The degree of a proper map between oriented manifolds #
Let M and N be topological manifolds of the same dimension n, oriented by o and o'
(TauCeti.HomologicalOrientation), and let f : M → N be continuous. For a compact subset
K ⊆ N whose preimage A = f⁻¹(K) is compact, f is a map of pairs
(M, M ∖ A) → (N, N ∖ K), so it carries the fundamental class [M]_A ∈ Hₙ(M, M ∖ A; ℤ) along A
to a class f_*[M]_A ∈ Hₙ(N, N ∖ K; ℤ). The source group need not be cyclic, and for A = ∅ the
class [M]_A is zero.
At a point y the target group Hₙ(N, N ∖ {y}; ℤ) is free of rank one on the local orientation
o' y, so f_*[M]_{f⁻¹(y)} = d • o' y for a unique integer d, the local degree
o.localDegree o' hE f y. For a proper map the local degree is locally constant, because local
homology is constant along a small compact ball in a chart. On a connected target the local
degree is therefore a single integer, the degree o.degree o' hE f, and it computes the image
of every fundamental class along a compact set:
f_*[M]_{f⁻¹(K)} = deg f • [N]_K for every compact K ⊆ N.
For compact M and N the case K = N reads f_*[M] = deg f • [N] in Hₙ(N; ℤ).
This is the classical homological definition of the degree of a proper map, which does not need
compactness of M or N nor any smoothness of f. The degree is 1 for the identity, changes
sign when either orientation is reversed, is multiplicative under composition, vanishes for maps
that are not surjective and is invariant under proper homotopies, that is homotopies
F : [0, 1] × M → N which are proper as maps on [0, 1] × M.
Main definitions #
TauCeti.HomologicalOrientation.localDegree o o' hE f y: the integerdwithf_*[M]_{f⁻¹(y)} = d • o' y, and0if the fibref⁻¹(y)is not compact.TauCeti.HomologicalOrientation.degree o o' hE f: the local degree at an arbitrary point ofN; it is the degree offwhenfis proper andNis connected.
Main results #
TauCeti.HomologicalOrientation.localDegree_eq_iff: the defining property of the local degree.TauCeti.HomologicalOrientation.isLocallyConstant_localDegree: the local degree of a proper map is locally constant, so it equals the degree on a preconnected target (TauCeti.HomologicalOrientation.localDegree_eq_degree).TauCeti.HomologicalOrientation.localHomologyMap_fundamentalClassOn_preimage:f_*[M]_{f⁻¹(K)} = deg f • [N]_Kfor every compactK, and its converseTauCeti.HomologicalOrientation.degree_eq_ifffor nonemptyK.TauCeti.HomologicalOrientation.degree_id,TauCeti.HomologicalOrientation.degree_neg_left,TauCeti.HomologicalOrientation.degree_neg_right,TauCeti.HomologicalOrientation.degree_comp: normalization, orientation reversal and multiplicativity.TauCeti.HomologicalOrientation.surjective_of_degree_ne_zero: a proper map of nonzero degree onto a preconnected manifold is surjective.TauCeti.HomologicalOrientation.degree_eq_of_homotopy: the degree is invariant under proper homotopies.TauCeti.HomologicalOrientation.mapFundamentalClass_eq_degree_smul: for compactMandN,f_*[M] = deg f • [N]inHₙ(N; ℤ), and more generallyTauCeti.HomologicalOrientation.mapFundamentalClass_comp_eq_degree_smul:(g ∘ f)_*[M] = deg f • g_*[N]for every continuousg : N → X.TauCeti.HomologicalOrientation.singularHomologyMap_eq_degree_smul: a self-map of a compact connected oriented manifold acts on top homology by multiplication by its degree, andTauCeti.HomologicalOrientation.bijective_singularHomologyMap_iff: it is bijective there exactly when its degree is±1.
References #
- A. Dold, Lectures on Algebraic Topology, Springer, 1972, Chapter VIII, §4.
