Grassmannian

In mathematics, the Grassmannian Gr(k, V) is a space that parameterizes all k-dimensional linear subspaces of the n-dimensional vector space V. For example, the Grassmannian Gr(1, V) is the space of lines through the origin in V, so it is the same as the projective space of one dimension lower than V.

For other uses, see Grassmannian (disambiguation).

When V is a real or complex vector space, Grassmannians are compact smooth manifolds.[1] In general they have the structure of a smooth algebraic variety, of dimension ${\displaystyle k(n-k).}$

The earliest work on a non-trivial Grassmannian is due to Julius Plücker, who studied the set of projective lines in projective 3-space, equivalent to Gr(2, R⁴) and parameterized them by what are now called Plücker coordinates. Hermann Grassmann later introduced the concept in general.

Notations vary between authors, with Gr_k(V) being equivalent to Gr(k, V), and some authors using Gr_k(n) or Gr(k, n) to denote the Grassmannian of k-dimensional subspaces of an unspecified n-dimensional vector space.

Motivation

By giving a collection of subspaces of some vector space a topological structure, it is possible to talk about a continuous choice of subspace or open and closed collections of subspaces; by giving them the structure of a differential manifold one can talk about smooth choices of subspace.

A natural example comes from tangent bundles of smooth manifolds embedded in Euclidean space. Suppose we have a manifold M of dimension k embedded in Rⁿ. At each point x in M, the tangent space to M can be considered as a subspace of the tangent space of Rⁿ, which is just Rⁿ. The map assigning to x its tangent space defines a map from M to Gr(k, n). (In order to do this, we have to translate the tangent space at each x ∈ M so that it passes through the origin rather than x, and hence defines a k-dimensional vector subspace. This idea is very similar to the Gauss map for surfaces in a 3-dimensional space.)

This idea can with some effort be extended to all vector bundles over a manifold M, so that every vector bundle generates a continuous map from M to a suitably generalised Grassmannian—although various embedding theorems must be proved to show this. We then find that the properties of our vector bundles are related to the properties of the corresponding maps viewed as continuous maps. In particular we find that vector bundles inducing homotopic maps to the Grassmannian are isomorphic. Here the definition of homotopic relies on a notion of continuity, and hence a topology.

Low dimensions

For k = 1, the Grassmannian Gr(1, n) is the space of lines through the origin in n-space, so it is the same as the projective space of n − 1 dimensions.

For k = 2, the Grassmannian is the space of all 2-dimensional planes containing the origin. In Euclidean 3-space, a plane containing the origin is completely characterized by the one and only line through the origin that is perpendicular to that plane (and vice versa); hence the spaces Gr(2, 3), Gr(1, 3), and P² (the projective plane) may all be identified with each other.

The simplest Grassmannian that is not a projective space is Gr(2, 4), which may be parameterized via Plücker coordinates.

The geometric definition of the Grassmannian as a set

Let V be an n-dimensional vector space over a field K. The Grassmannian Gr(k, V) is the set of all k-dimensional linear subspaces of V. The Grassmannian is also denoted Gr(k, n) or Gr_k(n).

The Grassmannian as a differentiable manifold

To endow the Grassmannian Gr_k(V) with the structure of a differentiable manifold, choose a basis for V. This is equivalent to identifying it with V = Kⁿ with the standard basis, denoted ${\displaystyle (e_{1},\dots ,e_{n})}$, viewed as column vectors. Then for any k-dimensional subspace w ⊂ V, viewed as an element of Gr_k(V), we may choose a basis consisting of k linearly independent column vectors ${\displaystyle (W_{1},\dots ,W_{k})}$. The homogeneous coordinates of the element w ∈ Gr_k(V) consist of the components of the n × k rectangular matrix W of maximal rank whose ith column vector is ${\displaystyle W_{i},\quad i=1,\dots ,k}$. Since the choice of basis is arbitrary, two such maximal rank rectangular matrices W and ${\displaystyle {\tilde {W}}}$ represent the same element w ∈ Gr_k(V) if and only if ${\displaystyle {\tilde {W}}=Wg}$ for some element g ∈ GL(k, K) of the general linear group of invertible k × k matrices with entries in K.

