Prove a Continuous Map Extends if and Only if F is Null Homotopic

Homotopic Map

Consequently every map homotopic to f has at least two fixed points, even though N(f) = 0, so P does not have the Wecken property.

From: History of Topology , 1999

Homotopy Theories and Model Categories

W.G. Dwyer , J. Spalinski , in Handbook of Algebraic Topology, 1995

PROOF OF 9.3

By Lemma 9.4, F identifies right homotopic maps between cofibrant objects of C and so induces a functor F′ : πC _c → D. By assumption, if g is a morphism of πC _c , which is represented by a weak equivalence in C then F′(g) is an isomorphism. Recall from 5.2 that there is a functor Q : C → πC _c with the property (5.1) that if f is a weak equivalence in C then g = Q(f) is a right homotopy class which is represented by a weak equivalence in C. It follows that the composite functor F′Q carries weak equivalences in C into isomorphisms in D. By the universal property (6.2) of Ho(C), the composite F′Q induces a functor Ho(C) → D, which we denote LF. There is a natural transformation t : (LF)γ → F which assigns to each X in C the map F(pX): LF(X) = F(QX) → F(X). If X is cofibrant then QX = X and the map t_X is the identity; in particular, t_X is an isomorphism.

We now have to show that the pair (LF, t) is universal from the left in the sense of 9.1. Let G : Ho(C) → D be a functor and s′ : Gγ → F a natural transformation. Consider a hypothetical natural transformation s′ : G → LF, and construct (for each object X of C) the following commutative diagram which in the horizontal direction involves the composite of s′ ˆ γ and t;

If s′ is to satisfy the condition of 9.1, then the composite across the top row of this diagram must be equal to s_QX , which gives the equality s′ _X = s _QX G(γpX)⁻¹ and proves that there is at most one natural transformation s′ which satisfies the required condition. However, it is obvious that setting s′ _X = s_QXG((γpX)⁻¹ does give a natural transformation Gγ → (LF)γ, and therefore (5.9) it also gives a natural transformation G → LF.

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Development of the Concept of Homotopy

Ria Vanden Eynde , in History of Topology, 1999

3.5 Homotopy in Brouwer's papers

In Brouwer's paper [7] we can find implicit occurrences of what we now call homotopic maps. This idea became more explicit in 1912 [8]. Then, Brouwer considers two continuous transformations of a surface into itself which can be continuously transformed into one another. The continuous modification of a continuous univalent transformation is described as follows (p. 527):

By a continuous modification of a univalent continuous transformation we understand in the following always the construction of a continuous series of univalent continuous transformations, i.e. a series of transformations depending in such a manner on a parameter, that the position of an arbitrary point is a continuous function of its initial position and the parameter.

The explicit statement which says that the position of a point is a continuous function of its initial position and the parameter is the mathematically rigorous definition of the deformation process. It marks the transition of the intuitive understanding of this process to a rigorously defined concept and allows for the extension of the homotopy concept from paths to maps in general. Tietze had already described this dependency in 1908, but in a narrower context: the "Gleichartigkeit" of homeomorphisms of manifolds onto themselves (we now call this isotopy). In [8] transformations which can be obtained from one another by continuous modification are said to belong to the same class. As Freudenthal indicated [8] it is the first time that the term class is used in the sense of homotopy class. It is in these papers that Brouwer also introduced simplicial maps ("simpliziale Abbildung") and simplicial approximation ("modifizierte simpliziale Abbildung", later called simplicial approximation). If a continuous map is given which maps a polyhedron into another one, a simplicial approximation allows the map to be replaced by a map which, perhaps after a subdivision of the polyhedra, is "sufficiently near to it" and is simplicial, i.e. it maps simplexes onto simplexes by "piecewise linear maps". Seen from a methodological point of view the idea of approximating a map by a "piecewise linear map", and not by a map which is piecewise constant as in the case in analysis, was an important new step. These new ideas were used by Brouwer to introduce the "degree" of a map (calculating the algebraic sum of the number of times a point is covered by its image, the sign being determined by the reversal or the preservation of the orientation) and to prove that homotopy classes of maps of the 2-sphere in itself are characterized by their degree. This result sets off the search for characterizing or counting the mapping classes of maps of higher dimensional spheres.

