pair disjoint 3-cycles except for ${123,456}$ is zero, and for the exceptional pair ${123,456}$ is $2z+1$. We prove a higher-dimensional analogue, which is a `converse of a lemma by Segal-Spie.z.
In 1983, Conway and Gordon proved that for every spatial complete graph on six vertices, the sum of the linking numbers over all of the constituent two-component links is odd, and that for every spatial complete graph on seven vertices, the sum of the Arf invariants over all of the Hamiltonian knots is odd. In 2009, the second author gave integral lifts of the Conway-Gordon theorems in terms of the square of the linking number and the second coefficient of the Conway polynomial. In this paper, we generalize the integral Conway-Gordon theorems to complete graphs with arbitrary number of vertices greater than or equal to six. As an application, we show that for every rectilinear spatial complete graph whose number of vertices is greater than or equal to six, the sum of the second coefficients of the Conway polynomials over all of the Hamiltonian knots is determined explicitly in terms of the number of triangle-triangle Hopf links.
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