# Equivariant Prequantization

## Abstract

If (*S, ω*) is a symplectic manifold, a prequantization of *S* is a principal circle bundle over *S* together with a connection form whose curvature is −*ω*. Such a circle bundle exists iff the period group of *ω* is contained in ℤ; i.e., the class [*ω*] ∈ *H* ^{2}(*S*, ℝ) comes from an integral class. If *S* is simply connected it follows from the universal coefficient theorem that the integral class is unique. Also note that for simply connected *S*, the period group of *ω* is in ℤ iff the spherical period group is in ℤ; i.e., *π* _{2}(*S*) ≅ *H* _{2}(*S*).^{1} If *S* is not simply connected it may have inequivalent prequantizations. Inequivalent prequantizations of *S* also induce inequivalent prequantizations of *S* × \(\bar S\), \(\bar S\) denoting (*S*, −*ω*). But one can show such prequantizations become equivalent when pulled back to the fundamental groupoid (\(\pi \left( S \right) = \tilde S \times \tilde S/{\pi _1}\left( S \right),\lambda :\tilde S \to S\) the universal cover, with the form induced from *S* × \(\bar S\) by λ × λ). Further if we only assume *ω* is integral on spherical classes, no prequantization may exist. In his preprint [**10**], Alan Weinstein gives a method for prequantizing the fundamental groupoid of a symplectic manifold (*S, ω*) when *ω* is integral on spherical classes, using connection theory. His result is equivalent to the statement (Corollary 1.3): For any symplectic manifold (*S, ω*) the period group of the fundamental groupoid *π*(*S*) is contained in ℤ iff the spherical period group of *S* is contained in ℤ. Since this is a statement about cohomology, Weinstein raises the question of giving a purely algebraic topology proof of this result.

## Keywords

Spectral Sequence Symplectic Manifold Connection Form Maximal Compact Subgroup Period Group## Preview

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