Homology of connected sum

From Topospaces
Jump to: navigation, search

This article describes the effect of the connected sum operation on the following invariant: homology groups


Suppose M_1 and M_2 are the connected manifolds of dimension n whose connected sum is being taken. Assume both We have:

Case for p Additional condition on M_1,M_2 What is known about H_p(M_1) and H_p(M_2)? Formula for H_p(M_1 \# M_2) in terms of homology groups of M_1 and M_2
0 none both isomorphic to \mathbb{Z} \mathbb{Z} (because both are connected, so is their connected sum).
Greater than 0, less than n - 1 none both are finitely generated abelian groups H_p(M_1) \oplus H_p(M_2)
n - 1 Both manifolds are compact, at least one of them is orientable both are finitely generated abelian groups, at least one is free abelian H_{n-1}(M_1) \oplus H_{n-1}(M_2)
n-1 other cases  ?  ?
n Both are compact and orientable both are \mathbb{Z} \mathbb{Z}
n other cases  ?  ?
Greater than n none both are zero groups 0

Euler characteristic

The Euler characteristics are related by the following formula when both M_1 and M_2 are compact connected manifolds:

\chi(M_1 \# M_2) = \chi(M_1) + \chi(M_2) - \chi(S^n)