Continuation across a birth--death pair adds an acyclic summand

Assume the Axiom of Choice (The Axiom of Choice). Let $M$ be a closed manifold and fix Morse--Smale pairs $(f^{-},g^{-})$ and $(f^{+},g^{+})$ related by a birth--death pair satisfying the isolated-trajectory hypothesis below: while $f^{-}$ has no critical points in an open set $U\subset M$, the function $f^{+}$ has in $U$ exactly two critical points $b$ of index $k$ and $c$ of index $k+1$; among the index-one trajectories of $f^{+}$ involving the new critical points, the only one is a single trajectory from $c$ to $b$, and all other critical points and index-one trajectories coincide with those of $f^{-}$; in the integral case choose orientations (positive rays in the critical orientation lines) at all critical points, compatible on the unchanged unstable manifolds (Morse functions and excellent Morse functions, Nondegenerate critical points, nullity, index, and coindex, Morse--Smale pairs).

Then the Morse complex of $f^{+}$ splits as a direct sum of the complex of $f^{-}$ and the two-term complex $\mathrm{\Lambda }\cdot c\stackrel{\pm 1}{\to }\mathrm{\Lambda }\cdot b,\mathrm{deg}c=k+1,\mathrm{deg}b=k,$ with unit differential ${\mathrm{\partial }}^{+}c=\pm b$ on the new generators and ${\mathrm{\partial }}^{+}b=0$ (The mod-two Morse differential, The signed Morse differential over the integers); this summand is acyclic, so adding it does not change homology. Any regular continuation datum from $(f^{-},g^{-})$ to $(f^{+},g^{+})$ adapted to this local model (The continuation chain map) is a chain map that is an isomorphism on homology by Reverse continuation is an inverse on Morse homology, and consequently the birth--death pair adds $t^k+t^{k+1}=t^k(1+t)$ to the Morse polynomial $c(t)$ of Morse homology recovers the Morse inequalities, equivalently it adds $t^k$ to the correction polynomial $Q(t)$ there (Morse homology of a Morse--Smale pair, Canonical Morse homology of a closed manifold).

The isolated-trajectory hypothesis above is a sufficient condition for a direct sum in the critical-point basis; it is not the geometry of a birth on the connected circle. A genuine circle birth changes one maximum and one minimum to two of each. Enumerate the latter in circular order as $p,q_1,c,q_2$, with $p,c$ maxima. Orient the two unstable intervals by increasing angle. Their outgoing arcs have opposite comparison signs, giving $\mathrm{\partial }p=q_1-q_2$ and $\mathrm{\partial }c=q_2-q_1$. In the integral bases $P=p+c$, $C=c$, $Q=q_1$, $B=q_2-q_1$, one has $\mathrm{\partial }P=0$ and $\mathrm{\partial }C=B$. Both changes of basis are unimodular, so they work over $\mathbb{Z}$ and $\mathbb{Z}\mathrm{/}2$. The homotopy $B\mapsto C$ contracts the new pair; the remaining complex on $P,Q$ is the zero-differential complex of the minimal circle function. Thus the same homology and polynomial conclusions hold, with $c^{-}=1+t$, $c^{+}=2+2t$, and $Q^{+}=Q^{-}+1$, although the direct sum appears only after this change of basis. A regular continuation map realizes the homology isomorphism by the general continuation theorem.