Algebraic topology

A dumbbell graph consists of two cycles joined by a path. Its connecting path contains bridges, unlike a Theta graph, although both the two-loop dumbbell and

Θ_{3}

have first homology

Z^{2}

Borsuk-Ulam theorem

Every continuous map

f : S^{n} \to R^{n}

takes the same value at some antipodal pair. Equivalently, every continuous antipodal map

S^{n} \to R^{n}

has a zero, and there is no antipodal map

S^{n} \to S^{n - 1}

Lusternik-Schnirelmann-Borsuk theorem

A cover of

S^{d}

d + 1

closed sets has a member containing an antipodal pair. The equivalent version for open sets follows by shrinking finite open covers on a compact metric space; the reverse implication follows by enlarging antipodal-pair-free closed sets slightly. This is a covering formulation of the Borsuk-Ulam theorem.

CW complex

A CW complex is built inductively by attaching discs along maps from their boundary spheres. Its

n

-skeleton contains all cells of dimension at most

n

CW pair ( $(X, A)$ )

A CW pair consists of a CW complex

X

and a subcomplex

A

. Its inclusion is a cofibration, so homotopies on

A

extend across

X

. The relative lifting property of a Serre fibration for such pairs allows specified boundary lifts to be retained while lifting a homotopy.

Cell attachment ( $X \cup_{ϕ} D^{q}$ )

Attaching a

q

-cell along

ϕ : S^{q - 1} \to X

forms the quotient of

X ⊔ D^{q}

identifying each boundary point with its image under

ϕ

. A constant attaching map gives

X \lor S^{q}

. An attaching map can affect the cohomology ring even when it does not change the additive cohomology groups.

The weak topology of a CW complex is determined by its closed cells: a subset is closed if and only if its inverse image under every characteristic map from a closed cell is closed. A map out of the complex is continuous when its restrictions along those characteristic maps are continuous. This is distinct from the weak topology of a vector space. Taking the product with a locally finite CW complex, such as a closed interval with its finite cell structure, retains the corresponding CW weak topology.

Topological graph

A topological graph is a CW complex with only zero-cells and one-cells, called vertices and edges. Edge endpoints attach to vertices; loops, multiple edges and infinitely many edges are allowed. A subgraph in this setting is a CW subcomplex. A connected acyclic graph is a tree and is a contractible space.

CW subcomplex

A CW subcomplex is a union of cells of a CW complex containing the attaching boundary of every included cell. In a combinatorial 2-complex, including a two-cell therefore includes every edge in its boundary. A connected CW subcomplex has a connected 1-skeleton, whose intrinsic path metric may differ from the ambient graph metric.

1-skeleton ( $X^{(1)}$ )

The one-skeleton

X^{(1)}

of a CW complex is the union of its zero-cells and one-cells. It is a graph. The unit-edge path metric measures the infimum of lengths of paths in this graph; it is different from metrics that allow paths through two-cell interiors.

2-complex

A two-dimensional CW complex has cells only in dimensions zero, one and two. In a combinatorial two-complex the two-cells are polygons attached by combinatorial edge paths. Its 1-skeleton carries the path metric assigning length one to each edge. Compact combinatorial complexes have finitely many cells, so their cell-boundary lengths have a finite maximum.

Disc diagram

A disc diagram over a combinatorial 2-complex is a finite contractible planar combinatorial complex together with a combinatorial map to that complex. Its exterior boundary circuit records a null-homotopic path. Tree portions and cut vertices are allowed; the diagram need not be an embedded disk or map injectively. Its area is its number of two-cells.

Disc diagram ladder

A ladder is a disc diagram whose blocks form a linear chain, each block a two-cell or a connecting edge, with cells attached successively along interior arcs. Its boundary consists of two opposite side paths between its ends. Every cell meets both side paths. If cell perimeters are bounded by

L

, each point of one side is within

L /2

of the other by traveling around a cell boundary; connecting edge paths coincide on the two sides.

Disc diagram spur

A spur is an exposed degree-one vertex and its incident edge in a disc diagram. The exterior boundary runs out along that edge and immediately back. Thus a spur in the interior of a boundary side contradicts that side being a reduced combinatorial path or a metric geodesic. A diagram with no two-cells may be a tree and have spurs rather than shells.

