Some properties of a higher-order coboundary operator.

We study properties of a higher-order coboundary operator, $\delta\sp{(a)}$, which increases dimension by a rather than just 1. Take the simplicial complex consisting of a single oriented $(n - 1)$-simplex, $\ V = \lbrack v\sb1,\...,v \sb{n}\rbrack.$ Let $C\sb{k}(\ V)$ be the usual group of $(k - 1)$-chains over $\doubz/(p).$ We define an operator, $\delta\sp{(a)}$, which maps $C\sb{k}(\ V)$ into $C\sb{k+a}(\ V)$ in such a way that $\delta\sp{(1)}$ is the usual coboundary operator. It turns out that $\delta\sp{(a)} \circ \delta\sp{(b)} = 0$ if both a and b are odd, or if there is a "carry" in one of the places when the base p expansion of $\lfloor{a\over2}\rfloor$ is added to the base p expansion of $\lfloor{b\over2}\rfloor.$. In order to calculate the dimension of the image of $\delta\sp{(a)}$, we examine a class of matrices $W\sb{r,s}(n).$ These matrices are a signed version of the incidence matrices discussed in "A Diagonal Form for the Incidence Matrices of t-subsets vs. k-subsets," by R. M. Wilson. We produce a diagonal form for the matrices $W\sb{r,s}(n)$ over $\doubz$ and use this form to calculate the rank of $W\sb{r,s}(n)$ over $\doubz/(p)$ for any prime p. With this result in hand, we are able to generalize four theorems found in "Cohomological Aspects of Hypergraphs," by F. R. K. Chung and R. L. Graham. First, we examine the kernel of $\delta\sp{(a)}$ in the case where a is even and the base p expansion of $a\over2$ has one non-zero term. In this instance we can find a b such that ker$\delta\sp{(a)}$ = im$\delta\sp{(b)}$. Next, we study the kernel of $\delta\sp{(a)}$ for general a and show that we can express any element in the kernel of $\delta\sp{(a)}$ as a sum of coboundaries. Following this, we examine what happens when a sum of coboundaries is equal to zero. Finally, we investigate the cohomology groups $$H\sbsp{k}{b,a} = {{\rm ker}\delta\sp{(a)}\over{\rm im}\delta\sp{(b)}}$$and give a formula for the dimension of $H\sbsp{k}{b,a}$ over $\doubz/(p).$ Under certain conditions, this formula has a nice expression as an alternating sum of binomial coefficients. We conclude the paper with simplified proofs for the theorems in Chung and Graham's paper and a discussion of some open problems related to our work.PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/105880/1/9226882.pdfDescription of 9226882.pdf : Restricted to UM users only