Now we define a coordinate atlas. For any n × k matrix W, we can apply elementary column operations to obtain its reduced column echelon form. If the first k rows of W are linearly independent, the result will have the form

${\displaystyle {\begin{bmatrix}1\\&1\\&&\ddots \\&&&1\\a_{1,1}&\cdots &\cdots &a_{1,k}\\\vdots &&&\vdots \\a_{n-k,1}&\cdots &\cdots &a_{n-k,k}\end{bmatrix}}.}$

The (n − k) × k matrix A = (a_ij) determines w. In general, the first k rows need not be independent, but for any W whose rank is ${\displaystyle k}$, there exists an ordered set of integers ${\displaystyle 1\leq i_{1}<\cdots <i_{k}\leq n}$ such that the submatrix ${\displaystyle W_{i_{1},\dots ,i_{k}}}$ consisting of the ${\displaystyle i_{1},\ldots ,i_{k}}$-th rows of W is nonsingular. We may apply column operations to reduce this submatrix to the identity, and the remaining entries uniquely correspond to w. Hence we have the following definition:

For each ordered set of integers ${\displaystyle 1\leq i_{1}<\cdots <i_{k}\leq n}$, let ${\displaystyle U_{i_{1},\dots ,i_{k}}}$ be the set of ${\displaystyle n\times k}$ matrices W whose k × k submatrix ${\displaystyle W_{i_{1},\dots ,i_{k}}}$ is nonsingular, where the jth row of ${\displaystyle W_{i_{1},\dots ,i_{k}}}$ is the i_jth row of W. The coordinate function on ${\displaystyle U_{i_{1},\dots ,i_{k}}}$ is then defined as the map ${\displaystyle A^{i_{1},\dots ,i_{k}}}$ that sends W to the (n − k) × k rectangular matrix whose rows are the rows of the matrix ${\displaystyle WW_{i_{1},\dots ,i_{k}}^{-1}}$ complementary to ${\displaystyle (i_{1},\dots ,i_{k})}$. The choice of homogeneous coordinate matrix W representing the element w ∈ Gr_k(V) does not affect the values of the coordinate matrix ${\displaystyle A^{i_{1},\dots ,i_{k}}}$ representing w on the coordinate neighbourhood ${\displaystyle U_{i_{1},\dots ,i_{k}}}$. Moreover, the coordinate matrices ${\displaystyle A^{i_{1},\dots ,i_{k}}}$ may take arbitrary values, and they define a diffeomorphism from ${\displaystyle U_{i_{1},\dots ,i_{k}}}$ onto the space of K-valued (n − k) × k matrices.

On the overlap

${\displaystyle U_{i_{1},\dots ,i_{k}}\cap U_{j_{1},\dots ,j_{k}}}$

of any two such coordinate neighborhoods, the coordinate matrix values are related by the transition relation

${\displaystyle A^{i_{1},\dots ,i_{k}}W_{i_{1},\dots ,i_{k}}=A^{j_{1},\dots ,j_{k}}W_{j_{1},\dots ,j_{k}},}$

where both ${\displaystyle W_{i_{1},\dots ,i_{k}}}$ and ${\displaystyle W_{j_{1},\dots ,j_{k}}}$ are invertible. Hence ${\displaystyle (U_{i_{1},\dots ,i_{k}},A^{i_{1},\dots ,i_{k}})}$ gives an atlas of Gr_k(V).

The Grassmannian as a homogeneous space

The quickest way of giving the Grassmannian a geometric structure is to express it as a homogeneous space. First, recall that the general linear group GL(V) acts transitively on the r-dimensional subspaces of V. Therefore, if H is the stabilizer of any of the subspaces under this action, we have

Gr(r, V) = GL(V)/H.

If the underlying field is R or C and GL(V) is considered as a Lie group, then this construction makes the Grassmannian into a smooth manifold. It also becomes possible to use other groups to make this construction. To do this, fix an inner product on V. Over R, one replaces GL(V) by the orthogonal group O(V), and by restricting to orthonormal frames, one gets the identity

Gr(r, n) = O(n)/(O(r) × O(n − r)).

In particular, the dimension of the Grassmannian is r(n − r).

Over C, one replaces GL(V) by the unitary group U(V). This shows that the Grassmannian is compact. These constructions also make the Grassmannian into a metric space: For a subspace W of V, let P_W be the projection of V onto W. Then

${\displaystyle d(W,W')=\lVert P_{W}-P_{W'}\rVert ,}$

where ||⋅|| denotes the operator norm, is a metric on Gr(r, V). The exact inner product used does not matter, because a different inner product will give an equivalent norm on V, and so give an equivalent metric.