Also in 1912, Brouwer gives a formal description of freely homotopic closed paths in a paper [9] in which he proves the topological invariance of closed plane curves as Schoenflies defined them. According to Schoenflies's [80] definition, a closed curve in the plane is a perfect bounded connected plane set which divides the plane in two regions of which it is the common boundary. As Freudenthal [8] mentions, Brouwer proves a broader result, namely the invariance of the number of domains determined by a bounded connected closed planar point set. Brouwer considers a bounded (h + 1)-foldly connected plane region ("Gebiet") g and an arbitrary point P in g. For such "Gebiete", he knows that there are h simple closed fundamental curves c ₁, c ₂,…, c _h through P such that any continuous closed curve σ (i.e. the continuous image of a circle) can be deformed continuously in g into a finite composition of the curves c _v (p. 523):

Let g be a bounded (h + 1)-foldly connected plane region, P any point on g. We can choose h, simple closed curves c ₁, c ₂,…, c _h through P, which are to be considered as fundamental curves, and which only intersect in P and have the property that each closed continuous curve σ in g can be transformed by continuous deformation in g into a canonic continuous curve ("kanonische stetige Kurve") φ consisting of a finite number of the curves c ^** _v .

^**This continuous deformation means that a plane region which is bounded by two concentric circles k ₁ and k ₂ can be continuously mapped in g such that σ corresponds to k ₁, and φ corresponds to k ₂.

This continuous deformation ("stetiger Abänderung") is an example of free homotopy. We now know that the fundamental group of the region g is the free group generated by h closed paths c ₁, c ₂,…, c _h each of which goes once round one of the h bounded regions determined by g. Since the fundamental group is a topological invariant the number of regions determined by g (number of generators of the fundamental group) does not change when a homeomorphism is applied to g. But it is exactly the construction of the generators and of the homotopic deformation which is most difficult. One has to define continuous maps α _i from the circle C into g for the paths c _i and for any given path σ one has to determine a continuous map F : C × [0, 1] → g : (x, t) → F(x, t). As seen above in Poincaré's and Tietze's work this problem was avoided by using what we now call a cell decomposition of the manifold under consideration. To define the fundamental group Tietze used a generator for each edge and the relations were obtained from running round the circumference of the 2-cells. Poincaré used what we now call the universal covering manifold and its automorphisms to justify an analogous method. If the fundamental group is defined in this way it has to be proved that the choice of a cell decomposition is irrelevant. Brouwer uses the concept of chains ("Ketten"), a concept used by Cantor [86] in the context of infinite plane sets. Brouwer describes them as follows (p. 523):

By a chain we mean a finite, cyclic ordered set of points, by an ɛ-chain a chain for which the distance between two subsequent points is smaller than ɛ.

He also introduces modifications of these chains (p. 523):

By an ɛ'-modification of a chain χ we understand firstly a displacement < ɛ' of a point of χ, by way of which χ goes over into an ɛ'-chain, secondly an addition of a new point between two subsequent points of χ, by way of which χ goes over into an ɛ'-chain. If two subsequent points of a chain coincide, they are considered as the same point. This way, after an ɛ'-modification, the number of points in the chain remains constant, increases with one or decreases with one.

Brouwer proves that for each ɛ > 0 small enough there exists an η > 0 and h ɛ-chains a _i such that to every ɛ-chain χ in π, a finite composition of the a _i can be associated which can be obtained from χ by ɛ-modifications. This is a property which is invariant under homeomorphisms, a result which follows from what we now call the uniform continuity of such maps on compact subsets of the plane. Brouwer's discussion is in terms of a metric space (the Euclidean plane) and the properties of continuous maps into such spaces. The result is put in the context of set theory. The use of ɛ-chains as introduced by Cantor [10], although he does not call them ɛ-chains yet, also reflects this. He thus obtains his result without having to rely on a cell decomposition of the set π; the existence of such a decomposition was proved later in 1925 by Rado [73]. Brouwer's paper also reflects the status of group theoretic thinking at that time. The ɛ₁-chains (more precisely the closed "Streckenzüge" determined by them) in g _ɛ through P which can be deformed into each other by ɛ-modifications can be considered as representing one element of a group, if during the modifications all intermediate "Züge" pass through P. This group is isomorphic to the fundamental group of g _ɛ. The fact that Brouwer uses free homotopy reflects that he did not think of using group theoretic terminology.