Disc diagram shell ( $\partial R = QP$ )

A shell is an exposed two-cell in a disc diagram, with perimeter split as an exterior boundary arc

Q

and an interior path

P

. An

i

-shell has

P

consisting of

i

interior pieces. Under

C^{'} (1/6)

, an

i

-shell with

i \leq 3

has

∣ P ∣ < ∣ \partial R ∣/2 < ∣ Q ∣

, so removal replaces a long exterior path by a strictly shorter path.

van Kampen lemma

A closed combinatorial path in a 2-complex is null-homotopic exactly when it is the boundary path of a disc diagram over that complex. A finite cellular null-homotopy can be arranged into such a planar diagram; conversely the contractible diagram supplies a null-homotopy. In a group presentation, this is the geometric form of expressing a trivial word as a product of conjugates of relators.

Reduced disc diagram

A disc diagram is reduced if it has no adjacent pair of two-cells that can be folded together across their common boundary and removed while preserving the outside boundary. A minimum-area diagram is reduced. Among diagrams of minimum area, minimizing the number of edges removes unnecessary tree portions. Internal common arcs in a reduced diagram satisfy the applicable small cancellation theory piece bounds.

Cellular map

A cellular map

f : X \to Y

between CW complexes satisfies

f (X^{q}) \subseteq Y^{q}

for every skeleton. It induces a chain map

f_{#} : C_{*}^{cell} (X) \to C_{*}^{cell} (Y)

Cellular approximation theorem

A continuous map between CW complexes is homotopic to a cellular map. If the map is already cellular on a subcomplex, the homotopy can be taken relative to that subcomplex. In particular, a map from a

d

-dimensional CW complex can be deformed into the

d

-skeleton of the target. This permits finite cellular chain complexes to compute induced homology maps, and turns maps to Complex projective space into maps to an appropriate finite-dimensional projective skeleton.

Finite CW complex

A finite CW complex has finitely many cells. Its cellular chain complex consists of finitely generated free abelian groups and vanishes outside finitely many degrees, so its integral homology groups are finitely generated.

CW approximation

A CW approximation of a space

X

is a CW complex

X_{cw}

with a weak homotopy equivalence

X_{cw} \to X

. The realization of the singular simplicial set of

X

supplies one functorially.

Finite type CW approximation ( $C ≃_{w} X$ )

For simply connected

X

with finitely generated integral homology in every degree, the Hurewicz theorem modulo a Serre class gives finitely generated homotopy groups. Realize finite generating sets of successive relative homotopy groups by attaching finitely many cells at each stage. The resulting weak equivalence has finitely many cells in each dimension.

Finite CW approximation from bounded homology ( $H_{i} (X) = 0 (i > n) \Rightarrow dim C \leq n + 1$ )

Start with a finite

n

-dimensional

n

-connected approximation. Its degree-

n

homology kernel is finite free. Realize a basis via the Relative Hurewicz theorem and attach

(n + 1)

-cells; their boundary is injective, so they kill precisely that kernel without adding top homology. The homological Whitehead theorem gives a finite weak equivalence of dimension at most

n + 1

CW filtration

A CW filtration builds a CW complex by its skeleta

\emptyset = X^{- 1} \subset X^{0} \subset X^{1} \subset \dots,

where each quotient

X^{n} / X^{n - 1}

is a wedge of

n

-spheres, one for each

n

-cell.

Moore space (algebraic topology)

For an abelian group

G

and an integer

m \geq 1

, a Moore space

M (G, m)

is a CW complex whose reduced homology is

G

in degree

m

and zero in every other degree.

Bockstein on a cyclic Moore space ( $β : H^{i} (M (Z / n, i); Z / n) \sim H^{i + 1} (M (Z / n, i); Z / n)$ )

Attach an

(i + 1)

-cell to

S^{i}

by a degree-

n

map, with

i \geq 1

and

n \geq 2

. The integral cellular differential is multiplication by

n

, while its modulo-

n

reduction is zero. Lifting the degree-

i

generator modulo

n^{2}

and applying the differential gives

n

, so the Bockstein homomorphism sends that generator to the degree-

(i + 1)

generator. This supplies a nonzero example in every positive degree.

Cyclic Moore space in dimension one ( $M (Z / n, 1)$ )

Attaching a two-cell to

S^{1}

by a map of degree

n

gives the Moore space

M (Z / n, 1)

. Its positive-dimensional cellular chain complex is

0 ⟶ Z n Z ⟶ 0,

H_{1} ≅ Z / n

and all its other reduced integral homology groups vanish.

Mapping cylinder

The mapping cylinder of

f : A \to X

M_{f} = (A \times [0, 1] ⊔ X) / ((a, 0) \sim f (a)) .