If the ground field k is arbitrary and GL(V) is considered as an algebraic group, then this construction shows that the Grassmannian is a non-singular algebraic variety. It follows from the existence of the Plücker embedding that the Grassmannian is complete as an algebraic variety. In particular, H is a parabolic subgroup of GL(V).

The Grassmannian as a scheme

In the realm of algebraic geometry, the Grassmannian can be constructed as a scheme by expressing it as a representable functor.[2]

Representable functor

Let ${\displaystyle {\mathcal {E}}}$ be a quasi-coherent sheaf on a scheme S. Fix a positive integer r. Then to each S-scheme T, the Grassmannian functor associates the set of quotient modules of

${\displaystyle {\mathcal {E}}_{T}:={\mathcal {E}}\otimes _{O_{S}}O_{T}}$

locally free of rank r on T. We denote this set by ${\displaystyle \mathbf {Gr} (r,{\mathcal {E}}_{T})}$.

This functor is representable by a separated S-scheme ${\displaystyle \mathbf {Gr} (r,{\mathcal {E}})}$. The latter is projective if ${\displaystyle {\mathcal {E}}}$ is finitely generated. When S is the spectrum of a field k, then the sheaf ${\displaystyle {\mathcal {E}}}$ is given by a vector space V and we recover the usual Grassmannian variety of the dual space of V, namely: Gr(r, V^∗).

By construction, the Grassmannian scheme is compatible with base changes: for any S-scheme S′, we have a canonical isomorphism

${\displaystyle \mathbf {Gr} (r,{\mathcal {E}})\times _{S}S'\simeq \mathbf {Gr} (r,{\mathcal {E}}_{S'})}$

In particular, for any point s of S, the canonical morphism {s} = Spec(k(s)) → S, induces an isomorphism from the fiber ${\displaystyle \mathbf {Gr} (r,{\mathcal {E}})_{s}}$ to the usual Grassmannian ${\displaystyle \mathbf {Gr} (r,{\mathcal {E}}\otimes _{O_{S}}k(s))}$ over the residue field k(s).

Universal family

Since the Grassmannian scheme represents a functor, it comes with a universal object, ${\displaystyle {\mathcal {G}}}$, which is an object of

${\displaystyle \mathbf {Gr} \left(r,{\mathcal {E}}_{\mathbf {Gr} (r,{\mathcal {E}})}\right),}$

and therefore a quotient module ${\displaystyle {\mathcal {G}}}$ of ${\displaystyle {\mathcal {E}}_{\mathbf {Gr} (r,{\mathcal {E}})}}$, locally free of rank r over ${\displaystyle \mathbf {Gr} (r,{\mathcal {E}})}$. The quotient homomorphism induces a closed immersion from the projective bundle ${\displaystyle \mathbf {P} ({\mathcal {G}})}$:

${\displaystyle \mathbf {P} ({\mathcal {G}})\to \mathbf {P} \left({\mathcal {E}}_{\mathbf {Gr} (r,{\mathcal {E}})}\right)=\mathbf {P} ({\mathcal {E}})\times _{S}\mathbf {Gr} (r,{\mathcal {E}}).}$

For any morphism of S-schemes:

${\displaystyle T\to \mathbf {Gr} (r,{\mathcal {E}}),}$

this closed immersion induces a closed immersion

${\displaystyle \mathbf {P} ({\mathcal {G}}_{T})\to \mathbf {P} ({\mathcal {E}})\times _{S}T.}$

Conversely, any such closed immersion comes from a surjective homomorphism of O_T-modules from ${\displaystyle {\mathcal {E}}_{T}}$ to a locally free module of rank r.[3] Therefore, the elements of ${\displaystyle \mathbf {Gr} (r,{\mathcal {E}})(T)}$ are exactly the projective subbundles of rank r in

${\displaystyle \mathbf {P} ({\mathcal {E}})\times _{S}T.}$

Under this identification, when T = S is the spectrum of a field k and ${\displaystyle {\mathcal {E}}}$ is given by a vector space V, the set of rational points ${\displaystyle \mathbf {Gr} (r,{\mathcal {E}})(k)}$ correspond to the projective linear subspaces of dimension r − 1 in P(V), and the image of ${\displaystyle \mathbf {P} ({\mathcal {G}})(k)}$ in

${\displaystyle \mathbf {P} (V)\times _{k}\mathbf {Gr} (r,{\mathcal {E}})}$

is the set

${\displaystyle \{(x,v)\in \mathbf {P} (V)(k)\times \mathbf {Gr} (r,{\mathcal {E}})(k)\mid x\in v\}.}$

