From d85eed46d4c3d03bdbcd48a068cee1d9a3b405bc Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Mon, 10 Mar 2025 15:46:05 -0300
Subject: Changes and comments addressing report #2

 - Indicated which of the requested changes were made.
 - Modified the manuscript to indicated that the function H must be
   Morse, i.e., not have any degenerate stationary points.
---
 topology.tex | 5 ++---
 1 file changed, 2 insertions(+), 3 deletions(-)

(limited to 'topology.tex')

diff --git a/topology.tex b/topology.tex
index d3d90d4..56af0c6 100644
--- a/topology.tex
+++ b/topology.tex
@@ -247,9 +247,8 @@ points, one must take pains to eliminate the sign of the determinant
 \cite{Fyodorov_2004_Complexity}. Here it is correct to preserve it.
 
 We need to choose a function $H$ for our calculation. Because $\chi$ is
-a topological invariant, any choice will work so long as it does not share some
-symmetry with the underlying manifold, i.e., that $H$ satisfies the Smale condition. Because our manifold of random
-constraints has no symmetries, we can take a simple height function $H(\mathbf
+a topological invariant, any choice will work so long as it does not have degenerate stationary points on the manifold, i.e., that it is a Morse function, and does not share some
+symmetry with the underlying manifold, i.e., that it satisfies the Smale condition. Because our manifold is random and has no symmetries, we can take a simple height function $H(\mathbf
 x)=\mathbf x_0\cdot\mathbf x$ for some $\mathbf x_0\in\mathbb R^N$ with
 $\|\mathbf x_0\|^2=N$. We call $H$ a height function because when $\mathbf x_0$ is
 interpreted as the polar axis of a spherical coordinate system, $H$ gives the height on the sphere relative to the equator.
-- 
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