geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
higher geometry / derived geometry
Ingredients
Concepts
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
Constructions
Examples
derived smooth geometry
Theorems
In the context of real-oriented cohomology theory, notably in KR-theory, by “real space” (“real manifold”) one means a topological space (a manifold) equipped with an action of the cyclic group of order 2 (a “non-linear real structure”).
In the context of string theory real spaces appear as orientifold target spacetimes. The involution fixed points here are known as O-planes.
There are three non-equivalent real structures on the circle , usually denoted
for the trivial involution;
for the reflection involution (identifying the two semi-circles);
for the antipodal involution (rotation by ).
Accordingly real-oriented cohomology theory is bigraded in a way modeled on this bigrading.
(This is standard notation, but maybe , would be more suggestive. Indeed the quotients in the first and the last case are actually circles, while in the second case it is the semi-circle.)
The complex -dimensional complexified cartesian space equipped with its conjugative involution is a real space. Explicitly, this involution sends to .
The complex -dimensional complex projective space equipped with a conjugation involution is a real space. For each choice of affine chart, the conjugation involution of this chart (which is biholomorphic to ) extends to a conjugation involution on . Any two conjugation involutions are -equivariantly diffeomorphic.
For example, the Riemann surface is diffeomorphic to the 2-sphere and its conjugation involution is the antipodal action.
For a square matrix, the determinant of its conjugation transpose equals the conjugate of its determinant. In symbols, . Hence sending a square matrix to its conjugate transpose is an involution on the complex general and special .
In particular, the real space equipped with this conjugate transpose involution is equivariantly diffeomorphic to , the circle equipped with its antipodal action (dicussed above).
There is a “real” analog of complex cobordism cohomology theory , the MR cohomology theory .
While is not the cobordism ring of real manifolds, still every real manifold does give a class in (Kriz 01, p. 13). For details see here: pdf.
Michael Atiyah, K-theory and reality, The Quarterly Journal of Mathematics. Oxford. Second Series 17 1 (1966) 367-386 [doi:10.1093/qmath/17.1.367, pdf, ISSN:0033-5606]
Igor Kriz, Real-oriented homotopy theory and an analogue of the Adams-Novikov spectral sequence, Topology 40 (2001) 317-399 (pdf)
Last revised on September 6, 2023 at 09:23:26. See the history of this page for a list of all contributions to it.