It deformation retracts onto the copy of

X

Mapping cone (topology, $C_{f}$ )

The topological mapping cone of

f : X \to Y

Y \cup_{f} CX

, where

CX

is the cone on

X

and its base is identified with

Y

through

f

. For nonempty

X

, the cone vertex supplies a distinguished basepoint. This construction is related to, but distinct from, the mapping cone of a chain map.

Cellular chain complex of a mapping cone ( $C_{q} (C_{f}) = C_{q} (Y) \oplus C_{q - 1} (X)$ )

For a cellular map between nonempty CW complexes, the reduced cellular chain complex of its topological mapping cone has differential

D (y, x) = (d_{Y} y + f_{#} x, - d_{X} x) .

Here

C_{- 1} (X) = 0

and the degree-zero group identifies a vertex

y

with the reduced chain

y - v

, where

v

is the cone vertex. Thus this is the mapping cone of

f_{#}

, in the displayed ordering of the summands.

Mapping cone exact sequence ( $\dots \to H_{q + 1} (C_{f}; G) \to H_{q} (X; G) \to H_{q} (Y; G) \to H_{q} (C_{f}; G) \to \dots$ )

For nonempty

X

, the topological mapping cone has this long exact sequence in homology, with ordinary homology for

X, Y

and reduced homology for

C_{f}

. In degree zero, the augmentation of

X

is accounted for by the cone vertex. The sequence follows from the Mayer–Vietoris theorem applied to neighborhoods of the cone and

Y

Mapping telescope

The mapping telescope of

X_{0} f_{0} X_{1} f_{1} \dots

glues the mapping cylinders of the

f_{i}

end to end. Its homology is naturally the direct limit

lim_{i} H_{*} (X_{i})

Mapping-telescope realization of the rational group with square-free denominators ( $H_{n} (T) ≅ Q_{sq}$ )

List the primes as

p_{0}, p_{1}, \dots

and form the mapping telescope of degree-

p_{i}

maps

S^{n} \to S^{n}

. Its reduced homology vanishes outside degree

n

, while

H_{n} (T; Z) ≅ lim (Z p_{0} Z p_{1} \dots) ≅ Q_{sq} .

Smash product ( $X \land Y$ )

For based spaces, the smash product is

X \land Y = (X \times Y) / (X \lor Y),

where the wedge

X \lor Y

consists of pairs having at least one basepoint coordinate.

Real projective space ( $RP^{n}$ )

 2  0

Real projective space is the antipodal quotient

RP^{n} = S^{n} / (x \sim - x)

Infinite-dimensional real projective space ( $RP^{\infty}$ )

This is the CW complex obtained as the increasing union of finite Real projective spaces under their standard inclusions. It has one cell in each nonnegative dimension. Its cellular chain complex has boundary multiplication by two in positive even dimensions and zero in odd dimensions. Consequently its integral cohomology is

Z

in degree zero,

Z /2

in positive even degrees and zero in odd degrees. Its mod-two cohomology is

Z_{2}

in every nonnegative degree.

Stunted real projective space ( $RP^{m} / RP^{k}$ )

Collapse the indicated standard skeleton to a point. Besides its basepoint, the quotient has one cell in dimensions

k + 1

through

m

. Its integral cellular boundary is two in even dimensions and zero in odd dimensions. Steenrod operations on its relative cohomology are inherited from real projective space.

Homotopy groups of the stunted projective space through degree nine ( $π_{7} (RP^{10} / RP^{6}) = π_{9} (RP^{10} / RP^{6}) = Z /2$ )

For this quotient, homotopy groups through degree six and in degree eight vanish. The degree-seven and degree-nine groups are

Z /2

. A map to

K (Z /2, 7)

induces an isomorphism on degree-nine integral homology because the relevant square of the bottom mod-two class is nonzero. Two relative Hurewicz steps then compute degrees eight and nine.

Steenrod squares on real projective space ( $Sq^{j} (x^{r}) = (j r) x^{r + j}$ )

For the degree-one mod-two generator,

Sq^{1} x = x^{2}

and higher squares of

x

vanish by instability. The Cartan formula gives the displayed binomial identity. Naturality applies it to relative classes of a stunted real projective space.

