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Cylinders give reflexivity of cobordism

Statement

For every closed smooth $n$ -manifold $M$ , the product $M \times [0, 1]$ with its product smooth structure and the collars $θ_{0} : [0, 1) \times M \to M \times [0, 1], θ_{0} (s, x) = (x, s),$ $θ_{1} : (- 1, 0] \times M \to M \times [0, 1], θ_{1} (s, x) = (x, 1 + s),$ is a bordism from $M$ to $M$ : take $(\partial (M \times [0, 1]))_{0} = M \times {0}$ and $(\partial (M \times [0, 1]))_{1} = M \times {1}$ (Unoriented smooth cobordism of closed manifolds).

If $(M, o)$ is oriented and $[0, 1]$ carries its standard orientation, then with the product orientation $o \otimes d t$ of $M \times [0, 1]$ the induced boundary orientation (Induced boundary orientation) on $M \times {0}$ is $(- 1)^{n + 1} o$ and that on $M \times {1}$ is $(- 1)^{n} o$ ; consequently $M \times [0, 1]$ oriented by $(- 1)^{n} (o \otimes d t)$ is an oriented bordism from $(M, o)$ to $(M, o)$ (Oriented smooth cobordism). Hence $M$ is cobordant to itself in both theories.

Facts & Assumptions

Given: A closed smooth $n$ -manifold $M$ , the product $W = M \times [0, 1]$ with its product smooth structure, and the maps $θ_{0} (s, x) = (x, s)$ , $θ_{1} (s, x) = (x, 1 + s)$ . In the oriented case, orientations $o$ of $M$ and the standard orientation of $[0, 1]$ .

[F1]

The boundaryless product atlas is given by Products of smooth manifolds have a canonical product smooth structure. For $M \times [0, 1]$ , use the same product charts with interval charts $t$ in the interior, $t$ near $0$ , and $1 - t$ near $1$ ; the last two take values in a half-space and their transitions extend smoothly, so Immersions and embeddings for manifolds with boundary supplies the usual product collars. The interval is compact by For $n \geq 1$ , every Euclidean closed ball and every Euclidean sphere of positive radius is compact after affine rescaling, and $M \times [0, 1]$ is compact by A product of finitely many compact spaces is compact in the product topology.

[F2]

If one factor of a product is closed, the boundary is the product of the other factor's boundary with the closed factor, and the orientation sign is $(- 1)^{\dim}$ of the closed factor: for oriented $M^{m}$ with $\partial N = \emptyset$ one has $\partial (M \times N) = \partial M \times N$ with the product boundary orientation, while if $\partial M = \emptyset$ then $M \times \partial N$ carries $(- 1)^{m}$ times the product orientation (Boundary orientation of a product with at most one boundary factor). On the interval with its standard orientation, $\partial [0, 1] = {1} - {0}$ (Boundary orientation is independent of the outward vector field).

[F3]

The product orientation is the tensor product of the selected rays under the ordered determinant isomorphism, and the induced boundary orientation is the outward-normal-first one (Product orientations, Induced boundary orientation).

[F4]

A bordism from $M_{0}$ to $M_{1}$ is data $(W, θ_{0}, θ_{1})$ with a decomposition of $\partial W$ into open and closed parts and collar embeddings onto open collar neighbourhoods; an oriented bordism additionally carries an orientation of $W$ whose induced boundary orientations are $- o_{0}$ on the incoming and $o_{1}$ on the outgoing face (Unoriented smooth cobordism of closed manifolds, Oriented smooth cobordism).

[F5]

The maps $θ_{0}, θ_{1}$ have injective differentials and are homeomorphisms onto their images, hence are smooth embeddings (Immersions and embeddings for manifolds with boundary, Diffeomorphisms and local diffeomorphisms of manifolds).

Proof

1.1F1F2F4F5

( $W$ with the two collars is a bordism.) The product charts and compactness in [F1] make $W$ a compact smooth manifold with boundary. The boundary of $W = M \times [0, 1]$ is $M \times \partial [0, 1] = M \times {0} ⊔ M \times {1}$ by [F2] applied with the second factor $[0, 1]$ ; the two parts are open and closed in the boundary. The maps $θ_{0}$ and $θ_{1}$ are smooth embeddings by [F5]; their images are $M \times [0, 1)$ and $M \times (0, 1]$ , which are open neighbourhoods of $M \times {0}$ and $M \times {1}$ in $W$ , and $θ_{0} ({0} \times M) = M \times {0}$ , $θ_{1} ({0} \times M) = M \times {1}$ . By [F4] the data $(W, θ_{0}, θ_{1})$ are a bordism from $M$ to $M$ ; the construction uses no choice.

1.2F2F3

(Induced orientations of the two faces.) Suppose $M$ is oriented by $o$ . Apply the second clause of [F2] to $M^{m} = M$ , $m = n$ , and $N = [0, 1]$ with its standard orientation: the boundary $M \times \partial [0, 1]$ carries $(- 1)^{n}$ times the product orientation $o \otimes o_{[0, 1]}$ . Since $\partial [0, 1] = {1} - {0}$ by [F2], the face $M \times {1}$ carries $(- 1)^{n} (o \otimes (+ 1)) = (- 1)^{n} o$ and the face $M \times {0}$ carries $(- 1)^{n} (o \otimes (- 1)) = (- 1)^{n + 1} o$ .

2.1F3F4step 1.2

( $W$ oriented by $(- 1)^{n} (o \otimes d t)$ is an oriented bordism.) Reverse the orientation of $W$ when $n$ is odd; that is, orient $W$ by $ω : = (- 1)^{n} (o \otimes d t)$ . Reversing an orientation reverses every induced boundary orientation, so by step 1.2 the face $M \times {0}$ now carries $- o$ and the face $M \times {1}$ carries $o$ . With $(\partial W)_{0} = M \times {0}$ and $(\partial W)_{1} = M \times {1}$ , this is exactly the oriented bordism condition $- o$ incoming, $o$ outgoing of [F4]: $(W, ω, θ_{0}, θ_{1})$ is an oriented bordism from $(M, o)$ to $(M, o)$ . For even $n$ the orientation $ω$ is the product orientation itself.

3.1F4step 1.1step 2.1∎

(Reflexivity in both theories.) Step 1.1 exhibits the cylinder as a bordism from $M$ to $M$ , so $M$ is cobordant to itself in the unoriented theory; step 2.1 exhibits, for every orientation $o$ of a closed smooth $M$ , an oriented bordism from $(M, o)$ to $(M, o)$ , so $(M, o)$ is oriented cobordant to itself. The empty manifold is covered ( $M = \emptyset$ , $W = \emptyset \times [0, 1] = \emptyset$ ). No choice principle is used anywhere: the data are the explicit collars of the product.

Depends on

Used by

Unoriented and oriented bordism groups Definition
Cartesian product makes bordism a graded ring Theorem
Disjoint union makes bordism classes abelian groups Theorem
Smooth cobordism is an equivalence relation Theorem

Dependency tree · two levels

55 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

Daniel S. Freed, Bordism: Old and New (lecture notes, UT Austin, Fall 2012) (standard reference, not scraped)
C. T. C. Wall, Differential Topology (Cambridge Studies in Advanced Mathematics 156, 2016) (standard reference, not scraped)