Homology Axioms Degree and Classical Applications — Examples

If $d>0$, the fibre of $1$ consists of the $d$ roots $e^{2\pi ik\mathrm{/}d}$, $0\le k<d$. In positively oriented angular coordinates centered at such a point and at $1$, the local map is $t\mapsto dt$. The homotopy $t\mapsto ((1-s)d+s)t$, $0\le s\le 1$, never sends a nonzero sufficiently small $t$ to zero, so it preserves the local pair and identifies the local action with that of the identity. Each local degree is therefore $+1$, and the finite-fibre formula gives $\mathrm{deg}{p}_d=d$.

If $d=-m<0$, the same $m$-point fibre has angular formula $t\mapsto -mt$. Homotoping its positive factor $m$ to $1$ leaves the reflection $t\mapsto -t$. Complex conjugation is a coordinate reflection on $S^1$, of degree $-1$; its singleton fibre and the local-sum formula give local sign $-1$. Thus $\mathrm{deg}{p}_{-m}=-m$. For $d=0$, $p_0=1$ is constant and has degree zero. In particular $p_1$ is the identity.

Take a regular $(n+1)$-simplex in ${\mathbb{R}}^{n+1}$ centered at the origin, with vertices $v_0,\dots ,v_{n+1}$. Radial projection of its boundary to $S^n$ is a homeomorphism: each ray meets its convex boundary exactly once, and the continuous bijection has compact source and Hausdorff target. The orthogonal symmetry interchanging $v_0,v_1$ and fixing the other vertices is reflection across the perpendicular bisector hyperplane. Rotate this configuration so that this hyperplane is the coordinate hyperplane of $r$; radial projection then commutes with the reflection.

With the orientation convention of [F2], put $c=\mathrm{\partial }[v_0,\dots ,v_{n+1}]={\sum }_{j=0}^{n+1}(-1{)}^j[v_0,\dots ,{\widehat{v}}_j,\dots ,v_{n+1}]$. Each codimension-two face occurs twice with opposite signs, so $\mathrm{\partial }c=0$. Conversely, cancellation along every common facet forces the coefficients of any top cycle to be a common integer multiple of those of $c$. There are no $(n+1)$-simplices in the boundary, so $c$ generates its top simplicial homology and hence its oriented fundamental class under the natural simplicial-to-singular isomorphism [F3].

The vertex transposition $\tau$ commutes with the boundary formula. In the filled simplex its action on $[v_0,\dots ,v_{n+1}]$ is multiplication by $-1$. Thus $\tau c=\mathrm{\partial }\tau [v_0,\dots ,v_{n+1}]=-c$. Radial projection carries this equality to $r_{\ast }[S^n]=-[S^n]$, independently of which sign was chosen for the generator.

In ${\mathbb{R}}^{n+1}$, negating all coordinates is the composite of its $n+1$ coordinate reflections. The antipodal formula in [F1] gives $(-1{)}^2=+1$ on $S^1$, $(-1{)}^3=-1$ on $S^2$, and $(-1{)}^4=+1$ on $S^3$.

By [F2], the odd dimensions $1$ and $3$ permit such a field and dimension $2$ does not. Concretely, identify ${\mathbb{R}}^2=\mathbb{C}$ and ${\mathbb{R}}^4={\mathbb{C}}^2$ and set $v(z)=iz$. It has norm $1$ on the sphere and real inner product $\mathrm{Re}\sum {\bar{z}}_{j}i{z}_j=0$ with $z$, so is tangent and nowhere zero.

Here is the sphere model and its orientation. The map $s(z)=(2\mathrm{Re}z,2\mathrm{Im}z,\mathrm{\mid }z{\mathrm{\mid }}^2-1)\mathrm{/}(1+\mathrm{\mid }z{\mathrm{\mid }}^2)$ lies on the unit sphere, and its inverse off the north pole is $(x,y,t)\mapsto (x+iy)\mathrm{/}(1-t)$. Both formulas are continuous and inverse by substitution. Moreover $s(z)$ approaches the north pole exactly as $\mathrm{\mid }z\mathrm{\mid }\to \mathrm{\infty }$, giving the topology with neighborhoods of infinity containing $\{\mathrm{\mid }z\mathrm{\mid }>R\}\cup \{\mathrm{\infty }\}$. Declare the finite coordinate $z$ positive and use $w=1\mathrm{/}z$ near infinity. At $z_0\mathrm{\ne }0$, writing $z=z_0+u$, the transition difference is $1\mathrm{/}(z_0+u)-1\mathrm{/}{z}_0=-u\mathrm{/}(z_0(z_0+u))$. On a sufficiently small disk its nonzero factor is homotopic through nonzero factors to $-1\mathrm{/}{z}_0^2$. Multiplication by a nonzero complex constant is a rotation followed by a positive dilation, homotopic through invertible real maps to the identity. Thus the two charts give compatible local orientations; no orientation-reversal is hidden at infinity.