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CONNECTIONS ON A PRINCIPAL FIBRE BUNDLE

YVONNE CHOQUET-BRUHAT , CÉCILE DEWITT-MORETTE , in Analysis, Manifolds and Physics, 2000

2 UNIVERSAL BUNDLE

The statement in lb shows that there is a way to classify the bundles with given group G, base M and fibre F if they can all be obtained as pull back of some "universal" bundle, with group G, fibre F and base some space X, by using the homotopy classes of maps M → X.

Show that if a principal bundle $U_G=(U,G,X)$ , with group G and base X is arcwise connected and such that its first n homotopy groups vanish:

(1) ${\pi }_1\left(U\right)={\pi }_{2}U=···={\pi }_n\left(U\right)=0$

then every principal bundle with group G and base a paracompact manifold of dimension ⩽ n can be obtained as the pull back of $U_G$ by a mapping f: M → X.

Such a bundle $U_G$ is called n-universal . Since the pulled back bundles are equivalent for homotopic maps M → X these maps are called classifying maps.

n-universal

classifying maps

Answer 2

We shall give an idea of the construction of the homomorphism $\tilde{h}:B\to U_G$ using (1).

The mapping $\tilde{h}$ , which will have to be continuous, can always be defined on fibres over isolated points of M, basis of B, by setting

$\tilde{h}|G_x=\zeta {\xi }^{-1},$

where ξ: G → G _x and ζ: G → G _y are respectively mappings from G onto the fibre G _x over x ∈ M and onto the fibre G _y over some point y ∈ X, which commute with the right action of G on the considered fibres.

Such an $\tilde{h}$ can be considered as a homomorphism from the restriction of B to fibres over the 0-cells of a complex decomposition (triangulation) of M (cf. Problem IV 1, Cohomology) into $U_G$ . We shall first show that $\tilde{h}$ can be extended to the restriction of B over the 1-cells.

Let D ₁ be a 1-cell in M; since D ₁ is contractible (homeomorphic to [0, 1]) the portion of B over D ₁ is equivalent to the trivial bundle D ₁ x G, and the restriction of B over ∂D ₁ is equivalent to ∂D ₁ x G. The boundary ∂D ₁ consists of two points x ₀, x ₁ ∈ M, above which the map $\tilde{h}$ has been defined, i.e., we know $\tilde{h}{|}_{\partial D_1\times G}:\partial D_1\times G\to U_G$ . We define f to be the mapping $\partial D_1\to U_G$ defined by $f(x)=\tilde{h}(x,e),x\in \partial D_1$ . Since $U_G$ is arcwise connected, f extends to a (continuous) map still denoted $f:D_1\to U_G$ ,

We define the map $h:D_1\times G\to U_G$ by

$\tilde{h}\left(x,g\right)=P\left(f\left(x\right),g\right),$

where P is the mapping

$\begin{array}{ccc}{U}_G\times G\to U_G& \text{by}& \left(y,g\right)\mapsto {\tilde{R}}_{g}y\end{array}.$

It is straightforward to check that $\tilde{h}$ takes the preassigned values on ∂D ₁ x G and is a homeomorphism $D_1\times G\to U_G$ .

Analogous reasoning will give the conclusion by induction for k cells, k ⩽ n. The boundary ∂D _k of a k-cell D _k is homeomorphic to a k − 1 sphere S _k−1 sphere, and maps from ∂D _k into $U_G$ can be extended to D _k because π _k−1 = 0.

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Characteristic Classes

P.B. Gilkey , ... S. Nikčević , in Encyclopedia of Mathematical Physics, 2006

Homotopy

Two smooth maps f ₀ and f ₁ from N to M are said to be homotopic if there exists a smooth map F: N × I → M so that f ₀(P) = F(P, 0) and so that f ₁(P) = F(P, 1). If f ₀ and f ₁ are homotopic maps from N to M, then $f_1^{*}V$ is isomorphic to $f_2^{*}V$ .