The Plücker embedding

Main article: Plücker embedding

The Plücker embedding is a natural embedding of the Grassmannian ${\displaystyle \mathbf {Gr} (k,V)}$ into the projectivization of the exterior algebra Λ^kV:

${\displaystyle \iota$ :\mathbf {Gr} (k,V)\to \mathbf {P} \left(\Lambda ^{k}V\right).}

Suppose that W is a k-dimensional subspace of the n-dimensional vector space V. To define ${\displaystyle \iota (W)}$, choose a basis {w₁, ..., w_k} of W, and let ${\displaystyle \iota (W)}$ be the wedge product of these basis elements:

${\displaystyle \iota (W)=[w_{1}\wedge \cdots \wedge w_{k}].}$

A different basis for W will give a different wedge product, but the two products will differ only by a non-zero scalar (the determinant of the change of basis matrix). Since the right-hand side takes values in a projective space, ${\displaystyle \iota }$ is well-defined. To see that ${\displaystyle \iota }$ is an embedding, notice that it is possible to recover W from ${\displaystyle \iota }$ as the span of the set of all vectors w such that ${\displaystyle w\wedge \iota (W)=0}$.

Plücker coordinates and the Plücker relations

The Plücker embedding of the Grassmannian satisfies some very simple quadratic relations called the Plücker relations. These show that the Grassmannian embeds as an algebraic subvariety of P(Λ^kV) and give another method of constructing the Grassmannian. To state the Plücker relations, fix a basis {e₁, ..., e_n} of V, and let W be a k-dimensional subspace of V with basis {w₁, ..., w_k}. Let (w_i1, ..., w_in) be the coordinates of w_i with respect to the chosen basis of V, let

$${\displaystyle {\mathsf {W}}={\begin{bmatrix}w_{11}&\cdots &w_{1n}\\\vdots &\ddots &\vdots \\w_{k1}&\cdots &w_{kn}\end{bmatrix}},}$$

and let {W₁, ..., W_n} be the columns of ${\displaystyle {\mathsf {W}}}$. For any ordered sequence ${\displaystyle 1\leq i_{1}<\cdots <i_{k}\leq n}$ of ${\displaystyle k}$ positive integers, let ${\displaystyle W_{i_{1},\dots ,i_{k}}}$ be the determinant of the ${\displaystyle k\times k}$ matrix with columns ${\displaystyle W_{i_{1}},\dots ,W_{i_{k}}}$. The set ${\displaystyle \{W_{i_{1},\dots ,i_{k}}:1\leq i_{1}<\cdots <i_{k}\leq n\}}$ is called the Plücker coordinates of the element ${\displaystyle W}$ of the Grassmannian (with respect to the basis {e₁, ..., e_n} of V). They are the linear coordinates of the image ${\displaystyle \iota (W)}$ of ${\displaystyle W}$ under the Plücker map, relative to the basis of the exterior power Λ^kV induced by the basis {e₁, ..., e_n} of V.

For any two ordered sequences ${\displaystyle 1\leq i_{1}<i_{2}\cdots <i_{k-1}\leq n}$ and ${\displaystyle 1\leq j_{1}<j_{2}\cdots <j_{k+1}\leq n}$ of ${\displaystyle k-1}$ and ${\displaystyle k+1}$ positive integers, respectively, the following homogeneous equations are valid and determine the image of Gr(k, V) under the Plücker embedding:

${\displaystyle \sum _{\ell =1}^{k+1}(-1)^{\ell }W_{i_{1},\dots ,i_{k-1},j_{\ell }}W_{j_{1},\dots ,{\widehat {j_{\ell }}},\dots j_{k+1}}=0,}$

where ${\displaystyle j_{1},\ldots ,{\widehat {j_{\ell }}},\ldots j_{k+1}}$ denotes the sequence ${\displaystyle j_{1},\ldots ,j_{k+1}}$ with the term ${\displaystyle j_{\ell }}$ omitted.

When dim(V) = 4, and k = 2, the simplest Grassmannian which is not a projective space, the above reduces to a single equation. Denoting the coordinates of P(Λ^kV) by W₁₂, W₁₃, W₁₄, W₂₃, W₂₄, W₃₄, the image of Gr(2, V) under the Plücker map is defined by the single equation

W₁₂W₃₄ − W₁₃W₂₄ + W₂₃W₁₄ = 0.