Standard affine atlas of real projective space

The

n + 1

manifold charts

U_{i} = {[X] : X_{i} \neq = 0}

have coordinates

X_{k} / X_{i}

, with

k \neq = i

. Their inverse inserts

1

as the

i

th homogeneous coordinate. On an overlap, changing the denominator from

X_{i}

X_{j}

divides each remaining coordinate by the nonzero coordinate

X_{j} / X_{i}

. These rational smooth transition maps give Real projective space its standard smooth manifold structure.

Integral cohomology ring of real projective space ( $H^{*} (RP^{n}; Z)$ )

For

m \geq 0

, the cohomology ring is

H^{*} (RP^{2 m}; Z) ≅ Z [a] / (2 a, a^{m + 1}), ∣ a ∣ = 2,

and

H^{*} (RP^{2 m + 1}; Z) ≅ Z [a, b] / (2 a, a^{m + 1}, ab, b^{2}), ∣ a ∣ = 2, ∣ b ∣ = 2 m + 1.

(2)

The generator

a

reduces to the square of the degree-one generator in the mod-two cohomology ring of real projective space;

b

is the integral orientation class in odd top dimension.

Integral cohomology of a product of finite real projective spaces

The cellular cochain complex of finite Real projective space consists of alternating maps zero and two. Its groups can be tensored using the Künneth theorem; the integral Künneth torsion polynomial efficiently counts all tensor and Tor functor summands. A product of factors of dimensions two, three and four has free-rank polynomial

1 + t^{3}

and torsion polynomial

3 t^{2} + 3 t^{3} + 5 t^{4} + 6 t^{5} + 5 t^{6} + 4 t^{7} + 2 t^{8} + t^{9}

Cellular homology of real projective space

Real projective space has one cell in every dimension from zero through

n

. With integral coefficients its cellular differential

C_{k} \to C_{k - 1}

is multiplication by

1 + (- 1)^{k}

, hence is zero for odd

k

and multiplication by two for even

k

. With coefficients in

F_{2}

, every cellular differential vanishes.

Mod-two cohomology ring of real projective space ( $H^{*} (RP^{m}; F_{2})$ )

The mod-two cohomology ring of real projective space is

H^{*} (RP^{m}; F_{2}) ≅ F_{2} [y] / (y^{m + 1}), ∣ y ∣ = 1.

Mapping-cylinder model of punctured real projective three-space

The complement of an open three-ball in

RP^{3}

is the mapping cylinder of the antipodal covering

S^{2} \to RP^{2}

. It deformation retracts to

RP^{2}

and has boundary

S^{2}

Integral homology of real projective three-space

The integral homology groups are

H_{i} (RP^{3}; Z) = ⎩ ⎨ ⎧ Z, Z /2, 0, i = 0, 3, i = 1, otherwise .

Double of punctured real projective three-space

Doubling

RP^{3}

minus an open ball along its spherical boundary gives

RP^{3} # RP^{3}

. Its fundamental group is

(Z /2) * (Z /2) ≅ D_{\infty},

and its universal cover is

S^{2} \times R

Mapping torus

The mapping torus of

f : F \to F

T_{f} = (F \times [0, 1]) / ((x, 1) \sim (f (x), 0)) .

f

is a homeomorphism, the projection to the circle

S^{1}

is a fiber bundle with fiber

F

. For a general continuous map, the mapping torus need not be a fiber bundle, but its homology still has the Wang sequence.

Inversion mapping torus of a torus ( $M_{n} = (T^{n - 1} \times [0, 1]) / ((x, 0) \sim (- x, 1))$ )

This mapping torus fibers over the circle with fiber

T^{n - 1}

and inversion as monodromy. The Wang sequence computes its additive cohomology. Coordinate projections to the Klein bottle expose its mod-two cup products.

Mod-two cohomology ring of an inversion mapping torus ( $H^{*} (M_{n}; F_{2}) = F_{2} [t, x_{1}, \dots, x_{n - 1}] / (t^{2}, x_{i}^{2} + t x_{i})$ )

The base class

t

and coordinate classes

x_{i}

all have degree one. Pullback of the mod-two intersection pairing of the Klein bottle gives

x_{i}^{2} = t x_{i}

. The Leray-Hirsch theorem gives the basis consisting of square-free products

x_{I}

and

t x_{I}

. Thus the additive dimensions match those of a torus, but the degree-one squares differ for

n \geq 2

Integral cohomology of an inversion mapping torus

For

r = n - 1

, the Wang sequence gives an additive splitting

H^{q} (M_{n}; Z) ≅ coker ((- 1)^{q - 1} - 1 : Z^{(q - 1 r)} \to Z^{(q - 1 r)}) \oplus ker ((- 1)^{q} - 1 : Z^{(q r)} \to Z^{(q r)}) .