Let [N, M] be the set of all homotopy classes of smooth maps from N to M. The association V → f*V induces a natural map

$[N,M]\times {\text{Vect}}_k(M,\mathbb{F})\to {\text{Vect}}_k(N,\mathbb{F})$

If M is contractible, then the identity map is homotopic to the constant map c. Consequently, V = Id*V is isomorphic to $c*V=M\times {\mathbb{F}}^k$ . Thus, any vector bundle over a contractible manifold is trivial. In particular, if $\left\{{\mathcal{O}}_{\alpha }\right\}$ is a simple cover of M and if $V\in \text{Vect}\left(M,\mathbb{F}\right)$ , then $V|{\mathcal{O}}_{\alpha }$ is trivial for each α. This shows that a simple cover is a trivializing cover for every $V\in \text{Vect}\left(M,\mathbb{F}\right)$ .

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The Stable Homotopy Theory of Finite Complexes

Douglas C. Ravenel , in Handbook of Algebraic Topology, 1995

DEFINITION 2.2.1

A covariant functor F from the category of topological spaces $T$ to some algebraic category $A$ (such as that of groups, rings, modules, etc.) is a function which assigns to each space X an object F(X) in $A$ and to each continuous map f: X → Y a homomorphism F(f): F(X) → F(Y) in such a way that F(f_g ) = F(f)F(g) and F sends identity maps to identity homomorphisms. A contravariant functor G is similar function which reverses the direction of arrows, i.e. G(f) is a homomorphism from G(Y) to G(X) instead of the other way around. In either case a functor is homotopy invariant if it takes isomorphic values on homotopy equivalent spaces and sends homotopic maps to the same homomorphism.

Familiar examples of such functors include ordinary homology, which is covariant and cohomology, which is contravariant. Both of these take values in the category of graded abelian groups. Definitions of them can be found in any textbook on algebraic topology. We will describe some less familiar functors which have proved to be extremely useful below.

These functors are typically used to prove that some geometric construction does not exist. For example one can show that the 2-sphere S ² and the torus T ² (doughnut-shapedsurface) are not homeomorphic by computing their homology groups and observing that they are not the same.

Each of these functors has that property that if the continuous map f is null homotopic then the homomorphism F(f) is trivial, but the converse is rarely true. Some of the best theorems in the subject concern special situations where it is. One such result is the nilpotence theorem (2.4.2), which is the main subject of this chapter.

Other results of this type in the past decade concern cases where at least one of the spaces is the classifying space of a finite or compact Lie group. A comprehensive book on this topic has yet to be written. A good starting point in the literature is the J.F. Adams issue of Topology (Vol. 31, No. 1, January 1992), specifically [16], [23], [32], [46], and [10].

The dream of every homotopy theorist is a solution to the following.

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K-Theory

V. Mathai , in Encyclopedia of Mathematical Physics, 2006

Some of the basic properties of K-theory are listed as follows. Details can be found in Karoubi (1978).

1.

PullbackIf f:N→M is a continuous map, then given a vector bundle π:E→M over M, the pullback vector bundle is defined as f*(E)={(x,v)∈N×E:f(x)=π(v)} over N. This induces a pullback homomorphism, f ^!:K ^•(M)→K ^•(N).

2.

Push-forward Let f:N→M be a smooth proper map between compact manifolds which is K-oriented, that is, TN⊕f*TM is a ${\text{spin}}^{\mathbb{C}}$ vector bundle over N. Then there is a pushforward homomorphism, also called a Gysin map, f _!:K ^•(N)→K ^•+d (M). where d=dimM− dimN, whose construction will be explained in the next section.

3.

HomotopyIf f:N→M and g:N→M are homotopic maps, then the pullback maps f ^!=g ^! are equal. If in addition, f and g are K-oriented, proper maps which are homotopic via proper maps, then the Gysin maps f _!=g _! are equal.

4.

ExcisionLet M ₁ be a closed subset of M and U be an open subset of M such that U is contained in the interior of M ₁. Then the inclusion of pairs (M\U,M ₁\U)↪(M,M ₁) induces an isomorphism in K-theory, K ^•(M,M ₁)≅K ^•(M\U,M ₁\U).

5.

ExactnessLet M ₁ be a closed subset of M. Then there is a six-term exact sequence in K-theory,

6.