In general, however, many more equations are needed to define the Plücker embedding of a Grassmannian in projective space.[4]

The Grassmannian as a real affine algebraic variety

Let Gr(r, Rⁿ) denote the Grassmannian of r-dimensional subspaces of Rⁿ. Let M(n, R) denote the space of real n × n matrices. Consider the set of matrices A(r, n) ⊂ M(n, R) defined by X ∈ A(r, n) if and only if the three conditions are satisfied:

• X is a projection operator: X² = X.
• X is symmetric: X^t = X.
• X has trace r: tr(X) = r.

A(r, n) and Gr(r, Rⁿ) are homeomorphic, with a correspondence established by sending X ∈ A(r, n) to the column space of X.

Duality

Every r-dimensional subspace W of V determines an (n − r)-dimensional quotient space V/W of V. This gives the natural short exact sequence:

0 → W → V → V/W → 0.

Taking the dual to each of these three spaces and linear transformations yields an inclusion of (V/W)^∗ in V^∗ with quotient W^∗:

0 → (V/W)^∗ → V^∗ → W^∗ → 0.

Using the natural isomorphism of a finite-dimensional vector space with its double dual shows that taking the dual again recovers the original short exact sequence. Consequently there is a one-to-one correspondence between r-dimensional subspaces of V and (n − r)-dimensional subspaces of V^∗. In terms of the Grassmannian, this is a canonical isomorphism

Gr(r, V) ≅ Gr(n − r, V^∗).

Choosing an isomorphism of V with V^∗ therefore determines a (non-canonical) isomorphism of Gr(r, V) and Gr(n − r, V). An isomorphism of V with V^∗ is equivalent to a choice of an inner product, and with respect to the chosen inner product, this isomorphism of Grassmannians sends an r-dimensional subspace into its (n − r)-dimensional orthogonal complement.

Schubert cells

The detailed study of the Grassmannians uses a decomposition into subsets called Schubert cells, which were first applied in enumerative geometry. The Schubert cells for Gr(r, n) are defined in terms of an auxiliary flag: take subspaces V₁, V₂, ..., V_r, with V_i ⊂ V_{i + 1}. Then we consider the corresponding subset of Gr(r, n), consisting of the W having intersection with V_i of dimension at least i, for i = 1, ..., r. The manipulation of Schubert cells is Schubert calculus.

Here is an example of the technique. Consider the problem of determining the Euler characteristic of the Grassmannian of r-dimensional subspaces of Rⁿ. Fix a 1-dimensional subspace R ⊂ Rⁿ and consider the partition of Gr(r, n) into those r-dimensional subspaces of Rⁿ that contain R and those that do not. The former is Gr(r − 1, n − 1) and the latter is a r-dimensional vector bundle over Gr(r, n − 1). This gives recursive formulas:

${\displaystyle \chi _{r,n}=\chi _{r-1,n-1}+(-1)^{r}\chi _{r,n-1},\qquad \chi _{0,n}=\chi _{n,n}=1.}$

If one solves this recurrence relation, one gets the formula: χ_r,n = 0 if and only if n is even and r is odd. Otherwise:

${\displaystyle \chi _{r,n}={\lfloor {\frac {n}{2}}\rfloor \choose \lfloor {\frac {r}{2}}\rfloor }.}$

Cohomology ring of the complex Grassmannian

Every point in the complex Grassmannian manifold Gr(r, n) defines an r-plane in n-space. Fibering these planes over the Grassmannian one arrives at the vector bundle E which generalizes the tautological bundle of a projective space. Similarly the (n − r)-dimensional orthogonal complements of these planes yield an orthogonal vector bundle F. The integral cohomology of the Grassmannians is generated, as a ring, by the Chern classes of E. In particular, all of the integral cohomology is at even degree as in the case of a projective space.

These generators are subject to a set of relations, which defines the ring. The defining relations are easy to express for a larger set of generators, which consists of the Chern classes of E and F. Then the relations merely state that the direct sum of the bundles E and F is trivial. Functoriality of the total Chern classes allows one to write this relation as

${\displaystyle c(E)c(F)=1.}$

The quantum cohomology ring was calculated by Edward Witten in The Verlinde Algebra And The Cohomology Of The Grassmannian. The generators are identical to those of the classical cohomology ring, but the top relation is changed to

${\displaystyle c_{k}(E)c_{n-k}(F)=(-1)^{n-r}}$

reflecting the existence in the corresponding quantum field theory of an instanton with 2n fermionic zero-modes which violates the degree of the cohomology corresponding to a state by 2n units.

