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Order of homology group of boundary is a square

Source: Alibaba Global Math Competition 2021, Problem 11

July 4, 2021

homologymanifoldsboundarytopologyalgebraic topologycollege contests

Problem Statement

Let

M

be a compact orientable

2n

-manifold with boundary, where

n \ge 2

. Suppose that

H_0(M;\mathbb{Q}) \cong \mathbb{Q}

and

H_i(M;\mathbb{Q})=0

for

i>0

. Prove that the order of

H_{n-1}(\partial M; \mathbb{Z})

is a square number.