nLab real space



Representation theory




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 /2\mathbb{Z}/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.


Real circles

There are three non-equivalent real structures on the circle S 1S^1, usually denoted

  • S 2,0S^{2,0} for the trivial involution;

  • S 1,1S^{1,1} for the reflection involution (identifying the two semi-circles);

  • S 0,2S^{0,2} for the antipodal involution (rotation by π\pi).

Accordingly real-oriented cohomology theory is bigraded in a way modeled on this bigrading.

(This is standard notation, but maybe S 1,0S^{1,0}, S 12,12,S 0,1S^{\tfrac{1}{2}, \tfrac{1}{2}}, S^{0,1} 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.)

Complexified cartesian spaces

The complex nn-dimensional complexified cartesian space n{\mathbb{C}}^n equipped with its conjugative involution is a real space. Explicitly, this involution sends (z 1,,z n)(z^1,\ldots, z^n) to (z 1¯,,z n¯)(\bar{z^1},\ldots, \bar{z^n}).

Complex projective spaces.

The complex nn-dimensional complex projective space n{\mathbb{P}}^n_{\mathbb{C}} 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 n{\mathbb{C}}^n) extends to a conjugation involution on n{\mathbb{P}}^n_{\mathbb{C}}. Any two conjugation involutions are /2{\mathbb{Z}}/2-equivariantly diffeomorphic.

For example, the Riemann surface 1{\mathbb{P}}^1_{\mathbb{C}} is diffeomorphic to the 2-sphere S 2S^2 and its conjugation involution is the antipodal action.

Complex general and special linear groups.

For AA a square matrix, the determinant of its conjugation transpose equals the conjugate of its determinant. In symbols, detA *=detA¯{\mathrm{det}} A^* = \overline{{\mathrm{det}} A}. Hence sending a square matrix to its conjugate transpose is an involution on the complex general GL(n,)GL(n,{\mathbb{C}}) and special SL(n,)SL(n,{\mathbb{C}}).

In particular, the real space SL(1,)SL(1,{\mathbb{C}}) equipped with this conjugate transpose involution is equivariantly diffeomorphic to S 0,2S^{0,2}, the circle equipped with its antipodal action (dicussed above).


Relation to real cobordism

There is a “real” analog of complex cobordism cohomology theory MUMU, the MR cohomology theory MM \mathbb{R}.

While π (M)\pi_\bullet(M \mathbb{R}) is not the cobordism ring of real manifolds, still every real manifold does give a class in MM \mathbb{R} (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.