Chern-Simons functional and the homology cobordism group

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Abstract: For each integral homology sphere

Y

, a function

G a m m a_{Y}

on the set of integers is constructed. It is established that

G a m m a_{Y}

depends only on the homology cobordism of

Y

and it recovers the Fr{o}yshov invariant. A relation between

G a m m a_{Y}

and Fintushel-Stern's

R

-invariant is stated. It is shown that the value of

G a m m a_{Y}

at each integer is related to the critical values of the Chern-Simons functional. Some topological applications of

G a m m a_{Y}

are given. In particular, it is shown that if

G a m m a_{Y}

is trivial, then there is no simply connected homology cobordism from

Y

to itself.

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