Associated measure

When V is n-dimensional Euclidean space, one may define a uniform measure on Gr(r, n) in the following way. Let θ_n be the unit Haar measure on the orthogonal group O(n) and fix W in Gr(r, n). Then for a set A ⊆ Gr(r, n), define

${\displaystyle \gamma _{r,n}(A)=\theta _{n}\{g\in \operatorname {O} (n):gW\in A\}.}$

This measure is invariant under actions from the group O(n), that is, γ_r,n(gA) = γ_r,n(A) for all g in O(n). Since θ_n(O(n)) = 1, we have γ_r,n(Gr(r, n)) = 1. Moreover, γ_r,n is a Radon measure with respect to the metric space topology and is uniform in the sense that every ball of the same radius (with respect to this metric) is of the same measure.

Oriented Grassmannian

This is the manifold consisting of all oriented r-dimensional subspaces of Rⁿ. It is a double cover of Gr(r, n) and is denoted by:

${\displaystyle {\widetilde {\mathbf {Gr} }}(r,n).}$

As a homogeneous space it can be expressed as:

${\displaystyle \operatorname {SO} (n)/(\operatorname {SO} (r)\times \operatorname {SO} (n-r)).}$

Applications

Grassmann manifolds have found application in computer vision tasks of video-based face recognition and shape recognition.[5] They are also used in the data-visualization technique known as the grand tour.

Grassmannians allow the scattering amplitudes of subatomic particles to be calculated via a positive Grassmannian construct called the amplituhedron.[6]

See also

• Schubert calculus
• For an example of the use of Grassmannians in differential geometry, see Gauss map and in projective geometry, see Plücker co-ordinates.
• Flag manifolds are generalizations of Grassmannians and Stiefel manifolds are closely related.
• Given a distinguished class of subspaces, one can define Grassmannians of these subspaces, such as the Lagrangian Grassmannian.
• Grassmannians provide classifying spaces in K-theory, notably the classifying space for U(n). In the homotopy theory of schemes, the Grassmannian plays a similar role for algebraic K-theory.[7]
• Affine Grassmannian
• Grassmann bundle

Notes

1. Milnor & Stasheff (1974), pp. 57–59.
2. Grothendieck, Alexander (1971). Éléments de géométrie algébrique. 1 (2nd ed.). Berlin, New York: Springer-Verlag. ISBN 978-3-540-05113-8., Chapter I.9
3. EGA, II.3.6.3.
4. Griffiths, Phillip; Harris, Joseph (1994), Principles of algebraic geometry, Wiley Classics Library (2nd ed.), New York: John Wiley & Sons, p. 211, ISBN 0-471-05059-8, MR 1288523, Zbl 0836.14001
5. Pavan Turaga, Ashok Veeraraghavan, Rama Chellappa: Statistical analysis on Stiefel and Grassmann manifolds with applications in computer vision, CVPR 23–28 June 2008, IEEE Conference on Computer Vision and Pattern Recognition, 2008, ISBN 978-1-4244-2242-5, pp. 1–8 (abstract, full text)
6. Arkani-Hamed, Nima; Trnka, Jaroslav (2013). "The Amplituhedron". Journal of High Energy Physics. 2014 (10). arXiv:1312.2007. Bibcode:2014JHEP...10..030A. doi:10.1007/JHEP10(2014)030.CS1 maint: ref=harv (link)
7. Morel, Fabien; Voevodsky, Vladimir (1999). "A¹-homotopy theory of schemes" (PDF). Publications Mathématiques de l'IHÉS. 90 (90): 45–143. doi:10.1007/BF02698831. ISSN 1618-1913. MR 1813224. Retrieved 2008-09-05., see section 4.3., pp. 137–140

References

• Hatcher, Allen (2003). "Vector Bundles & K-Theory" (2.0 ed.). Cite journal requires |journal= (help) section 1.2
• Milnor, John W.; Stasheff, James D. (1974). Characteristic classes. Annals of Mathematics Studies. 76. Princeton, NJ: Princeton University Press. ISBN 0-691-08122-0. see chapters 5–7
• Harris, Joe (1992). Algebraic Geometry: A First Course. New York: Springer. ISBN 0-387-97716-3.
• Mattila, Pertti (1995). Geometry of Sets and Measures in Euclidean Spaces. New York: Cambridge University Press. ISBN 0-521-65595-1.