Are loops not allowed when gluing together simplices in a simplicial complex?

Question

I'm taking a graduate geometric topology class and our professor made a quick remark that we're not allowed to make stuff like these by gluing together simplexes in a simplicial complex. I didn't get time to ask him to clarify, but does it mean that loops are not allowed when gluing together simplices to form a simplicial complex? Any ideas?

Also, is there any good reference for this material? Thanks!

Answer 1

• A 1-dimensional simplex is a line segment with two end points. The line segment in the first figure does not have two end points.
• In a simplicial complex, the intersection of two of the simplices is either empty or is a single face of both. In the second figure, the intersection between the two 1-dimensional simplices consists of the two points, whereas a single face would be a single point.
• The third figure suffers from the same problems.