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Relative homology of a single handle pair

Statement

Assume ${A C}_{ω}$ and the one-critical-point compact-band hypotheses, with critical index $k$ . For every abelian group $G$ and $i \geq 0$ , $H_{i} (M^{b}, M^{a}; G) ≅ G$ if $i = k$ and zero otherwise. In particular this holds for the additive group of any coefficient ring. No orientation of $M$ is needed.

Facts & Assumptions

[F1]

One critical point handle attachment: Assume ${A C}_{ω}$ . Let $f : M \to R$ be smooth on a boundaryless $n$ -manifold and let $a < b$ be regular values. If $f^{- 1} ([a, b])$ is compact and has exactly one critical point $p$ , nondegenerate of index $k$ , then $M^{b}$ is diffeomorphic to $M^{a}$ with one $k$ -handle attached and corners rounded. No orientation or Morse–Smale hypothesis is required.

[F2]

Relative homology of the standard handle pair: For any abelian group $G$ , integers $0 \leq k \leq n$ , and $i \geq 0$ , the standard handle pair has $H_{i} (D^{k} \times D^{n - k}, S^{k - 1} \times D^{n - k}; G) ≅ G$ if $i = k$ and zero otherwise. Here $D^{0}$ is a point and $S^{- 1} = \emptyset$ .

[F3]

Collar neighborhood theorem: Assume ${A C}_{ω}$ . Every smooth manifold with boundary has a smooth collar.

[F4]

Excision for singular homology: If $Z \subseteq X$ and $\overline{Z} \subseteq {int}_{X} (A)$ , then inclusion $(X - Z, A - Z) \to (X, A)$ induces isomorphisms $H_{n} (X - Z, A - Z; G) ≅ H_{n} (X, A; G)$ for every $n$ .

[F5]

The singular chain homotopy formula: Let $H : X \times I \to Y$ be a homotopy from $f$ to $g$ . Then the prism operator $P_{H}$ of def-prism-operator-for-a-homotopy satisfies $g_{#} - f_{#} = \partial P_{H} + P_{H} \partial$ as homomorphisms $C_{n} (X; G) \to C_{n} (Y; G)$ for every $n \geq 1$ and every abelian group $G$ . In degree $0$ , the same identity reduces to $g_{#, 0} - f_{#, 0} = \partial P_{H} : C_{0} (X; G) \to C_{0} (Y; G) .$

Proof

Given: The objects and hypotheses in the statement.

1.1

Use the smooth handle description and its lower-collar comparison to replace the sublevel pair, up to homotopy of pairs, by $(Y, A)$ , where $Y = A \cup_{h} (D^{k} \times D^{n - k})$ before rounding. Undoing the local rounding is a homeomorphism with the chosen collared model. Collar compression identifies the lower sublevel inclusions as in the theorem. The prism identity on quotient chains makes these pair homotopies induce homology isomorphisms.

F1F3F5

2.1

For $k > 0$ , put $r = ∣ u ∣$ in the handle and choose $0 < r_{0} < R < 1$ . Let $A^{'} = A \cup_{h} {r > r_{0}}$ . This is open in $Y$ and contains the closed set $A$ : near its attaching seam it contains a whole handle collar, while outside the seam the ambient space is locally just $A$ . The radial collar homotopy sends $r$ to $(1 - t) r + t$ and fixes $A$ , so $A^{'}$ retracts to $A$ . In the exact quotient-chain sequence for $A \subset A^{'} \subset Y$ , the relative complex $C_{*} (A^{'}, A; G)$ is acyclic by this retraction. Consequently the quotient map induces a homology isomorphism: lift a cycle in $C_{*} (Y, A^{'}; G)$ ; its boundary in the acyclic kernel can be filled there and subtracted to obtain a cycle lift. If a lifted cycle bounds in the quotient, lift a bounding chain and fill the remaining cycle in the kernel. This proves surjectivity and injectivity, including degree zero.

F5step 1.1algebra

3.1

Excise $Z = A$ , since $\overline{Z} = A \subset {int}_{Y} (A^{'}) = A^{'}$ . The remaining pair is $({r < 1} \times D^{n - k}, {r_{0} < r < 1} \times D^{n - k})$ . Truncate radii at $R$ by $r \mapsto \min (r, R)$ ; its straight radial homotopy preserves the annular subspace. Next the radial map $r \mapsto \min (R, R r / r_{0})$ sends the annular subspace to the sphere of radius $R$ . Its homotopy to the identity preserves that annulus; on the sphere it is the identity. These maps exhibit a homotopy equivalence of pairs with $(D_{R}^{k} \times D^{n - k}, S_{R}^{k - 1} \times D^{n - k})$ . Rescale $R$ to one.

F4F5step 2.1algebra

4.1

The standard-handle calculation now applies. If $k = 0$ , the attachment is disjoint; every singular simplex, being connected, lies in one summand, and the relative chains are exactly those of $D^{n}$ . Their homology is the same standard-pair result with empty attaching subspace. The constructions preserve degree zero and work for $G = 0$ , $k = 1$ , $k = n$ and $n = 0$ without any ambient orientation.

F2step 3.1algebra∎

Depends on

Used by

Relative homology of a handle by excision Example

Dependency tree · two levels

25 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Nicolaescu, An Invitation to Morse Theory (standard reference, not scraped)
Audin–Damian, Morse Theory and Floer Homology (standard reference, not scraped)