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The homology effect of surgery away from the middle dimensions

Statement

Assume ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $M$ be a closed smooth $m$ -manifold, $0 \leq p \leq m - 1$ , $q = m - p$ , let $φ$ be a framed embedded surgery sphere with trace $W_{φ}$ and surgered manifold $M_{φ}$ , and let $A$ be an abelian group. Then:

(i) $H_{i} (W_{φ}, M; A) = 0$ for $i \neq p + 1$ and $H_{p + 1} (W_{φ}, M; A) ≅ A$ , identified with the coefficient classes of the core disk;

(ii) $H_{i} (W_{φ}, M_{φ}; A) = 0$ for $i \neq q$ and $H_{q} (W_{φ}, M_{φ}; A) ≅ A$ , identified with the coefficient classes of the cocore disk;

(iii) consequently $H_{i} (M; A) \to H_{i} (W_{φ}; A)$ is an isomorphism for $i \notin {p, p + 1}$ , $H_{i} (M_{φ}; A) \to H_{i} (W_{φ}; A)$ is an isomorphism for $i \notin {q - 1, q}$ , and therefore $H_{i} (M; A) ≅ H_{i} (M_{φ}; A)$ for every $i \notin {p, p + 1, q - 1, q}$ ;

(iv) define $s_{A} : A \to H_{p} (M; A)$ by $s_{A} (a) = (φ_{0})_{*} [S^{p}]_{a}$ , where $[S^{p}]_{a}$ is the sphere fundamental cycle with coefficient $a$ ; for $p = 0$ this means the reduced cycle $a [x_{+}] - a [x_{-}]$ . Under the core identification in (i), the connecting map is $\pm s_{A}$ , and $H_{p} (W_{φ}; A) ≅ H_{p} (M; A) / im s_{A} .$ Dually define $t_{A} (a) = β_{*} [S^{q - 1}]_{a}$ for the belt sphere $β$ , using the reduced cycle when $q = 1$ . The dual connecting map is $\pm t_{A}$ , and $H_{q - 1} (W_{φ}; A) ≅ H_{q - 1} (M_{φ}; A) / im t_{A}$ . For $A = Z$ , the image is the cyclic subgroup generated by the sphere class. An arbitrary abelian coefficient group need not have a generator.

The only degrees in which the homology can change are $p, p + 1, q - 1, q$ , exactly as the two relative computations allow.

Facts & Assumptions

Given: the closed smooth $m$ -manifold $M$ , integers $0 \leq p \leq m - 1$ and $q = m - p$ , a framed embedded surgery sphere $φ$ , the trace $W_{φ}$ , the surgered manifold $M_{φ}$ , and an abelian group $A$ .

[F3]

The upper boundary of the surgery trace is the surgered manifold: the outgoing face is identified with $M_{φ}$ , and there are homotopy equivalences of pairs relative to the indicated faces, $(W_{φ}, M) ≃ (M \cup_{φ_{0}} D^{p + 1}, M)$ and $(W_{φ}, M_{φ}) ≃ (M_{φ} \cup_{β} D^{q}, M_{φ})$ , where $β$ is the belt-sphere embedding. The characteristic disks include collar paths to the respective faces.

[F4]

Relative homology of the standard handle pair: for any abelian group $G$ and integers $0 \leq k \leq n$ , the standard handle pair satisfies $H_{i} (D^{k} \times D^{n - k}, S^{k - 1} \times D^{n - k}; G) ≅ G$ for $i = k$ and $0$ otherwise.

[F5]

Excision for singular homology: if $\overline{Z} \subseteq int (Y)$ in a pair $(X, Y)$ , removing $Z$ induces relative homology isomorphisms. One first enlarges the base across a short attaching collar by deformation retraction, then excises its portion outside the cell collar; this satisfies the interior condition and leaves a disk relative to a boundary collar, which retracts to the disk-boundary pair.

[F6]

Long exact sequence of a pair: for a pair $(Z, Y)$ there is the exact sequence $\dots \to H_{i + 1} (Z, Y; A) \to H_{i} (Y; A) \to H_{i} (Z; A) \to H_{i} (Z, Y; A) \to \dots .$

[F7]

Naturality of the homology connecting morphism: the connecting homomorphism of the long exact sequence of a pair is natural with respect to maps of pairs.

[F8]

Relative fundamental class and boundary orientation: the relative fundamental class of a compact oriented manifold with boundary restricts on the boundary to the fundamental class of the boundary in the outward-normal-first convention; in particular the connecting homomorphism of the pair $(D^{p + 1}, S^{p})$ sends the relative fundamental class to $\pm [S^{p}]$ (Homology of spheres).

