Smooth orientation sign is the local integral homology multiplier

A homotopy $H:(X\times I,A\times I)\to (Y,B)$ gives the same induced relative homology map at both ends. Indeed every simplex in $A$ has all its prism simplices mapped into $B$, so the operator of [F3] passes to quotient chains. Its identity $g_{\mathrm{\#}}-f_{\mathrm{\#}}=\mathrm{\partial }P+P\mathrm{\partial }$ then says that the two images of any relative cycle differ by a relative boundary. The same reasoning includes degree zero and unnormalized degenerate simplices. Also the boundary map [F2] commutes with maps of pairs, because $\mathrm{\partial }{f}_{\mathrm{\#}}c=f_{\mathrm{\#}}\mathrm{\partial }c$ on each representative.

Let $J_i$ be reflection of one coordinate of ${\mathbb{R}}^n$. Since ${\mathbb{R}}^n$ is contractible, its pair sequence [F2] identifies $H_n({\mathbb{R}}^n,{\mathbb{R}}^n\setminus 0)$ naturally with ${\tilde{H}}_{n-1}({\mathbb{R}}^n\setminus 0)$. The radial deformation retraction [F6] identifies the latter group with ${\tilde{H}}_{n-1}(S^{n-1})$ and commutes with $J_i$, because radial normalization satisfies $r(J_{i}x)=J_{i}r(x)$. For $n\ge 2$ this action is $-1$ by [F5]. For $n=1$, the two components of the punctured line have classes $[+]$ and $[-]$, and the augmentation kernel is generated by $[+]-[-]$; reflection interchanges them and negates their difference. Hence every $J_i$ acts as $-1$ on the infinite cyclic local group [F1], regardless of which generator was chosen.

Compute the action of the elementary matrices in [F8] on the pair $({\mathbb{R}}^n,{\mathbb{R}}^n\setminus 0)$. A shear $I+c{E}_{ij}$, $i\mathrm{\ne }j$, is homotopic to the identity through $I+tc{E}_{ij}$, whose inverse is $I-tc{E}_{ij}$; thus its action is $+1$ by step 1.1. A positive coordinate scaling $c>0$ is homotopic to the identity by replacing $c$ with $(1-t)+tc>0$, so also acts as $+1$. A negative scaling is a positive scaling followed by $J_i$, hence acts as $-1$ by [F4] and step 2.1. A swap of coordinates $i,j$ is conjugate to a coordinate reflection: use the basis with vectors $e_i+e_j$ and $e_i-e_j$ in that plane and the other standard basis vectors elsewhere; the swap has eigenvalues $+1,-1$ on those two specified vectors. Conjugation does not change its local multiplier, since every invertible change of basis induces an automorphism of the same cyclic group by [F4], and conjugating multiplication by $-1$ leaves it $-1$. These multipliers equal the determinant signs in [F9].

Factor any $L\in {\mathrm{GL}}_n(\mathbb{R})$ into finitely many elementary matrices by [F7]. Functoriality [F4] and step 3.1 multiply their local multipliers, while [F10] multiplies their determinant signs. Thus $L_{\ast }{e}_n=\mathrm{sgn}\det (L)e_n\mathrm{.}$ The empty factorization gives the identity multiplier $+1$. This proof requires no connectedness theorem for the general linear or orthogonal group and selects only a finite factorization of the one matrix.

Let $f$ be a centred smooth coordinate representative with $f(0)=0$ and invertible $L=Df(0)$. By [F12], choose $C>0$ with $\mathrm{\mid }{L}^{-1}v\mathrm{\mid }\le C\mathrm{\mid }v\mathrm{\mid }$, hence $\mathrm{\mid }Lx\mathrm{\mid }\ge C^{-1}\mathrm{\mid }x\mathrm{\mid }$. By [F11] choose a small ball about zero contained in the coordinate domain and on which $\mathrm{\mid }r(x)\mathrm{\mid }\le (2C{)}^{-1}\mathrm{\mid }x\mathrm{\mid }$, where $r(x)=f(x)-Lx$. Then $H(x,t)=Lx+tr(x),\mathrm{\mid }H(x,t)\mathrm{\mid }\ge (2C{)}^{-1}\mathrm{\mid }x\mathrm{\mid }(x\mathrm{\ne }0,0\le t\le 1)\mathrm{.}$ Thus this is a homotopy of pairs from the linear map to $f$, into $({\mathbb{R}}^n,{\mathbb{R}}^n\setminus 0)$, and $f$ has no other zero in the ball. By [F16], inclusion of this small ball identifies its local group with the whole Euclidean local group; shrinking again does not change the map by [F4]. Steps 1.1 and 4.1 prove that the germ multiplier is $\mathrm{sgn}\det Df(0)$. No local inverse theorem is needed for this homology calculation.

At $p$ in an oriented smooth manifold choose a positive chart centred at $p$ and use [F16] to pull $e_n$ back to ${\mu }_p$. Two such charts are related by a smooth transition fixing zero with positive derivative determinant, by [F13]. Step 5.1 says the transition induces multiplication by $+1$ on the local group, so the two values ${\mu }_p$ agree. This defines one generator at every point by a unique chart-independent value; it does not select a chart at every point.

Verify local continuity in the precise sense of [F15]. Inside one positive chart choose concentric balls $K=\bar{B}(0,r)\subset B(0,s)$. By [F14], choose the unique class over $K$ whose restriction at zero is the generator from step 6.1. Let $\rho$ satisfy $r<\rho <s$. Excision [F16] identifies the supported pair with $(B(0,s),B(0,s)\setminus K)$; since the ball is contractible, the boundary map [F2] identifies its degree-$n$ relative group with reduced degree-$(n-1)$ homology of the annulus. Radial deformation onto $S_{\rho }$ shows that the boundary class is represented by a sphere class there. For $x\in \mathrm{int}K$, restriction to the local pair at $x$ and translation of the target by $-x$ sends this sphere map to $u\mapsto u-x$. The homotopy $u\mapsto u-tx$ avoids zero because $\mathrm{\mid }x\mathrm{\mid }<r<\rho$, so step 1.1 and naturality of [F2] identify its class with $u\mapsto u$. This is exactly the generator defined using the chart centred at $x$ in step 6.1. The calculation works for $n=1$ on reduced $H_0$ as well. Hence the restrictions of the one ball class are all the ${\mu }_x$, proving that this is a continuous generator section by [F15].

For the given germ $F$ choose positive centred charts on source and target. The local map in these charts is exactly the one in step 5.1, so $F_{\ast }{\mu }_p=\mathrm{sgn}\det (Df(0)){\mu }_q=\mathrm{sgn}(d{F}_p){\mu }_q$ by [F13]. The local map is independent of shrinking and charts by [F4], [F16] and step 6.1. Changing the common reference $e_n$ to $-e_n$ negates all source and target generators and leaves this multiplier unchanged.

When $n=0$, each singleton chart is open and its local group is $\mathbb{Z}[p]$ by [F1]. The section $\epsilon (p)[p]$ is continuous on the discrete manifold. The unique local point map sends $[p]$ to $[q]$, so its multiplier in these signed generators is $\epsilon (p)\epsilon (q)$, as in [F13]. Empty manifolds impose an empty section condition, and no germ at an absent point. Singular derivatives are excluded; the remainder estimate in step 5.1 explicitly excludes zero along the entire homotopy except at its fixed origin. All homotopy endpoints and the identity/empty-factorization case are included. One reference generator for the fixed dimension and finitely many witnesses for one germ suffice; no family of generators over dimensions, charts, or points is chosen, and no AC occurs.