Cup productThere is a canonical map given by external tensor product, K ⁱ (M)⊗K ^j (N)→K ^i+j (M×N). When N=M, one can compose this with the homomorphism induced by the diagonal map M→M×M given by x→(x,x), to get a cup product, K ^p (M)⊗K ^q (M)→K ^p+q (M).

7.

Bott periodicityThis is arguably the most important property of K-theory. It says that the zero-section embedding ${\iota }^M:M\hookrightarrow M\times {\mathbb{R}}^n$ induces a Gysin isomorphism, ${\iota }^M!:K^{•}\left(M\right)\stackrel{\cong }{\to }{K}^{•+n}\left(M\times {\mathbb{R}}^n\right)$ , which is given as follows. Let ${\pi }_M:M\times {\mathbb{R}}^n\to M$ and ${\pi }_{{\mathbb{R}}^n}:M\times {\mathbb{R}}^n\to R^n$ denote the projections onto the factors, and $b={\iota }_{!}1\in K^n\left({\mathbb{R}}^n\right)$ the Bott element, where $\iota \{0\}\hookrightarrow {\mathbb{R}}^n$ is the inclusion of the origin. Then the Bott periodicity isomorphism is given by ${\iota }^M!(x)={\pi }_M^{!}(x)\cup {\pi }_{{\mathbb{R}}^n}^{!}(b)\in K^{•+n}\left(M\times {\mathbb{R}}^n\right)$ for all x∈K ^•(M).

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Cohomology Theories

U. Tillmann , in Encyclopedia of Mathematical Physics, 2006

Cohomology of Smooth Manifolds

A smooth manifold M of dimension n can be triangulated by smooth simplices $\sigma :{\text{Δ}}^n\to M$ . If M is compact, oriented, without boundary, the sum of these simplices define a homology cycle [M], the fundamental class of M. The most remarkable property of the cohomology of manifolds is that they satisfy "Poincaré duality": taking cap product with [M] defines an isomorphism:

[14] $D:=[M]\cap :H^k(M;\mathbb{Z}){\displaystyle \stackrel{\simeq }{\to }}{H}_{n-k}(M;\mathbb{Z})\text{for all}k$

In particular, for connected manifolds, $H^n\left(M;\mathbb{Z}\right)\simeq \mathbb{Z}$ ; and every map $f:M\prime \to M$ between oriented, compact closed manifolds of the same dimension has a degree: $f*:H*(M;\mathbb{Z})\to H*(M\prime ;\mathbb{Z})$ is multiplication by an integer deg(f  ), the degree of f. For smooth maps, the degree is the number of points in the inverse image of a generic point $p\in M$ counted with signs:

$\text{deg}(f)={\displaystyle \sum _{p\prime \in f^{-1}(p)}}\text{sign}(p\prime )$

where $\text{sign}(p\prime )$ is +1 or −1 depending on whether f is orientation preserving or reversing in a neighborhood of p′. For example, a complex polynomial of degree d defines a map of the two-dimensional sphere to itself of degree d: a generic point has n points in its inverse image and the map is locally orientation preserving. On the other hand, a map of $S^{n-1}$ induced by a reflection of ${\mathbb{R}}^n$ reverses orientation and has degree −1. Thus, as degrees multiply on composing maps, the antipodal map $x\mapsto -x$ has degree ${\left(-1\right)}^n$ . As an application we prove:

Every tangent vector field on an even-dimensional sphere Sⁿ⁻¹ has a zero.

Proof

Assume v(x) is a vector field which is nonzero for all $x\in S^{n-1}$ . Then x is perpendicular to v(x), and after rescaling, we may assume that v(x) has length 1. The function $F\left(x,t\right)=\text{cos}\left(t\right)x+\text{sin}\left(t\right)v\left(x\right)$ is a well-defined homotopy from the identity map $\left(t=0\right)$ to the antipodal map $\left(t=\pi \right)$ . But this is impossible as homotopic maps induce the same map in (co)homology and we have already seen that the degree of the identity map is 1 while the degree of the antipodal map is ${\left(-1\right)}^n=-1$ when n is odd.

•

It is well known that two self-maps of a sphere of any dimension are homotopic if and only if they have the same degree, that is, ${\pi }_n\left(S^n\right)\simeq \mathbb{Z}$ for $n\ge 1$ .

