Compute the fundamental group of the space obtained by identifying the faces of the solid prism shown in Problem 6 of the source sheet.
The face-identification diagram appears on page 1 of the source sheet.
Solution
1. The \(Q\)-labelled top and bottom triangular faces are identified preserving their markings, so before the side-face identifications the quotient is \(\Delta^2\times S^1\).
This is the mapping torus of the identity on the triangular cross-section.
2. The three side faces carry the same \(F\) marking. Their identification is constant in the \(S^1\) direction and, on the boundary of the triangular cross-section, is precisely the standard dunce-cap identification of all three sides.
The common face marking identifies corresponding vertical coordinates, while the three edges of the triangular cross-section are identified with the orientations encoded by the repeated \(F\) glyph. This is the same three-edge pattern as the source sheet’s triangle example.
3. Hence the quotient is homeomorphic to\[D\times S^1,\]where \(D\) is the dunce cap.
The identifications in <1>1–<1>2 are products of the cross-section identifications with the circle coordinate.
4. The dunce cap is contractible, so\[\boxed{\pi_1(X)\cong\pi_1(S^1)\cong\mathbb Z.}\]
Projection \(D\times S^1\to S^1\) is a homotopy equivalence because \(D\) is contractible.