- A. Hatcher, Algebraic Topology, Section 3.3, Lemma 3.27.
The local degree of f : M → N at y ∈ N: the integer d with
f_*[M]_{f⁻¹(y)} = d • o' y in Hₙ(N, N ∖ {y}; ℤ), where [M]_{f⁻¹(y)} is the fundamental class
of M along the fibre (TauCeti.HomologicalOrientation.localDegree_eq_iff). It is 0 when the
fibre is not compact; for a proper map every fibre is compact.
Instances For
At a point with compact fibre, f carries the fundamental class along the fibre to the local
degree times the local orientation.
The local degree vanishes at a point whose fibre is not compact.
The defining property of the local degree: at a point y with compact fibre, the local
degree is the integer c with f_*[M]_{f⁻¹(y)} = c • o' y.
The local degree at a point outside the range of f vanishes: the fibre is empty, and the
fundamental class along the empty set is zero.
Let K ⊆ N have compact preimage. Then the restriction of f_*[M]_{f⁻¹(K)} to a point
y ∈ K is the local degree at y times the local orientation.
The local degree of a proper map is locally constant.
The degree of f : M → N: its local degree at an arbitrary point of N, and 0 when N
is empty. For a proper map onto a preconnected manifold the local degree is the same at every
point (TauCeti.HomologicalOrientation.localDegree_eq_degree), and the degree is characterized by
f_*[M]_{f⁻¹(K)} = deg f • [N]_K for compact K
(TauCeti.HomologicalOrientation.localHomologyMap_fundamentalClassOn_preimage).
Instances For
On a nonempty target, the degree is the local degree at the chosen point h.some.
The degree of a map to an empty manifold vanishes.
The local degree of a proper map onto a preconnected manifold is its degree at every point.
The degree computes the image of every fundamental class: for a proper map f onto a
preconnected manifold and every compact K ⊆ N, f_*[M]_{f⁻¹(K)} = deg f • [N]_K. In particular
the degree may be read off from any nonempty compact K
(TauCeti.HomologicalOrientation.degree_eq_iff).
The degree is determined by any nonempty compact set: for a proper map f onto a
preconnected manifold and nonempty compact K ⊆ N, the degree is the integer c with
f_*[M]_{f⁻¹(K)} = c • [N]_K.
A proper map of nonzero degree onto a preconnected manifold is surjective.
The identity has local degree 1 at every point.
The identity of a nonempty manifold has degree 1.
Reversing the orientation of the source negates the local degree.
Reversing the orientation of the target negates the local degree.
Reversing the orientation of the source negates the degree.
Reversing the orientation of the target negates the degree.
Local degrees multiply under composition: for proper maps f : M → N and g : N → P with
N preconnected, the local degree of g ∘ f at z is deg f times the local degree of g at
z.
The degree is multiplicative: for proper maps f : M → N and g : N → P with N
preconnected, deg (g ∘ f) = deg g * deg f.
Invariance of the local degree under proper homotopies: if H is a homotopy from f₀
to f₁ which is proper as a map [0, 1] × M → N, then f₀ and f₁ have the same local degree at
every point.
Invariance of the degree under proper homotopies: maps joined by a homotopy which is
proper as a map [0, 1] × M → N have the same degree.
The degree computes the image of the fundamental class: for a map f : M → N between
compact oriented n-manifolds with N preconnected, f_*[M] = deg f • [N] in Hₙ(N; ℤ).
The degree of a composite with a further map: for f : M → N between compact oriented
n-manifolds with N preconnected and any continuous g : N → X,
(g ∘ f)_*[M] = deg f • g_*[N] in Hₙ(X; ℤ).
A self-map acts on top homology by its degree: for a self-map f of a compact connected
oriented n-manifold, f_* is multiplication by the degree of f on Hₙ(M; ℤ).
A self-map is bijective on top homology exactly when its degree is ±1, for a compact
connected oriented manifold.