Proof

Given: the objects and hypotheses of the statement.

1.1F3

By [F3] the incoming pair has the cell model $(M \cup_{φ_{0}} D^{p + 1}, M)$ and the outgoing pair has $(M_{φ} \cup_{β} D^{q}, M_{φ})$ . Use the cylinder paths in that model when referring to a characteristic disk with boundary in a face. These are homotopy equivalences of pairs, not a handle attachment directly to the boundaryless manifold $M$ .

2.1F3F4F5F8step 1.1algebra

In the incoming cell model, enlarge $M$ to $U = M \cup {u : ∣ u ∣ > 1 / 2}$ in the attached disk. The radial boundary collar retracts $U$ onto $M$ , so replacing $M$ by $U$ does not change relative homology. Excise $Z = M \cup {u : ∣ u ∣ > 3 / 4}$ ; its closure is contained in the interior of $U$ , as required by [F5]. The remaining pair is a closed disk of radius $3 / 4$ relative to its collar $1 / 2 < ∣ u ∣ \leq 3 / 4$ , whose collar retracts to its boundary. Thus the relative groups are those of $(D^{p + 1}, S^{p})$ , or the standard handle pair by contraction of its second factor. By [F4] they are $A$ in degree $p + 1$ and zero otherwise. The identification sends each $a \in A$ to the disk's oriented relative cycle with coefficient $a$ , not to a claimed generator of $A$ . This proves (i).

3.1F3F4F5F8step 1.1step 2.1

Apply exactly the collar enlargement and excision of step 2.1 to the outgoing $q$ -cell model of step 1.1. The remaining pair is $(D^{q}, S^{q - 1})$ , equivalently the handle pair with the disk factors exchanged. By [F4] its homology is $A$ in degree $q$ and zero otherwise; the identification uses the oriented cocore relative cycle with each coefficient $a \in A$ . This includes $q = 1$ , whose boundary is a two-point sphere. This proves (ii).

4.1F6step 2.1step 3.1

In the exact sequence of [F6] for the pair $(W_{φ}, M)$ , the relative groups vanish outside degree $p + 1$ by (i): hence $H_{i} (M; A) \to H_{i} (W_{φ}; A)$ is an isomorphism for $i \notin {p, p + 1}$ . Similarly, using (ii), $H_{i} (M_{φ}; A) \to H_{i} (W_{φ}; A)$ is an isomorphism for $i \notin {q - 1, q}$ . Comparing the two through $H_{i} (W_{φ}; A)$ gives $H_{i} (M; A) ≅ H_{i} (M_{φ}; A)$ outside ${p, p + 1} \cup {q - 1, q}$ . This proves (iii).

5.1F6F7F8step 2.1step 3.1algebra∎

The incoming characteristic disk, including its cylinder collar, is a map of pairs $(D^{p + 1}, S^{p}) \to (W_{φ}, M)$ with boundary $φ_{0}$ . For each $a \in A$ , the boundary of its relative fundamental cycle is $[S^{p}]_{a}$ by [F8]; at $p = 0$ it is the difference of the two endpoint cycles. Naturality [F7] gives $\partial (a) = \pm s_{A} (a)$ . Since $H_{p} (W_{φ}, M; A) = 0$ , [F6] makes $H_{p} (M; A) \to H_{p} (W_{φ}; A)$ surjective with kernel $im s_{A}$ , giving the stated quotient. Applying the same calculation to the outgoing characteristic disk gives $\partial (a) = \pm t_{A} (a)$ and the dual quotient. For integral coefficients the image is generated by the image of $1 \in Z$ ; no cyclicity is asserted for general $A$ . This proves (iv).

Depends on

Used by

Middle-dimensional surgery can change an intersection form Counterexample
One-surgery on a three-manifold as framed knot surgery Example
Middle-dimensional surgery has an intersection-form obstruction Remark

Dependency tree · two levels

51 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

Wolfgang Lück, A Basic Introduction to Surgery Theory (lecture notes, Münster, 27 October 2004) (standard reference, not scraped)
Andrew Ranicki, Algebraic and Geometric Surgery (Oxford Mathematical Monographs, 2002; electronic copy) (standard reference, not scraped)
C. T. C. Wall, Differential Topology (Cambridge Studies in Advanced Mathematics 156, Cambridge University Press 2016) (standard reference, not scraped)