The kernel is free, so the splitting exists as groups. The map in even degree is zero; the map in odd degree is multiplication by minus two.

Integral cup products in the four-dimensional inversion mapping torus ( $H^{*} (M_{4}; Z)$ )

Choose the base class

a

, degree-two torsion classes

τ_{i} = β (x_{i})

, and degree-two free lifts

b_{ij}

reducing to

x_{i} x_{j}

. A generator

κ

of the top

Z /2

reduces to

t x_{1} x_{2} x_{3}

. The mod-two cohomology ring of an inversion mapping torus determines products in top integral degree, since reduction there is injective. The products of distinct

b_{12}, b_{13}, b_{23}

equal

κ

; so does

b_{ij} τ_{k}

when

i, j, k

are all distinct. Their other degree-four products vanish. Also

a τ_{i} = 0

because degree-three integral cohomology is torsion-free.

Fiber monodromy ( $ϕ$ )

A fiber monodromy is the return homeomorphism of the fiber of a bundle over

S^{1}

. Its mapping torus reconstructs the total space.

Homological monodromy ( $ϕ_{*}$ )

Homological monodromy is the induced automorphism of the homology of a fiber. For a fibered knot, its action on the first homology group presents the Alexander module.

Heisenberg nilmanifold as a torus mapping torus

The quotient of the real Heisenberg group by the Integer Heisenberg group is the mapping torus of the torus automorphism

f : S^{1} \times S^{1} \to S^{1} \times S^{1}, f (z, w) = (z w, w) .

Its universal cover is the contractible real Heisenberg group, so its fundamental group is the integer Heisenberg group.

Wang sequence

For the mapping torus of

f : F \to F

, the Wang exact sequence contains

\dots \to H_{q} (F) 1 - f_{*} H_{q} (F) \to H_{q} (T_{f}) \to H_{q - 1} (F) 1 - f_{*} H_{q - 1} (F) \to \dots .

Mapping-torus homology from Mayer–Vietoris ( $1 - f_{*}$ )

For a continuous self-map

f : X \to X

, the mapping torus has the Wang sequence even when

f

is not a homeomorphism. Cover the time circle by a seam neighborhood and an interval away from the seam. Their inverse images retract to

X

: the seam neighborhood is a mapping cylinder, and the other is a product. Their intersection retracts to two copies of

X

. The Mayer–Vietoris sequence map on homology is, after consistent choices of signs,

(a, b) ⟼ (a + f_{*} b, - a - b) .

Invertible changes of coordinates reduce it to the direct sum of

id - f_{*}

and an isomorphism. Canceling the isomorphism summands gives the Wang sequence. This proof does not assume a fiber bundle structure for a noninvertible

f

Homology

Homology assigns abelian groups

H_{i} (X)

to a space, measuring cycles modulo boundaries in each dimension.

Primitive homology class ( $Δ x = x \otimes 1 + 1 \otimes x$ )

A positive-degree class whose reduced homology diagonal vanishes. In a torsion-free homology Hopf algebra, powers of an even primitive class have binomial diagonals. Dualizing that diagonal produces the multiplication of a divided power algebra.

Acyclic space

A nonempty space is acyclic with the chosen coefficients if its reduced homology vanishes in every degree. Equivalently it has the homology of a point. It need not be contractible.

Homology of a directed union ( $H_{i} (⋃_{a} X_{a}) ≅ lim_{a} H_{i} (X_{a})$ )

Suppose

X = ⋃_{a} X_{a}

is directed by inclusion and every compact subset of

X

lies in some

X_{a}

. Every finite singular chain then lies in one

X_{a}

, so

C_{*} (X) = lim_{a} C_{*} (X_{a})

. Exactness of filtered colimits gives

H_{i} (X) ≅ lim_{a} H_{i} (X_{a}) .

Countability of the homology of an open Euclidean subset

Every open subset

U \subseteq R^{N}

is the directed union of finite unions of closed rational cubes contained in

U

. Every compact subset of

U

lies in one such finite polyhedron, whose homology is finitely generated. The homology of a directed union therefore expresses each

H_{i} (U; Z)

as a direct limit over a countable family of countable groups, so it is countable.

