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

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*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.

