On Homeomorphism Type of Symmetric Products of Compact Riemann Surfaces with Punctures

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Publication date: 1 June 2016

Abstract: Let

M_{g, k}^{2}

and

M_{g^{'}, k^{'}}^{2}

be compact Riemann surfaces with punctures (

g, g^{'} g e 0

- genuses,

k, k^{'} g e 1

- number of punctures). For any Hausdorff space

X

the quotient space

m a t h r m {S y m}^{n} X := X^{n} / S_{n}

is the

n

-th symmetric product of

X, n g e 2

. It is well known, that

m a t h r m {S y m}^{n} M_{g, k}^{2}

is a smooth quasi-projective variety. Open manifolds

m a t h r m {S y m}^{n} M_{g, k}^{2}

and

m a t h r m {S y m}^{n} M_{g^{'}, k^{'}}^{2}

are homotopy equivalent iff

2 g + k = 2 g^{'} + k^{'}

. Blagojevi'{c}-Gruji'{c}-v{Z}ivaljevi'{c} Conjecture (2003). Fix any

n g e 2

, and two pairs

(g, k)

and

(g^{'}, k^{'})

with the condition

2 g + k = 2 g^{'} + k^{'}

. If

g e g^{'}

, then open manifolds

m a t h r m {S y m}^{n} M_{g, k}^{2}

and

m a t h r m {S y m}^{n} M_{g^{'}, k^{'}}^{2}

are not continuously homeomorphic. The conjecture was proved in 2003 in the paper by P.Blagojevi'{c}, V.Gruji'{c} and R.v{Z}ivaljevi'{c} for the case

m a t h r m m a x (g, g^{'}) g e f r a c n 2

(this implies the case

n = 2

). As far as the author knows, up to this moment there were no results if

m a t h r m m a x (g, g^{'}) < f r a c n 2

. The aim of this paper is to prove the conjecture in full generality.

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