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    • CommentRowNumber1.
    • CommentAuthorAndrew Stacey
    • CommentTimeJun 21st 2011

    I got a question by email about the equivariant tubular neighbourhoods in loop spaces (as opposed to those defined using propagating flows so I figured it was time to nLabify that section of differential topology of mapping spaces. Of course, in so doing I figured out a generalisation: given a fibre bundle EBE \to B, everything compact, we consider smooth maps EME \to M which are constant on fibres. This is a submanifold of the space of all smooth maps EME \to M. Assuming we can put a suitable measure on the fibres of EE, then we can define a tubular neighbourhood of this submanifold.

    Details at equivariant tubular neighbourhoods. Title may be a bit off now, but it’s that because the original case was for the fibre bundle S 1S 1S^1 \to S^1 with fibre n\mathbb{Z}_n.

    This entry is also notable because I produced it using a whole new LaTeX-to-iTeX converter. Details on the relevant thread.

    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 22nd 2011

    The displayed equation above lemma 1 (in version 1) is a bit wonky on the spacing, but apart from that a quick scan showed it to be very nice. Well done!

    • CommentRowNumber3.
    • CommentAuthorAndrew Stacey
    • CommentTimeJun 22nd 2011

    Aagh. I keep forgetting that \text behaves ever so slightly differently with regard to swallowing spaces. Fixed.

    Also added a picture to try to illustrate some of the maps.