and $j$ with $i+j \leq d$ then $\sum_{k=0}^d h_k$ is even.

-   Suppose $\partial h_n = 0$ for some $1 \leq n < \frac d 2$. If $(h_\ell + h_{\ell-1} + \cdots + h_{\ell-n+1}) -\partial h_\ell < n$ for some $ n \leq \ell \leq d-n$ then $\sum_{k=0}^d h_k$ is even.

-   Suppose $\partial h_i=0$ and $h_j=0$ for some integers $i$ and $j$ with $0< i < \frac d 2$ and $d-i  < j <d$. If $\partial h_\ell \leq \ell$ for some $\ell \leq d-j$ then $\sum_{k=0}^d h_k$ is even.

Note that conditions (5), (6) and (7) are unnecessary when $d$ is odd. We also note that some special cases of the above theorem are due to Kolins [@Ko]. For necessity, (1), (2), (3) and some special cases of (4), (5) and (7) (the condition (4) for $n=1$, the condition (5) for $i=1,2$ and $j=1$, and the condition (7) for $\ell=1$) were proved in [@Ko Sections 3, 4 and 5]. Also, when $\partial {\mathbf{h}}$ is positive or $\sum_{k=0}^d h_k$ is even, the sufficiency of the theorem was proved in [@Ko Theorem 16]. In particular, for even dimensional simplicial cell balls, the sufficiency of the theorem is due to Kolins.

A 