Does Z^n as the First Homology Group Imply R^n with Real Coefficients?

  • Context: Graduate 
  • Thread starter Thread starter petergreat
  • Start date Start date
Click For Summary
SUMMARY

The first homology group of a manifold being Z^n definitively implies that the first homology group with real coefficients is R^n, according to the Universal Coefficient Theorem for homology. This theorem establishes that if the integral homology is free and has no torsion, the homology groups with other coefficients will also be free and of the same dimension. Therefore, the relationship between Z^n and R^n is established through the properties of free groups in homology.

PREREQUISITES
  • Understanding of homology groups in algebraic topology
  • Familiarity with the Universal Coefficient Theorem for homology
  • Knowledge of torsion in homology
  • Basic concepts of singular chains and coefficients in homology
NEXT STEPS
  • Study the Universal Coefficient Theorem for homology in detail
  • Explore the implications of torsion in homology groups
  • Learn about singular chains and their role in homology
  • Investigate the differences between integer and real coefficients in homology
USEFUL FOR

Mathematicians, algebraic topologists, and students studying homology theory will benefit from this discussion, particularly those interested in the relationships between different coefficient systems in homology.

petergreat
Messages
266
Reaction score
4
If the first homology group of a manifold is Z^n, does it imply that the first homology group with real coefficients (obtained from singular chains with real coefficients) is R^n?
 
Physics news on Phys.org
i think so. in general the group with integer coefficients should have more information than with other coefficients. look up universal coefficients:

in general, if the integral homology has no torsion, i.e. is free, then the other homologies are free of the same dimension.

please forgive me for answering in my dotage without verifying any of this. but i am probably not too far wrong.
 
petergreat said:
If the first homology group of a manifold is Z^n, does it imply that the first homology group with real coefficients (obtained from singular chains with real coefficients) is R^n?

Look at the Universal Coefficient Theorem for homology.
 

Similar threads

Undergrad SL(n,R) Lie group as submanifold of GL(n,R)
  • · Replies 36 ·
2
Replies
36
Views
6K
Graduate Why Are Homology Groups Not MUCH Larger?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Spherically symmetric manifold
  • · Replies 10 ·
Replies
10
Views
2K
Graduate Reduced homology of sphere cross reals?
  • · Replies 3 ·
Replies
3
Views
2K
Graduate Differential structure of the group of automorphism of a Lie group
  • · Replies 0 ·
Replies
0
Views
3K
Graduate Formal Computations of Euler Classes
  • · Replies 19 ·
Replies
19
Views
7K
Graduate Change of Coefficients in Homology and Individual Classes
  • · Replies 16 ·
Replies
16
Views
6K
Graduate Visualising singular cohomology using cap product.
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Induced orientation on boundary of ##\mathbb{H}^n## in ##\mathbb{R}^n##
  • · Replies 6 ·
Replies
6
Views
3K
Graduate Differential forms on R^n vs. on manifold
  • · Replies 4 ·
Replies
4
Views
3K