The leading-term estimate gives $\mathrm{\mid }P(z)\mathrm{\mid }\ge \mathrm{\mid }{a}_d\mathrm{\mid }\mathrm{\mid }z{\mathrm{\mid }}^d\mathrm{/}2$ for all sufficiently large $\mathrm{\mid }z\mathrm{\mid }$: divide the sum of lower-degree terms by $\mathrm{\mid }z{\mathrm{\mid }}^d$, which tends to zero. It proves continuity at infinity. For a finite $a$, polynomial division yields $P(a+u)-P(a)=u^{m}q(a+u)$ with $m\ge 1$ and $q(a)\mathrm{\ne }0$. Shrink the disk until $\mathrm{\mid }q(a+u)-q(a)\mathrm{\mid }<\mathrm{\mid }q(a)\mathrm{\mid }$. The homotopy $u^m((1-s)q(a+u)+sq(a))$ has zero only at $u=0$ for every $0\le s\le 1$. Uniform boundedness of the factors permits a source disk mapping into one fixed target coordinate disk. Hence this is a homotopy of punctured local pairs to $q(a)u^m$.

For a disk $D$ centered at zero, the pair exact sequence [F3] identifies $H_2(D,D\setminus \{0\};\mathbb{Z})$ with $H_1(D\setminus \{0\};\mathbb{Z})$; radial deformation identifies the latter with the counterclockwise circle generator. The normalized map of a small circle for $c{u}^m$, $c\mathrm{\ne }0$, is a rotation times $e^{it}\mapsto e^{imt}$. The latter has $m$ preimages of $1$. Near each preimage its angular formula is $t\mapsto mt$, homotopic through positive linear slopes to $t\mapsto t$, so its local degree is $+1$. Applying [F1] in dimension one gives angular degree $m$. Rotation acts trivially by its rotation homotopy. Naturality [F4] of the disk pair boundary therefore gives local degree $m$ for $c{u}^m$, and step 2.1 gives the claimed finite local multiplicities.

In the coordinates $w=1\mathrm{/}z$ and $v=1\mathrm{/}P(z)$ at the two infinities, the map is $v=w^d\mathrm{/}(a_d+a_{d-1}w+\cdots +a_0{w}^d)$, extended by $v(0)=0$. Its denominator is nonzero on a small disk. The same nonvanishing-factor homotopy and disk-boundary calculation give local degree $d$ at infinity. The value is positive with the compatible chart orientations of step 1.1.

For any finite $b$, [F2] applied to the degree-$d$ polynomial $P-b$ says its distinct roots have positive multiplicities summing to $d$. They form the entire finite fibre, since infinity maps to infinity. By step 3.1 and [F1], $\mathrm{deg}\widehat{P}$ is the sum of these local degrees, hence $d$. This also agrees with applying [F1] to the singleton fibre of infinity and step 4.1. Repeated roots and $d=1$ require no change.

Apply [F1] with $G=\mathbb{Z}$ and with $G=\mathbb{Z}\mathrm{/}2$. Both yield ordinary theories, and their degree-zero groups at a point are respectively $\mathbb{Z}$ and $\mathbb{Z}\mathrm{/}2$.

Any natural equivalence would in particular supply an isomorphism between these two groups at that point. No such isomorphism exists: every element of $\mathbb{Z}\mathrm{/}2$ is annihilated by $2$, whereas $2\cdot 1\mathrm{\ne }0$ in $\mathbb{Z}$. The coefficient-isomorphism hypothesis of [F2] is therefore not satisfied, and its uniqueness assertion makes no equivalence claim for these two theories.

Choose two disjoint closed round $n$-balls $D_{+},D_{-}$ in $S^n$, each with nonempty interior. Collapse the complement of their interiors to a point. As in the pinch construction of [F3], this gives a continuous quotient to $(D_{+}\mathrm{/}\mathrm{\partial }{D}_{+})\vee (D_{-}\mathrm{/}\mathrm{\partial }{D}_{-})\cong S^n\vee S^n$. On the first quotient use an orientation-preserving homeomorphism to the target sphere, taking the collapsed boundary to a chosen basepoint $b$. On the second use an orientation-reversing homeomorphism with the same basepoint image; obtain it by composing an orientation-preserving one with a reflection fixing $b$. Such a reflection has degree $-1$ by [F2].

The two maps agree at the wedge point, so they descend to a continuous map $f:S^n\to S^n$. Each ball quotient already covers the target, so $f$ is surjective and nonconstant. A point $y\mathrm{\ne }b$ has precisely two preimages, one in each ball interior. With the inherited local orientations their local degrees are $+1$ and $-1$: restriction of each oriented quotient homeomorphism to its interior preserves the local generator, and the inserted reflection reverses it. Thus [F1] gives $\mathrm{deg}f=1-1=0$. This refutes constancy, without making any assertion that a degree-zero map cannot be nullhomotopic.

