Zero-dimensional bordism groups

(The zeroth homology of a compact one-manifold, and the evaluation of the pushforward.) Let $W$ be a compact smooth one-manifold and let $R$ be $\mathbb{Z}$ or ${\mathbb{F}}_2$. By [F2] the group $H_0(W;\mathbb{Z})$ is free on the path components with generators $e_C$, the classes of single points; for $R={\mathbb{F}}_2$ the same description holds: the augmentation $ϵ$ that sums the coefficients of a $0$-chain over each path component is surjective, and its kernel is exactly the image of ${\mathrm{\partial }}_1$, because every singular $1$-simplex has its two endpoints in a single path component (so its boundary has componentwise coefficient sum $0$), while a finite $0$-chain with componentwise sums $0$ is a finite sum of terms $[x]-[b_C]$ with $x,b_C$ in the same component, each of which is the boundary of a singular $1$-simplex along a path from $b_C$ to $x$; the finitely many paths and basepoints are selected by finite choice [F6]. Consequently $H_0(W;R)\cong {⨁}_{C}R{e}_C$, and for a finite set $M\subseteq \mathrm{\partial }W$ with coefficients ${ϵ}_x\in R$, the pushforward is $i_{\ast }[M]={\sum }_C\left({\sum }_{x\in M\cap C}{ϵ}_x\right)e_C$.

(Invariance of the signed count.) Let $W$ be an oriented bordism from $(M_0,o_0)$ to $(M_1,o_1)$ between closed oriented $0$-manifolds, and let $f(M):={\sum }_{x\in M}{ϵ}_x\in \mathbb{Z}$ be the signed count. The induced boundary orientation on $\mathrm{\partial }W$ is $(-o_0)\bigsqcup o_1$, so the boundary fundamental class is the sum of the classes of the points of $M_0$ with signs $-{ϵ}_x$ and of $M_1$ with signs ${ϵ}_x$. By [F1] with $R=\mathbb{Z}$ and step 1.1, $i_{\ast }[\mathrm{\partial }W]=0$ gives for every path component $C$ of $W$ that the signed number of boundary points lying in $C$ is $0$, and summing over $C$ gives $-f(M_0)+f(M_1)=0$. Hence $f$ is unchanged by oriented cobordism and descends to a map $f:{\mathrm{\Omega }}_0^{SO}\to \mathbb{Z}$.

(Invariance of the parity.) Let $W$ be a bordism from $M_0$ to $M_1$ between closed (unoriented) $0$-manifolds. Use the canonical mod-two orientation of $W$ and of its boundary ([F1] with $R={\mathbb{F}}_2$). By [F1] and step 1.1 over ${\mathbb{F}}_2$, every path component of $W$ contains an even number of boundary points, so the cardinality of $\mathrm{\partial }W=M_0\bigsqcup M_1$ is even; hence $\mathrm{\mid }{M}_0\mathrm{\mid }\equiv \mathrm{\mid }{M}_1\mathrm{\mid }(\mathrm{m}\mathrm{o}\mathrm{d}2)$, and the parity descends to a map ${\mathrm{\Omega }}_0^O\to \mathbb{Z}\mathrm{/}2\mathbb{Z}$.

(Completeness: vanishing invariant implies null-cobordant.) (i) Let $M$ be a finite set of points of even cardinality. By [F6] pair the points of $M$; for each pair $\{a,b\}$ use a copy $W_{\{a,b\}}$ of $[0,1]$, affinely diffeomorphic to $B^1$, identify $a$ with $0$ and $b$ with $1$, and give its whole boundary the collar $\theta (s,a)=s\mathrm{/}2$, $\theta (s,b)=1-s\mathrm{/}2$ for $0\le s<1$; the images $[0,1\mathrm{/}2)$ and $(1\mathrm{/}2,1]$ are disjoint open neighbourhoods of the endpoints; then $W_{\{a,b\}}$ is a compact smooth one-manifold whose whole boundary is $\{a,b\}$, so the disjoint union over the pairs is a compact one-manifold with boundary $M$. Hence $M$ is null-cobordant. (ii) Let $(M,o)$ be a finite oriented point set with signed count $0$. Then the number of positively oriented points equals the number of negatively oriented points; by [F6] pair each positive point $p$ with a negative point $n$. On the interval $[0,1]$ with its standard orientation, the collars $\theta (s,p)=s\mathrm{/}2$ and $\theta (s,n)=1-s\mathrm{/}2$ for $0\le s<1$, with disjoint images $[0,1\mathrm{/}2)$ and $(1\mathrm{/}2,1]$, exhibit the whole boundary $\{p,n\}$ as the incoming part and induce on it the orientation $\{1\}-\{0\}=\{n\}-\{p\}$, that is, the negative of the orientation with $p$ positive and $n$ negative ([F5]). Hence each pair, and therefore all of $M$, is null-cobordant. In both cases the construction uses finitely many intervals and no infinite selection.

(The isomorphisms and the bounding criterion.) The signed count is additive under disjoint union and $f({\mathrm{p}\mathrm{t}}_{+})=1$, so it is a surjective homomorphism ${\mathrm{\Omega }}_0^{SO}\to \mathbb{Z}$; by step 2.1 it is well defined and by step 2.3 its kernel is zero, so it is injective and hence an isomorphism of abelian groups with generator the positively oriented point. The parity is additive under disjoint union and equals $1$ on a one-point manifold, so it is a surjective homomorphism ${\mathrm{\Omega }}_0^O\to \mathbb{Z}\mathrm{/}2\mathbb{Z}$; it is well defined by step 2.2, and step 2.3(i) makes its kernel zero, so it is an isomorphism with generator the class of a point. Finally, a compact zero-manifold bounds a compact one-manifold if and only if it is null-cobordant, which by [F3] and [F4] happens exactly when its invariant vanishes: the signed count is $0$ in the oriented theory, and the cardinality is even in the unoriented theory. The proof used only the boundary pushforward, the elementary description of $H_0$, finitely many explicit intervals and finite choice, so it is choice-free and does not use the classification of compact one-manifolds.