•

When M is not orientable, [M] still defines a cycle in homology with ${\mathbb{Z}}_2$ -coefficients, and $\left[M\right]\cap$ defines an isomorphism between the cohomology and homology with ${\mathbb{Z}}_2$ coefficients.

•

As [M] represents a homology class, so does every other closed (orientable) submanifold of M. It is however not the case that every homology class can be represented by a submanifold or linear combinations of such.

Cohomology is a contravariant functor. Poincaré duality however allows us to define, for any $f:M\prime \to M$ between oriented, compact, closed manifolds of arbitrary dimensions, a "transfer" or "Umkehr map,"

$f^{!}:=D^{-1}{f}_{*}D\prime :H*(M\prime ;\mathbb{Z})\to H^{*-c}(M;\mathbb{Z})$

which lowers the degree by $c=\text{dim}M\prime -\text{dim}M$ . It satisfies the formula

$f^{!}\left(f*\left(x\right)\cup y\right)=x\cup f^{!}\left(y\right)$

for all $x\in H*\left(M;\mathbb{Z}\right)$ and $y\in H*(M\prime ;\mathbb{Z})$ . When f is a covering map then $f^{!}$ can be defined on the chain level by

$f^{!}(x)(\sigma ):=x\left({\displaystyle \sum _{f({\displaystyle \tilde{\sigma }})=\sigma }}{\displaystyle \tilde{\sigma }}\right)$

where $x\in C*(M\prime )$ and $\sigma \in C_{*}\left(M\right)$ .

de Rham Cohomology

If $x_1,\dots ,x_n$ are the local coordinates of ${\mathbb{R}}^n$ , define an algebra Ω* to be the algebra generated by symbols $\text{d}{x}_1,\dots ,\text{d}{x}_n$ subject to the relations $\text{d}{x}_i\text{d}{x}_j=-\text{d}{x}_j\text{d}{x}_i$ for all i, j. We say $\text{d}{x}_{i_1}\cdots \text{d}{x}_{i_q}$ has degree q. The differential forms on ${\mathbb{R}}^n$ are the algebra

$\text{Ω}*\left({\mathbb{R}}^n\right):=\left\{C^{\infty }\text{functions on}{\mathbb{R}}^n\right\}{\otimes }_{\mathbb{R}}\text{Ω}*$

The algebra $\text{Ω}*({\mathbb{R}}^n)={\oplus }_{q=0}^n{\text{Ω}}^q({\mathbb{R}}^n)$ is naturally graded by degree. There is a differential operator $d:{\text{Ω}}^q\left({\mathbb{R}}^n\right)\to {\text{Ω}}^{q+1}\left({\mathbb{R}}^n\right)$ defined by

1.

if $f\in {\text{Ω}}^0\left({\mathbb{R}}^n\right)$ , then $\text{d}f=\sum \left(\partial f/\partial x_i\right)\text{d}{x}_i$

2.

if $\omega =\sum f_I\text{d}{x}_I$ , then $\text{d}\omega =\sum \text{d}{f}_I\text{d}{x}_I$

I stands here for a multi-index. For example, in ${\mathbb{R}}^3$ the differential assigns to 0-forms (=functions) the gradient, to 1-forms the curl, and to 2-forms the divergence. An easy exercise shows that $d^2=0$ and the qth de Rham cohomology of ${\mathbb{R}}^n$ is the vector space

$H_{\text{de R}}^q\left({\mathbb{R}}^n\right)=\frac{\text{ker}d:{\text{Ω}}^q\left({\mathbb{R}}^n\right)\to {\text{Ω}}^{q+1}\left({\mathbb{R}}^n\right)}{\text{im}d:{\text{Ω}}^{q-1}\left({\mathbb{R}}^n\right)\to {\text{Ω}}^q\left({\mathbb{R}}^n\right)}$

More generally, the de Rham complex $\text{Ω}*\left(M\right)$ and its cohomology $H_{\text{de R}}^{*}\left(M\right)$ can be defined for any smooth manifold M.

