### Questions about Terminology.

1. I have a surface in three-space, and am looking for a term to describe if it's possible to determine which side a viewer is on. I was thinking about the 33336 tiling, how viewed from one side it has a left twist, and the other side has a right twist, so I thought chiral vs. achiral would work, but if I have a surface that is all black on one side, and all white on the other, compared to one that is white on both sides, that terminology is wrong. So is there a term to describe if a surface looks the same from both sides, or has a different appearance.

2. In a tiling, if A and B both belong to the same Transitivity class, is there an equivalence term to say that. When I've been writing I've been saying that A and B are Transitive, but I'm not sure it that's valid, or correct, so I want to make sure before I use it any more.

Thanks

Nekura

### Relationship between L-polynomial and the Stiefel-Whitney class

Let * M *be a closed manifold of odd dimension * n=2k+1*. Let * L(p)* denote the Hirzebruch L-polynomial in the Pontrjagin classes of *M*. Denote by * L _{2k}(p) *the component of

*L(p)*in

*H*and by

^{2k}(M,**Z**)*w*the

_{2k}Î H^{2k}(M,**Z**_{2})*2k*-th Stiefel-Whitney class of

*M.*I have strong reasons to believe that the reduction of

*L*modulo 2 is equal to to

_{2k}(p)*w*. Is it really so? And, if it is, how to prove it?

_{2k. }**Remark**: if *k * is odd, so that *2k * is not divisible by 4, then, of course, *L _{2k}(p) = 0 * by dimensional reason. In this case, Massey [Amer. Jornal of Math.

**82**, 92-102] showed that

*w*Thus, if

_{2k}=0.*k*is odd, the answer to my question is positive.

### (no subject)

Do you probably know how to:

1. Find furthest-neighbor or furthest-site diagram out of Voronoi Diagram? (Fast and efficient way for each site to know which one is farthest site from it)?

2. Let P1...Pk be a collection of pairwise disjoint simple polygons with a total of n edges, all enclosed in a given square. Find in O(n*logn) time a largest disk that can be inscribed in this square so that it is disjoint from all the interiors of the polygons Pi ------------ how do I solve this? Provided I know how to find largest empty circle in Voronoi diagram, how can I take care that it does not intersect the EDGES of polygons (staying disjoint with their interior)?

Thanks a lot!!!

### Question on Haar Measure/Integration

If G is a locally compact group and K subse G is compact, and

lim n --> inf int_K f_n(x) dx = 0, then does lim n --> inf int_{K^-1} f_n(x^-1) dx = 0? (f_n subset L^p(G) for some p >= 1) I actually ultimately need this question asked for integration being Bochner integration, but is this true even for Lebesgue integrals? I don't think it is, since the modular function will screw things up even if it's a 'mild' function.

### Non-Metrical Geometry

It is mentioned in A. N. Whitehead's "Process and Reality" (pages 490s), in discussion of infinitesimals (Weierstrass), presentational immediacy, strain locus, projection (opposite of injection, I think), God and mentality. Intuitively, it brings me back to the 470s (pages); "Its is to be remembered that two points determine a complete straight line, that three non-colinear points determine a complete plane, and that four non-coplanar points determine a complete three dimensional flat locus." ("Process and Reality", page 472)

### (no subject)

How are you using the special math text that you used in your first post to this community? Can you give me a link to how to use it?

Would graph theory be included in your interests?

Thank you very much!

### naive defitinion of normalisation of complex manifolds

are they correct at all ? I am willing to assume everything lies a very nice ambient space, say

holomorphically convex manifold.

\def\hatZ{{\hat Z}}

Let $\nnn:\hatZ \ra Z$ be the normalisation of a variety $Z$, and

let $\Zz k(\C) = \{z\in Z(\C): \#\nnn\inv(z)\geq k\}$ be a closed

subvariety of points which have $k$ preimages or more.

\begin{fact}\label{fact:zknrm} $\Zz k$ is a closed subvariety of $Z$, for all $k$;

$\Zz {\deg \nnn + 1}$ is empty, $\Zz 1 = Z$. \end{fact} \bp

Reference!!!! \ep

\begin{fact}\label{fact:214} Around each point $z\in Z(\C)$ there exists a

sufficiently small neighbourhood $z\in V$ open in complex topology

such that

$$V\cap \Zz k= \bigcup\limits_{i_1<..<i_k} V_{i_1}\cap ...\cap V_{i_k},$$ where $$Z\cap V=V_1\cup ...\cup V_n$$ is the irreducible decomposition of $Z\cap V$.\end{fact} \bp not [Grauert, ???] [Abhyankar, Local Analytic Geometry, p.402, Cl 7] \ep

### Flat connection vs. representation of the fundamental group

_{1}(M) →

**C**

^{n}of its fundamental group gives rise to a flat vector bundle E

_{α}= M×

_{α}

**C**whose monodromy is isomorphic to α

**.**Given a continuous family of connections α(t) the bundles E

_{α(t) }are isomorphic (as vector bundles without a connection). Thus it is not difficult to see that there exists one bundle E → M and a continuous family of connections (t) such that the monodromy of (t) is isomorphic to α(t). Moreover if α(t) is differentiable then (t) can be chosen to be differentiable. I have 2 questions: </p>

1. Is there any book or paper where these simple facts are proven?

2. How far this relationship between families of representations and families of connections can be pushed? For example, if α(t) is analytic can one find an analytic family (t)?

### Reference for variational formula for the determinant of an elliptic operator

δ Det(A) := δ (-∂

_{s}Tr ( A

^{-s}) |

_{s=0})

_{ }= ∂

_{s}( s Tr[(δ A ) A

^{-s-1}])|

_{s=0}

cf., for example, Burghelea, Friedlander, Kappeler,

*Meyer-Vietoris type formula for determinants of elliptic differential operators.*

**J. Funct. Anal**. 107 (1992), 34--65, or Kontsevich, Vishik ,

*Geometry of determinants of elliptic operators*. (of course, to define A

^{-s}one uses a spectral cut, which I did not write explicitly to simplify the notation).

**Is it proven anywhere in the literature?**(I do know how to prove it, but I need a reference)