= Morse inequalities
{c}
{title2=$\sum_{i=0}^j(-1)^{j-i}(c_i-b_i)\ge0$}
For a closed <manifold> and a <Morse function>, the displayed strong inequalities hold for every $j$, with equality at the top dimension. Indeed the <Morse chain complex> rank formula makes the alternating sum equal to $r_{j+1}\ge0$. Adding the inequalities at $j$ and $j-1$ gives $c_j\ge b_j$. The weak inequalities, even together with equality of <Euler characteristics>, are less restrictive: $b=(1,0,0,1)$ and $c=(2,0,0,2)$ satisfy those numerical conditions but violate the strong inequality at $j=1$.
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