Orientability

Orientability

The index theorem

Published in Peter B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, 2018

In this section, we will prove the Atiyah Singer theorem for a general elliptic complex by interpreting the index as a map in K theory. Let M be a compact smooth manifold without boundary of dimension m we impose no restrictions on either the parity of m or on the orientability of M. We adopt the notation of §3.6. We give T*M ⊕ 1 the symplectic orientation to orient ΣM = S(T*M ⊕ 1).

Design in the service of topology and the classification theorem

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Journal Information

Published in International Journal of Mathematical Education in Science and Technology, 2023

Shahar Rozenstin, Shai Gul

One of the fundamental ideas discussed in this manuscript is orientability. This property has been discussed in detail in Braselton et al. (2002). Famous non-orientable objects include the Möbius strip and the Klein bottle. Séquin (2013) presents a wide variety of different geometric realizations of the Klein bottle. Flórez and Mukherjee (2020) describe some classic experiments on the Möbius strip, the projective plane, and the Klein bottle in a way that seeks to intrigue and encourage readers to learn more about the properties of these non-orientable surfaces.

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The index theorem

Design in the service of topology and the classification theorem