Reduced homology ( $H_{*} (X)$ )

Reduced homology modifies degree zero so that a one-point space has zero homology in every degree. For a nonempty space,

H_{k} (X) = H_{k} (X)

when

k > 0

, while

H_{0} (X) ≅ H_{0} (X) \oplus Z

Homology group

A homology group is one of the abelian groups produced by a homology theory. Its elements are homology classes, represented in singular homology by cycles modulo boundaries.

Zeroth homology group ( $H_{0} (X)$ )

The zeroth homology group is the free abelian group on the path components of

X

. In particular, if

X

has finitely many path components, then

H_{0} (X; Z) ≅ Z^{# π_{0} (X)} .

Homology class

A homology class is an equivalence class of chain cycles modulo chain boundaries in a chain complex.

Functoriality of homology

A continuous map

f : X \to Y

induces homomorphisms

f_{*} : H_{n} (X) \to H_{n} (Y)

satisfying

(g \circ f)_{*} = g_{*} \circ f_{*}

and

(id_{X})_{*} = id_{H_{n} (X)}

Homotopy invariance of homology

Homotopic maps induce the same homomorphism on every homology group. Consequently, homotopy equivalent spaces have isomorphic homology groups.

First homology ( $H_{1} (X)$ )

The first homology group

H_{1} (X)

is the group of one-dimensional cycles modulo boundaries. For a path-connected space it is the abelianization of the fundamental group.

Singular homology

Singular homology forms a chain complex from formal sums of continuous maps from standard simplices into a topological space. Its homology groups are invariant under homotopy.

Betti number ( $b_{k} = dim_{R} H_{k} (M; R)$ )

A Betti number is the dimension of a real singular homology group; on a smooth manifold it is also the dimension of the corresponding de Rham cohomology group. For a finite-dimensional cohomology group on a closed Riemannian manifold, it equals the dimension of its space of harmonic representatives. Betti numbers can increase under a finite cover, so equality is a separate hypothesis, not a general covering-space theorem.

Singular chain ( $C_{n} (X)$ )

A singular

n

-chain is a finite formal integer linear combination of singular simplices. Its support is contained in a compact subset of the ambient space.

Singular chain group ( $C_{q} (X; Z)$ )

The singular chain group is the free abelian group on all continuous maps

Δ^{q} \to X

, the singular simplices. In particular it is free even when the set of simplices is infinite. This freeness makes applying

Hom (C_{q} (X), -)

to a short exact coefficient sequence exact.

Singular simplex

A singular

n

-simplex in a space

X

is a continuous map

σ : Δ^{n} \to X

from the standard

n

-simplex. Its alternating restrictions to the faces form its boundary.

Intersection form ( $Q_{M}$ )

For an oriented compact manifold of dimension

4 k

, the intersection form is the bilinear pairing on middle-dimensional homology obtained by representing classes by transverse cycles and counting their signed intersection points. On an oriented four-manifold it is equivalently the cup-product pairing on

H^{2}

evaluated on the fundamental class.

Unimodular intersection pairing ( $L \sim Hom (L, Z)$ )

An integral intersection form on a finite-rank free abelian group

L

is unimodular if its adjoint

x \mapsto λ (x, -)

is an isomorphism to the integral dual. In an integral basis, this is equivalent to the pairing matrix having determinant

1

- 1

, not merely nonzero determinant. For example, projective lines generate a unimodular intersection form on the Complex projective plane, with matrix

(1)

in its complex orientation.

Novikov additivity

When compact oriented four-manifolds are glued along an entire closed boundary three-manifold, the signature of the result is the sum of their intersection form signatures. It applies to

X_{0} \cup_{\partial X_{0}} (- X_{1})

with the difference of the two signatures. Gluing along only part of a boundary can require a correction; that is a different situation.

Intersection pairing ( $H_{k} (W) \times H_{n - k} (W, \partial W) \to F$ )

An intersection pairing counts signed intersections of transverse representatives of complementary homology classes, using one relative representative when the manifold has boundary. With

F_{2}

coefficients it is defined without orientation choices. Poincare-Lefschetz duality makes this a perfect pairing for compact manifolds.

Half-lives-half-dies theorem ( $dim ker i_{*} = \frac{1}{2} dim H_{1} (\partial W; F_{2})$ )

For a compact three-manifold

W

, let

L = ker (H_{1} (\partial W; F_{2}) \to H_{1} (W; F_{2}))

. The boundary intersection pairing satisfies

L^{⊥} = L

: the long exact sequence in relative homology identifies

L

with boundaries of relative surfaces, and the boundary-interior adjoint identity identifies its orthogonal complement with the same kernel. Hence

2 dim L = dim H_{1} (\partial W; F_{2})

. This also implies that a closed surface with odd first mod-two Betti number, such as

RP^{2}

, cannot be the entire boundary of a compact three-manifold.

