Does the Einstein field equation uniquely determines the topology of spacetime? [duplicate]

This question already has an answer here:

I am trying to understand whether the Einstein field equation uniquely determines the topology of spacetime. As far as I know, given a metric, we can always find the induced topology. However, I was wondering if this topology is unique. To put it another way, metric completely encodes the local data about the manifold but does it also completely encode the global data about the manifold?

For example, $$\mathbb{R}^2$$ and $$T^2$$ both admit a flat metric. Is it possible to distinguish these two topologies using the field equation?

marked as duplicate by Ben Crowell general-relativity StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); May 13 at 2:12

• The field equations all reduce to $0=0$ for your two empty spacetimes, so the answer is No. The field equations do not determine the topology. However, I think some features of the topology are encoded in the spectrum of differential operators on the spacetimes. – G. Smith May 12 at 21:05