Write $S^0=\{a,b\}$ and use the point classes $e_a,e_b$ as the ordered basis of $H_0$, consistently with [F2]. A map sends a point class to the class of its image. Hence, with columns recording images, the identity, transposition, constant-$a$, and constant-$b$ maps induce respectively $\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$, $\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$, $\left(\begin{array}{cc}1& 1\\ 0& 0\end{array}\right)$, and $\left(\begin{array}{cc}0& 0\\ 1& 1\end{array}\right)$. These are all four maps of a two-point discrete space. In particular the transposition matrix is not an integer multiple of the identity; the rank-one hypothesis of [F1] is absent.

The augmentation of [F3] is $(u,v)\mapsto u+v$. Its kernel has generator $e_b-e_a$. The four matrices in step 1.1 act on this generator as $+1,-1,0,0$, respectively. Thus reduced degree exists in this case, but is a different convention from the unreduced definition restricted to positive-dimensional spheres.

For a family of abelian groups $M_i$, denote ${\prod }_{i\ge 0}{M}_i\mathrm{/}{⨁}_{i\ge 0}{M}_i$ by $Q(M)$. Degreewise maps induce maps of $Q$, giving functoriality. If $A_i\stackrel{u_i}{\to }{B}_i\stackrel{v_i}{\to }{C}_i$ is exact at every $B_i$, then $Q(A)\to Q(B)\to Q(C)$ is exact: when $v(b)$ is finitely supported, replace the finitely many bad coordinates of $b$ by zero; the resulting equivalent tuple lies in $\prod \ker v_i$. Choose coordinate lifts in $A_i$ and obtain a preimage in $Q(A)$. The converse inclusion follows from $v_i{u}_i=0$. This uses ordinary choice for the countable family of nonempty lift sets. Adding, deleting, or altering finitely many initial coordinates has no effect on $Q$.

Apply step 1.1 to the ordinary singular pair sequence supplied by [F2]. Define the $E$ boundary by sending a representative $(x_i{)}_{i\ge 0}$ to the tuple whose $j$th coordinate is the singular boundary $\mathrm{\partial }{x}_{j+1}\in H_j(A;\mathbb{Z})$. Dropping the unused initial coordinate is precisely the shift identification of step 1.1. At each term the coordinatewise singular exact sequence and that shift prove the required exactness, even though $E_n$ is independent of $n\in \mathbb{Z}$. Homotopies and excision induce coordinatewise equal maps and isomorphisms, so they do so on $Q$. Apply [F1] here only to the ordinary singular theory [F2]: its quotient identifications and suspension isomorphisms give coordinatewise reduced cofiber sequences and suspension maps. Applying $Q$ and step 1.1 gives the reduced $E$ axioms and identifies them with the displayed pair construction. Replacing reduced absolute groups by unreduced absolute groups changes only coordinate zero and therefore does not change $Q$. All structure maps are natural.

A point, the empty space under the zero reduced-empty convention, and $S^0$ have reduced homology in at most one degree. Their $Q$ groups vanish; hence the coefficient of $E$ is zero. For any finite family, the product over degrees commutes with its finite direct sum, as does the finite-support subgroup. Thus finite disjoint-sum additivity holds for pairs and finite wedge additivity for reduced spaces. For a finite-dimensional CW pair of dimension at most $N$, [F3] and [F4] applied to singular homology give $H_i(X,A;\mathbb{Z})=0$ outside $0\le i\le N$, regardless of the number of cells. Therefore $E_n(X,A)=0$ for every integer $n$.

Give $W={\bigvee }_{i\ge 1}{S}^i$ its CW structure with one vertex and one cell in each positive dimension, all attached by constant maps. The cellular differentials vanish, including the degree-one terminal-minus-initial differential. On each skeleton [F3] and [F4] compute one copy of $\mathbb{Z}$ in each positive degree present. Applying [F5] to ordinary singular homology, not to $E$, gives ${\tilde{H}}_i(W;\mathbb{Z})=\mathbb{Z}$ for every $i\ge 1$ and zero in degree zero. Hence $E_n(W)={\prod }_{i\ge 1}\mathbb{Z}\mathrm{/}{⨁}_{i\ge 1}\mathbb{Z}\mathrm{\ne }0$: the all-ones tuple is not finitely supported. Yet each $E_n(S^i)=0$ by step 2.2, so the canonical infinite wedge map from the direct sum of these zero groups is not surjective.

Taking the direct sum of the pair functors $H$ and $E$, with componentwise structure maps, preserves homotopy, exactness, excision, dimension, and finite additivity. Its coefficient at a point is $\mathbb{Z}\oplus 0=\mathbb{Z}$, and step 2.2 identifies it with $H$ on all finite-dimensional pairs. On $W$, ordinary $H_{-1}(W;\mathbb{Z})=0$ by the nonnegative singular chain complex, whereas $(H\oplus E{)}_{-1}(W)=E_{-1}(W)\mathrm{\ne }0$ by step 3.1. Thus even an abstract graded-group equivalence fails there. The ordinary all-CW uniqueness theorem requires arbitrary additivity, exactly the axiom that this construction fails.