Degree constraint from intersection forms ( $P^{T} Q_{M} P = (de g f) Q_{N}$ )

For a map

f : M \to N

of connected closed oriented four-manifolds, naturality of the cup product gives

Q_{M} (f^{*} u, f^{*} v) = (de g f) Q_{N} (u, v) .

When

de g f \neq = 0

, nondegeneracy makes

f^{*} : H^{2} (N; Q) \to H^{2} (M; Q)

injective. Consequently the target intersection form, multiplied by the mapping degree, must embed into the source form. For equal second Betti numbers, a matrix

P

for pullback satisfies

P^{T} Q_{M} P = (de g f) Q_{N}

; taking determinants often rules out nonzero degrees. In particular, a two-dimensional indefinite source form cannot realize a nonzero multiple of a two-dimensional definite target form.

Real de Rham intersection form in dimension four ( $Q : H_{dR}^{2} (N; R)^{2} \to R$ )

On a compact oriented smooth manifold

N

of dimension four without boundary, set

Q ([α], [β]) = \int_{N} α \land β

for closed real two-forms. The Stokes theorem makes the expression independent of representatives, and the even degrees make it symmetric. It is the real de Rham cohomology version of the intersection form. The Hodge decomposition theorem and Hodge star operator express it as

Q (h, k) = (h, * k)_{L^{2}}

on harmonic representatives.

Signature of the intersection form from harmonic duality ( $signature Q = (b^{+}, b^{-})$ )

On a closed oriented Riemannian manifold of dimension four, the Hodge star operator preserves harmonic two-forms and splits them into orthogonal spaces

H^{+}

and

H^{-}

with eigenvalues

+ 1

and

- 1

. The real de Rham intersection form in dimension four equals the positive

L^{2}

inner product on

H^{+}

and its negative on

H^{-}

. It is nondegenerate, since

Q (h, * h) = ∥ h ∥_{2}^{2}

. Its signature pair is

(dim H^{+}, dim H^{-})

Integral homology ( $H_{*} (X; Z)$ )

Integral homology is homology with coefficients in

Z

. It retains both the free and torsion parts of each homology group.

Homology of a sphere ( $H_{i} (S^{n}; Z)$ )

For

n > 0

H_{i} (S^{n}; Z) ≅ {Z, 0, i = 0, n, otherwise .

Writing

S^{n}

as two contractible hemispheres whose intersection deformation retracts to

S^{n - 1}

gives the calculation inductively through the Mayer–Vietoris theorem.

Degree of a continuous mapping ( $de g f$ )

For a map

f : S^{m} \to S^{m}

, the degree is the integer by which

f_{*}

multiplies a chosen generator of

H_{m} (S^{m}; Z)

. It is invariant under homotopy.

Degrees of maps of the zero-sphere ( $de g f \in {- 1, 0, 1}$ )

Define mapping degree for

S^{0}

using its rank-one reduced homology group

H_{0} (S^{0}; Z)

. Its generator is the difference of its two points. The four maps of that two-point set consist of the identity, the interchange, and two constant maps; their degree multipliers are

1, - 1, 0, 0

. Thus sphere maps of arbitrary integer degree require positive sphere dimension.

Relative mapping degree ( $F_{*} [D^{m}, S^{m - 1}] = d [D^{m}, S^{m - 1}]$ )

For a map of oriented disk pairs, the multiplier on top relative homology is its relative mapping degree. Naturality of the connecting homomorphism identifies it with the mapping degree of the boundary map.

Graph intersection formula for mapping degree ( $de g (F ∣_{\partial D^{m}}) = (D^{m} \times {0}) \cdot Γ_{F}$ )

For a smooth map of disk pairs whose graph of a function is transverse to the zero slice, orient the graph by its domain. With the tangent space of the zero slice ordered before that of the graph, the local smooth intersection number at a zero is

sgn det D F

. Summing these signs gives the relative mapping degree and hence the boundary mapping degree. Reversing the order of these two

m

-dimensional tangent spaces changes the sign by

(- 1)^{m}

Quotient-sphere degree identity ( $de g F = de g (F ∣_{\partial D^{m}})$ )

Collapsing the boundary identifies the top relative homology of a disk pair with the top reduced homology of its quotient sphere. Together with the boundary connecting homomorphism, this shows that the induced quotient map and the boundary map have the same mapping degree, using the corresponding orientations.