Let σ be a smooth, singular, real $\left(q+1\right)$ -chain on M, and let $\omega \in {\text{Ω}}^q\left(M\right)$ . Stokes theorem then says

${\int }_{\partial \sigma }\omega ={\int }_{\sigma }\text{d}\omega$

and therefore integration defines a pairing between the qth singular homology and the qth de Rahm cohomology of M. This pairing is exact and thus de Rahm cohomology is isomorphic to singular cohomology with real coefficients:

$H_{\text{de R}}^{*}\left(M\right)\simeq \left(H_{*}\left(M;\mathbb{R}\right)\right)*\simeq H*\left(M;\mathbb{R}\right)$

Let ${\text{Ω}}_c^{*}\left(M\right)$ denote the subcomplex of compactly supported forms and $H_c^{*}\left(M\right)$ its cohomology. Integration with respect to the first i coordinates defines a map

${\text{Ω}}_c^{*}\left({\mathbb{R}}^n\right)\to {\text{Ω}}_c^{*-i}\left({\mathbb{R}}^{n-i}\right)$

which induces an isomorphism in cohomology; note in particular $H_c^n\left({\mathbb{R}}^n\right)=\mathbb{R}$ . More generally, when $E\to M$ is an i-dimensional orientable, real vector bundle over a compact, orientable manifold M, integration over the fiber gives the "Thom isomorphism":

$H_c^{*}\left(E\right)\simeq H_c^{*-i}\left(M\right)\simeq H_{\text{de R}}^{*-i}\left(M\right)$

For orientable fiber bundles $F\to M\prime {\displaystyle \stackrel{f}{\to }}M$ with compact, orientable fiber F, integration over the fiber provides another definition of the transfer map

$f^{!}:H_{\text{de R}}^{*}(M\prime )\to H_{\text{de R}}^{*-i}(M)$

Hodge Decomposition

Let M be a compact oriented Riemannian manifold of dimension n. The Hodge star operator, *, associates to every q-form an $\left(n-q\right)$ -form. For ${\mathbb{R}}^n$ and any orthonormal basis $\{e_1,\dots ,e_n\}$ , it is defined by setting

$*(e_1\wedge \cdots \wedge e_q):=\pm e_{p+1}\wedge \cdots \wedge e_n$

where one takes + if the orientation defined by $\{e_1,\dots ,e_n\}$ is the same as the given one, and − otherwise. Using local coordinate charts this definition can be extended to M. Clearly, * depends on the chosen metric and orientation of M. If M is compact, we may define an inner product on the q-forms by

$(\omega ,\omega \prime ):={\int }_M\omega \wedge *\omega \prime$

With respect to this inner product * is an isometry. Define the codifferential via

$\delta :={(-1)}^{\text{<mi mathvariant="italic" is="true">np</mi>}+n+1}*d*:{\text{Ω}}^q(M)\to {\text{Ω}}^{q-1}(M)$

and the Laplace–Beltrami operator via

$\text{Δ}:=\delta d+d\delta$

The codifferential satisfies ${\delta }^2=0$ and is the adjoint of the differential. Indeed, for q-forms ω and $\left(q+1\right)$ -forms $\omega \prime$ :

[15] $(d\omega ,\omega \prime )=(\omega ,\delta \omega \prime )$

It follows easily that Δ is self-adjoint, and furthermore,

[16] $\text{Δ}\omega =0\text{if and only if}d\omega =0\text{and}\delta \omega =0$

A form ω satisfying $\text{Δ}\omega =0$ is called "harmonic." Let ${\mathcal{H}}^q$ denote the subspace of all harmonic q-forms. It is not hard to prove the "Hodge decomposition theorem":

${\text{Ω}}^q={\mathcal{H}}^q\oplus \text{im}d\oplus \text{im}\delta$

Furthermore, by adjointness [15], a form ω is closed only if it is orthogonal to im δ. On calculating the de Rham cohomology we can also ignore the summand im d and find that:

Each de Rham cohomology class on a compact oriented Riemannian manifold M contains a unique harmonic representative, that is, $H_{\text{de R}}^q\left(M\right)\simeq {\mathcal{H}}^q$ .

Warning:This is an isomorphism of vector spaces and in general does not extend to an isomorphism of algebras.

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Source: https://www.sciencedirect.com/topics/mathematics/homotopic-map

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