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The handle matrix of a simple acyclic presentation

Example

The matrix $A = (\begin{array}{cc} 2 & 1 \\ 1 & 1 \end{array})$ lies in ${GL}_{2} (Z)$ (its determinant is $1$ ). It is carried to the identity by the following elementary operations, each of the kinds realised geometrically by handle slides, renaming and reorientation: interchange the two rows, giving $(\begin{array}{cc} 1 & 1 \\ 2 & 1 \end{array})$ ; subtract twice row $1$ from row $2$ , giving $(\begin{array}{cc} 1 & 1 \\ 0 & - 1 \end{array})$ ; subtract column $1$ from column $2$ , giving $(\begin{array}{cc} 1 & 0 \\ 0 & - 1 \end{array})$ ; multiply row $2$ by $- 1$ , giving $I_{2}$ . Consequently, under ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )), a connected simply connected h-cobordism of dimension $n + 1 \geq 6$ , presented in indices $k, k + 1$ , $2 \leq k \leq n - 2$ , with two middle handles of each of two adjacent indices and intersection matrix $A$ can be slid to a presentation with matrix $I_{2}$ , after which the Whitney trick produces a cancelling pair configuration and the pairs cancel; this is the finite model for the diagonalisation and cancellation steps of the theorem (The middle-handle intersection matrix of an h-cobordism).

Facts & Assumptions

Given: The integer matrix $A = (\begin{array}{cc} 2 & 1 \\ 1 & 1 \end{array})$ and the square matrices $I_{2}$ , together with the displayed sequence of row and column operations.

[F1]

The determinant of a $2 \times 2$ matrix is $a d - b c$ , and a square integer matrix is invertible over $Z$ exactly when its determinant is a unit; the identity matrix satisfies $\det I_{2} = 1$ (For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix, Rectangular matrix multiplication and the identity matrix $I_{n}$ , including zero-sized shapes).

[F2]

An invertible integer matrix is carried to the identity by finitely many row and column additions, interchanges and sign changes, and the general Smith normal form theorem gives the diagonal target, while the handle-matrix reduction lemma proves its finite elementary implementation over $Z$ (Every matrix over a PID has a Smith normal form, Handle slides, renumberings and reorientations reduce a unimodular middle-handle matrix to the identity).

[F3]

Under countable choice and the simply connected h-cobordism dimension/index hypotheses, a presentation with handles only in two adjacent middle indices $k, k + 1$ and invertible middle-handle matrix is unimodular, and each elementary row or column operation is realised on the presentation by a handle slide, a renumbering or a reorientation, all preserving the presented manifold relative to the incoming boundary (Acyclicity makes the simply connected middle-handle matrix unimodular, Handle slides, renumberings and reorientations reduce a unimodular middle-handle matrix to the identity).

Verification

technique · direct

1.1F1given

The determinant of $A$ is $2 \cdot 1 - 1 \cdot 1 = 1$ , so by [F1] the matrix $A$ is invertible over $Z$ with $A \in {GL}_{2} (Z)$ and inverse $(\begin{array}{cc} 1 & - 1 \\ - 1 & 2 \end{array})$ , whose product with $A$ is the identity.

1.2F1F2given

The displayed operations transform $A$ into $I_{2}$ : interchanging the rows gives $(\begin{array}{cc} 1 & 1 \\ 2 & 1 \end{array})$ ; subtracting twice the first row from the second gives $(\begin{array}{cc} 1 & 1 \\ 0 & - 1 \end{array})$ ; subtracting the first column from the second gives $(\begin{array}{cc} 1 & 0 \\ 0 & - 1 \end{array})$ ; multiplying the second row by $- 1$ gives $(\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}) = I_{2}$ . Each step is a row or column addition, an interchange or a sign change, so the sequence is exactly the reduction of [F2].

2.1F2F3step 1.2∎

If a two-index h-cobordism presentation with matrix $A$ satisfies the stated dimension, simple-connectivity and countable-choice hypotheses, [F3] realizes exactly these four operations by slides, relabeling and reorientation, preserving the manifold relative to the incoming face. Its matrix becomes $I_{2}$ . Then The Whitney trick realizes algebraic middle-handle cancellation geometrically and Middle-handle pairs with one geometric intersection cancel give geometric cancellation and the product presentation. This is conditional on such a handle presentation; invertibility of an arbitrary integer matrix alone is not an existence construction of a cobordism.

Depends on

Used by

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Sources

John Milnor, Lectures on the h-Cobordism Theorem (notes by L. Siebenmann and J. Sondow, Princeton University Press 1965; scanned edition with searchable text layer) (standard reference, not scraped)
Andrew Ranicki, Algebraic and Geometric Surgery (Oxford Mathematical Monographs, Oxford University Press 2002; complete electronic copy) (standard reference, not scraped)