Local degree of a continuous map ( $de g_{x} f$ )

Let

f : M \to N

be a continuous map between oriented

d

-dimensional manifolds, with

d \geq 1

. Suppose

x

is an isolated point of the fiber over

y = f (x)

. Choose an open neighborhood

U

with

U \cap f^{- 1} (y) = {x}

. The map of pairs

(U, U ∖ {x}) ⟶ (N, N ∖ {y})

induces a map between rank-one degree-

d

local homology groups. Relative to the local orientation of a manifold, this map is multiplication by an integer, the local degree. Excision makes it independent of the neighborhood. A smooth map between manifolds with invertible derivative at

x

has local degree

sign det D f_{x}

, by the inverse function theorem and the chosen orientations.

Degree as a sum of local degrees ( $de g f = \sum_{x \in f^{- 1} (y)} de g_{x} f$ )

For a continuous map

f : M \to N

of closed connected oriented

d

-dimensional manifolds, suppose

f^{- 1} (y) = {x_{1}, \dots, x_{s}}

is finite. Then

de g f = \sum_{j = 1}^{s} de g_{x_{j}} f .

By excision, the degree-

d

relative homology group

H_{d} (M, M ∖ f^{- 1} (y); Z)

is the direct sum of the local groups at

x_{j}

. The fundamental class maps to the tuple of local orientation generators. Mapping this tuple to

H_{d} (N, N ∖ {y}; Z)

adds their local degrees. Naturality identifies the result with the image of

(de g f) [N]

. If the fiber is empty, the same argument gives degree zero.

Degree under suspension

f : S^{m} \to S^{m}

, then its suspension

Σ f : S^{m + 1} \to S^{m + 1}

has the same degree. This follows from naturality of the suspension isomorphism in reduced homology.

Sphere maps of arbitrary integer degree ( $f_{n, m} = S^{n - 1} (z \mapsto z^{m}), n \geq 1$ )

Every integer

m

is the mapping degree of a continuous map

S^{n} \to S^{n}

in every dimension

n \geq 1

. Start with the circle map

z \mapsto z^{m}

, whose argument winds

m

times, and use degree under suspension repeatedly. Negative

m

reverses the winding, and

m = 0

is constant before suspending. The degrees of maps of the zero-sphere are a genuine exceptional case.

Degree of a map between oriented manifolds ( $de g f$ )

For a map

f : M \to N

between closed connected oriented

d

-manifolds, its degree is the integer determined by

f_{*} [M] = (de g f) [N]

in top-dimensional homology.

Multiplicativity of mapping degree ( $de g (g \circ f) = de g g de g f$ )

For maps of oriented connected closed manifolds of the same dimension, apply functoriality of homology to

f_{*} [M] = (de g f) [N]

and

g_{*} [N] = (de g g) [L]

. The composite coefficient is their product. This recovers the multiplication of signed sheet counts for compositions of finite covering maps.

Homotopy invariance of mapping degree ( $f_{0} ≃ f_{1} ⟹ de g f_{0} = de g f_{1}$ )

For maps between oriented connected closed manifolds of the same dimension, a homotopy induces equal maps on top-dimensional homology, hence equal coefficients on the fundamental class. For smooth maps and a smooth homotopy, the degree by integration of a pullback volume form gives another proof: the Generalized Stokes theorem on the homotopy cylinder makes the difference of endpoint pullback integrals zero because the target top form is closed. This is an obstruction to extending a degree-one sphere identity over a ball.

Degree does not classify general manifold maps

Although topological degree classifies maps

S^{n} \to S^{n}

up to homotopy, it is not a complete invariant for other manifolds. On the torus, the identity and the integer shear

(1011)

both have degree one, but induce distinct maps on

H_{1} (T^{2}; Z)

and cannot be homotopic.

Degree by integration of a pullback volume form ( $de g f = \int_{M} f^{*} ω, \int_{N} ω = 1$ )

For a smooth map

f : M \to N

between oriented connected closed manifolds of the same dimension, a normalized volume form on the target gives

de g f = \int_{M} f^{*} ω

. The formula agrees with the degree as a sum of local degrees. A change of normalized top-form adds an exact form by top-dimensional de Rham cohomology, whose pullback has zero integral by Stokes theorem.

Spherical degree by area pullback ( $de g g = \frac{1}{4 π} \int_{S^{2}} g^{*} ω$ )