The arms race between the united states and the soviet union began in 1949 when the soviet union tested its own a weapon on which the united states had a monopoly until that time.

atomic bomb.

Detail:

The Soviet Union had conducted espionage concerning the United States' "Manhattan Project," which was the US government's development of atomic bombs during World War II.  The US tested its first atomic bomb in July, 1945, and then used two such bombs to force Japan to surrender in August, 1945.

The USA would not share its nuclear weapons technology with the USSR, even though the two nations were allies during World War II.  But by 1949, the the Soviet Union had developed and tested its own atomic bomb.  From that point on, an arms race truly began between the two superpowers.  

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